Advances in Statistical Multisource-Multitarget Information Fusion 9781608077984, 1608077985

Table of contents :
Contents
Preface
Acknowledgments
Chapter 1 Introduction to the Book
1.1 OVERVIEW OF FINITE-SET STATISTICS
1.2 RECENT ADVANCES IN FINITE-SET STATISTICS
1.3 ORGANIZATION OF THE BOOK
Part I Elements of Finite-Set Statistics
Chapter 2 Random Finite Sets
2.1 INTRODUCTION
2.2 SINGLE-SENSOR, SINGLE-TARGET STATISTICS
2.3 RANDOM FINITE SETS (RFSs)
2.4 MULTIOBJECT STATISTICS IN A NUTSHELL
Chapter 3 Multiobject Calculus
3.1 INTRODUCTION
3.2 BASIC CONCEPTS
3.3 SET INTEGRALS
3.4 MULTIOBJECT DIFFERENTIAL CALCULUS
3.5 KEY FORMULAS OF MULTIOBJECT CALCULUS
Chapter 4 Multiobject Statistics
4.1 INTRODUCTION
4.2 BASIC MULTIOBJECT STATISTICAL DESCRIPTORS
4.3 IMPORTANT MULTIOBJECT PROCESSES
4.4 BASIC DERIVED RFSs
Chapter 5 Multiobject Modeling and Filtering
5.1 INTRODUCTION
5.2 THE MULTISENSOR-MULTITARGET BAYES FILTER
5.3 MULTITARGET BAYES OPTIMALITY
5.4 RFS MULTITARGET MOTION MODELS
5.5 RFS MULTITARGET MEASUREMENT MODELS
5.6 MULTITARGET MARKOV DENSITIES
5.7 MULTISENSOR-MULTITARGET LIKELIHOOD FUNCTIONS
5.8 THE MULTITARGET BAYES FILTER IN p.g.fl. FORM
5.9 THE FACTORED MULTITARGET BAYES FILTER
5.10 APPROXIMATE MULTITARGET FILTERS
Chapter 6 Multiobject Metrology
6.1 INTRODUCTION
6.2 MULTIOBJECT MISS DISTANCE
6.3 MULTIOBJECT INFORMATION FUNCTIONALS
Part II RFS Filters: StandardMeasurement Model
Chapter 7 Introduction to Part II
7.1 SUMMARY OF MAJOR LESSONS LEARNED
7.2 STANDARD MULTITARGET MEASUREMENT MODEL
7.3 AN APPROXIMATE STANDARD LIKELIHOOD FUNCTION
7.4 STANDARD MULTITARGET MOTION MODEL
7.5 STANDARD MOTION MODEL WITH TARGET SPAWNING
7.6 ORGANIZATION OF PART II
Chapter 8 Classical PHD and CPHD Filters
8.1 INTRODUCTION
8.2 A GENERAL PHD FILTER
8.3 ARBITRARY-CLUTTER PHD FILTER
8.4 CLASSICAL PHD FILTER
8.5 CLASSICAL CARDINALIZED PHD (CPHD) FILTER
8.6 ZERO FALSE ALARMS (ZFA) CPHD FILTER
8.7 PHD FILTER FOR STATE-DEPENDENT POISSON CLUTTER
Chapter 9 Implementing Classical PHD/CPHDFilters
9.1 INTRODUCTION
9.2 “SPOOKY ACTION AT A DISTANCE”
9.3 MERGING AND SPLITTING FOR PHD FILTERS
9.4 MERGING AND SPLITTING FOR CPHD FILTERS
9.5 GAUSSIAN MIXTURE (GM) IMPLEMENTATION
9.6 SEQUENTIAL MONTE CARLO (SMC) IMPLEMENTATION
Chapter 10 Multisensor PHD and CPHD Filters
10.1 INTRODUCTION
10.2 THE MULTISENSOR-MULTITARGET BAYES FILTER
10.3 THE GENERAL MULTISENSOR PHD FILTER
10.4 THE MULTISENSOR CLASSICAL PHD FILTER
10.5 ITERATED-CORRECTOR MULTISENSOR PHD/CPHD FILTERS
10.6 PARALLEL COMBINATION MULTISENSOR PHD AND CPHD FILTERS
10.7 AN ERRONEOUS “AVERAGED” MULTISENSOR PHD FILTER
10.8 PERFORMANCE COMPARISONS
Chapter 11 Jump-Markov PHD/CPHD Filters
11.1 INTRODUCTION
11.2 JUMP-MARKOV FILTERS: A REVIEW
11.3 MULTITARGET JUMP-MARKOV SYSTEMS
11.4 JUMP-MARKOV PHD FILTER
11.5 JUMP-MARKOV CPHD FILTER
11.6 VARIABLE STATE SPACE JUMP-MARKOV CPHD FILTERS
11.7 IMPLEMENTING JUMP-MARKOV PHD/CPHD FILTERS
11.8 IMPLEMENTED JUMP-MARKOV PHD/CPHD FILTERS
Chapter 12 Joint Tracking and Sensor-Bias Estimation
12.1 INTRODUCTION
12.2 MODELING SENSOR BIASES
12.3 OPTIMAL JOINT TRACKING AND REGISTRATION
12.4 THE BURT-PHD FILTER
12.5 SINGLE-FILTER BURT-PHD FILTERS
12.6 IMPLEMENTED BURT-PHD FILTERS
Chapter 13 Multi-Bernoulli Filters
13.1 INTRODUCTION
13.2 THE BERNOULLI FILTER
13.3 THE MULTISENSOR BERNOULLI FILTER
13.4 THE CBMEMBER FILTER
13.5 JUMP-MARKOV CBMEMBER FILTER
Chapter 14 RFS Multitarget Smoothers
14.1 INTRODUCTION
14.2 SINGLE-TARGET FORWARD-BACKWARD SMOOTHER
14.3 GENERAL MULTITARGET FORWARD-BACKWARD SMOOTHER
14.4 BERNOULLI FORWARD-BACKWARD SMOOTHER
14.5 PHD FORWARD-BACKWARD SMOOTHER
14.6 ZTA-CPHD SMOOTHER
Chapter 15 Exact Closed-Form Multitarget Filter
15.1 INTRODUCTION
15.2 LABELED RFSS
15.3 EXAMPLES OF LABELED RFSS
15.4 MODELING FOR THE VO-VO FILTER
15.5 CLOSURE OF MULTITARGET BAYES FILTER
15.6 IMPLEMENTATION OF THE VO-VO FILTER: SKETCH
15.7 PERFORMANCE RESULTS
Part III RFS Filters for UnknownBackgrounds
Chapter 16 Introduction to Part III
16.1 INTRODUCTION
16.2 OVERVIEW OF THE APPROACH
16.3 MODELS FOR UNKNOWN BACKGROUNDS
16.4 ORGANIZATION OF PART III
Chapter 17 RFS Filters for Unknown pD
17.1 INTRODUCTION
17.2 THE PD-CPHD FILTER
17.3 BETA-GAUSSIAN MIXTURE (BGM) APPROXIMATION
17.4 BGM IMPLEMENTATION OF THE PD-PHD FILTER
17.5 BGM IMPLEMENTATION OF THE PD-CPHD FILTER
17.6 THE PD-CBMEMBER FILTER
17.7 IMPLEMENTATIONS OF PD-AGNOSTIC RFS FILTERS
Chapter 18 RFS Filters for Unknown Clutter
18.1 INTRODUCTION
18.2 A GENERAL MODEL FOR UNKNOWN BERNOULLI CLUTTER
18.3 CPHD FILTER FOR GENERAL BERNOULLI CLUTTER
18.4 THE λ-CPHD FILTER
18.5 THE κ-CPHD FILTER
18.6 MULTISENSOR κ-CPHD FILTERS
18.7 THE κ-CBMEMBER FILTER
18.8 IMPLEMENTED CLUTTER-AGNOSTIC RFS FILTERS
18.9 CLUTTER-AGNOSTIC PSEUDOFILTERS
18.10 CPHD/PHD FILTERS WITH POISSON-MIXTURE CLUTTER
18.11 RELATED WORK
Part IV RFS Filters for Nonstandard Measurement Models
Chapter 19 RFS Filters for Superpositional Sensors
19.1 INTRODUCTION
19.2 EXACT SUPERPOSITIONAL CPHD FILTER
19.3 HAUSCHILDT’S APPROXIMATION
19.4 THOUIN-NANNURU-COATES (TNC) APPROXIMATION
Chapter 20 RFS Filters for Pixelized Images
20.1 INTRODUCTION
20.2 THE IO MULTITARGET MEASUREMENT MODEL
20.3 IO MOTION MODEL
20.4 IO-CPHD FILTER
20.5 IO-MEMBER FILTER
20.6 IMPLEMENTATIONS OF IO-MEMBER FILTERS
Chapter 21 RFS Filters for Cluster-Type Targets
21.1 INTRODUCTION
21.2 EXTENDED-TARGET MEASUREMENT MODELS
21.3 EXTENDED-TARGET BERNOULLI FILTERS
21.4 EXTENDED-TARGET PHD/CPHD FILTERS
21.5 EXTENDED-TARGET CPHD FILTER: APB MODEL
21.6 CLUSTER-TARGET MEASUREMENT MODEL
21.7 CLUSTER-TARGET PHD AND CPHD FILTERS
21.8 MEASUREMENT MODELS FOR LEVEL-1 GROUP TARGETS
21.9 PHD/CPHD FILTERS FOR LEVEL-1 GROUP TARGETS
21.10 MEASUREMENT MODELS FOR GENERAL GROUP TARGETS
21.11 PHD/CPHD FILTERS FOR LEVEL-ℓ GROUP TARGETS
21.12 A MODEL FOR UNRESOLVED TARGETS
21.13 MOTION MODEL FOR UNRESOLVED TARGETS
21.14 THE UNRESOLVED-TARGET PHD FILTER
21.15 APPROXIMATE UNRESOLVED-TARGET PHD FILTER
21.16 APPROXIMATE UNRESOLVED-TARGET CPHD FILTER
Chapter 22 RFS Filters for Ambiguous Measurements
22.1 INTRODUCTION
22.2 RANDOM SET MODELS OF AMBIGUOUS MEASUREMENTS
22.3 GENERALIZED LIKELIHOOD FUNCTIONS (GLFS)
22.4 UNIFICATION OF EXPERT-SYSTEM THEORIES
22.5 GLFS FOR IMPERFECTLY CHARACTERIZED TARGETS
22.6 GLFS FOR UNKNOWN TARGET TYPES
22.7 GLFS FOR INFORMATION WITH UNKNOWN CORRELATIONS
22.8 GLFS FOR UNRELIABLE INFORMATION SOURCES
22.9 USING GLFS IN MULTITARGET FILTERS
22.10 GLFS IN RFS MULTITARGET FILTERS
22.11 USING GLFS WITH CONVENTIONAL MULTITARGET FILTERS
Part V Sensor, Platform, and Weapons Management
Chapter 23 Introduction to Part V
23.1 BASIC ISSUES IN SENSOR MANAGEMENT
23.2 INFORMATION THEORY AND INTUITION: AN EXAMPLE
23.3 SUMMARY OF RFS SENSOR CONTROL
23.4 ORGANIZATION OF PART V
Chapter 24 Single-Target Sensor Management
24.1 INTRODUCTION
24.2 EXAMPLE: MISSILE-TRACKING CAMERAS
24.3 SINGLE-SENSOR, SINGLE-TARGET CONTROL: MODELING
24.4 SINGLE-SENSOR, SINGLE-TARGET CONTROL: SINGLE-STEP
24.5 SINGLE-SENSOR, SINGLE-TARGET CONTROL: OBJECTIVE
24.6 SINGLE-SENSOR, SINGLE-TARGET CONTROL: HEDGING
24.7 SINGLE–SENSOR, SINGLE-TARGET CONTROL: OPTIMIZATION
24.8 SPECIAL CASE 1: IDEAL SENSOR DYNAMICS
24.9 SIMPLE EXAMPLE: LINEAR-GAUSSIAN CASE
24.10 SPECIAL CASE 2: SIMPLIFIED NONIDEAL DYNAMICS
Chapter 25 Multitarget Sensor Management
25.1 INTRODUCTION
25.2 MULTITARGET CONTROL: TARGET AND SENSOR STATE SPACES
25.3 MULTITARGET CONTROL: CONTROL SPACES
25.4 MULTITARGET CONTROL: MEASUREMENT SPACES
25.5 MULTITARGET CONTROL: MOTION MODELS
25.6 MULTITARGET CONTROL: MEASUREMENT MODELS
25.7 MULTITARGET CONTROL: SUMMARY OF NOTATION
25.8 MULTITARGET CONTROL: SINGLE STEP
25.9 MULTITARGET CONTROL: OBJECTIVE FUNCTIONS
25.10 MULTISENSOR-MULTITARGET CONTROL: HEDGING
25.11 MULTISENSOR-MULTITARGET CONTROL: OPTIMIZATION
25.12 SENSOR MANAGEMENT WITH IDEAL SENSOR DYNAMICS
25.13 SIMPLIFIED NONIDEAL MULTISENSOR DYNAMICS
25.14 TARGET PRIORITIZATION
Chapter 26 Approximate Sensor Management
26.1 INTRODUCTION
26.2 SENSOR MANAGEMENT WITH BERNOULLI FILTERS
26.3 SENSOR MANAGEMENT WITH PHD FILTERS
26.4 SENSOR MANAGEMENT WITH CPHD FILTERS
26.5 SENSOR MANAGEMENT WITH CBMEMBER FILTERS
26.6 RFS SENSOR MANAGEMENT IMPLEMENTATIONS
Appendix A Glossary of Notation and Terminology
A.1 TRANSPARENT NOTATIONAL SYSTEM
A.2 GENERAL MATHEMATICS
A.3 SET THEORY
A.4 FUZZY LOGIC AND DEMPSTER-SHAFER THEORY
A.5 PROBABILITY AND STATISTICS
A.6 RANDOM SETS
A.7 MULTITARGET CALCULUS
A.8 FINITE-SET STATISTICS
A.9 GENERALIZED MEASUREMENTS
Appendix B Bayesian Analysis of Dynamic Systems
B.1 FORMAL BAYES MODELING IN GENERAL
B.2 THE BAYES FILTER IN GENERAL
Appendix C Rigorous Functional Derivatives
C.1 NONCONSTRUCTIVE DEFINITION OF THE FUNCTIONAL DERIVATIVE
C.2 THE CONSTRUCTIVE RADON-NIKOD´YM DERIVATIVE
C.3 CONSTRUCTIVE DEFINITION OF THE FUNCTIONAL DERIVATIVE
Appendix D Partitions of Finite Sets
D.1 COUNTING PARTITIONS
D.2 RECURSIVE CONSTRUCTION OF PARTITIONS
Appendix E Beta Distributions
Appendix F Markov Time Update of Beta Distributions
Appendix G Normal-Wishart Distributions
G.1 PROOF OF (G.8)
G.2 PROOF OF (G.22)
G.3 PROOF OF (G.23)
G.4 PROOF OF (G.29)
Appendix H Complex-Number Gaussian Distributions
Appendix I Statistics of Level-1 Group Targets
Appendix J FISST Calculus and Moyal’s Calculus
J.1 A “POINT PROCESS” FUNCTIONAL CALCULUS
J.2 VOLTERRA FUNCTIONAL DERIVATIVES
J.3 MOYAL’S FUNCTIONAL CALCULUS OF p.g.fl.s
Appendix K Mathematical Derivations
References
About the Author
Index

Citation preview

Ronald P. S. Mahler is a senior staff research scientist at Lockheed Martin Advanced Technology Laboratories in Eagan, MN. He earned his Ph.D. in mathematics from Brandeis University, Waltham, MA. He is recipient of the 2005 IEEE AESS Harry Rowe Mimno Award, the 2007 IEEE AESS M. Barry Carlton Award, and the 2007 JDL-DFG Joseph Mignogna Data Fusion Award.

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Advances in Statistical Multisource-Multitarget

Contents Overview: Introduction; Elements of Finite-Set Statistics; RFS Filters: Standard Measurement Model; RFS Filters for Unknown Backgrounds; RFS Filters for Nonstandard Measurement Models; Sensor, Platform, and Weapons Management.

Information Fusion

Since 2007, FISST has inspired a considerable amount of research conducted in more than a dozen nations and reported in nearly a thousand publications. This sequel addresses the most intriguing practical and theoretical advances in FISST, for the first time aggregating and systematizing them into a coherent, integrated, and deep-dive picture. Special emphasis is given to computationally fast exact closed-form implementation approaches. The book also includes the first complete and systematic description of RFS-based sensor/platform management and situation assessment.

Mahler

This is the sequel to the 2007 Artech House best-selling title, Statistical Multisource-Multitarget Information Fusion. That earlier book was a comprehensive resource for an in-depth understanding of finite-set statistics (FISST), a unified, systematic, and Bayesian approach to information fusion. The cardinalized probability hypothesis density (CPHD) filter, which was first systematically described in the earlier book, has since become a standard multitarget detection and tracking technique, especially in research and development.

Advances in Statistical Multisource-Multitarget

Information Fusion Ronald P. S. Mahler

ISBN 13: 978-1-60807-798-4 ISBN: 1-60807-798-5

ARTECH HOUSE BOSTON I LONDON

www.artechhouse.com

PMS 4975

PMS 1585

Advances in Statistical Multisource-Multitarget Information Fusion

For a complete listing of titles in the Artech House Electronic Warfare Library, turn to the back of this book.

Advances in Statistical Multisource-Multitarget Information Fusion Ronald P. S. Mahler

Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the U.S. Library of Congress. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Cover design by John Gomes

ISBN 13: 978-1-60807-798-4

© 2014 ARTECH HOUSE 685 Canton Street Norwood, MA 02062

All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark.

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Contents Preface

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Acknowledgments

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Chapter 1 Introduction to the Book 1.1 Overview of Finite-Set Statistics 1.1.1 The Philosophy of Finite-Set Statistics 1.1.2 Misconceptions About Finite-Set Statistics 1.1.3 The Measurement-to-Track Association Approach 1.1.4 The Random Finite Set (RFS) Approach 1.1.5 Extension to Nontraditional Measurements 1.2 Recent Advances in Finite-Set Statistics 1.2.1 Advances in Conventional PHD and CPHD Filters 1.2.2 Multitarget Smoothers 1.2.3 PHD and CPHD Filters for Unknown Backgrounds 1.2.4 PHD Filters for Nonpoint Targets 1.2.5 Advances in Classical Multi-Bernoulli Filters 1.2.6 RFS Filters for “Raw-Data” Sensors 1.2.7 Theoretical Advances 1.2.8 Advances in Fusing Nontraditional Measurements 1.2.9 Advances Toward Fully Unified Systems 1.3 Organization of the Book

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Elements of Finite-Set Statistics

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Chapter 2 Random Finite Sets 2.1 Introduction 2.1.1 Organization of the Chapter 2.2 Single-Sensor, Single-Target Statistics 2.2.1 Basic Notation 2.2.2 State Spaces and Measurement Spaces 2.2.3 Random States and Measurements, Probability-Mass Functions, and Probability Densities 2.2.4 Target Motion Models and Markov Densities 2.2.5 Measurement Models and Likelihood Functions 2.2.6 Nontraditional Measurements 2.2.7 The Single-Sensor, Single-Target Bayes Filter 2.3 Random Finite Sets (RFSs) 2.3.1 RFSs and Point Processes 2.3.2 Examples of RFSs 2.3.3 Algebraic Properties of RFSs 2.4 Multiobject Statistics in a Nutshell

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Chapter 3 Multiobject Calculus 3.1 Introduction 3.2 Basic Concepts 3.2.1 Set Functions 3.2.2 Functionals 3.2.3 Functional Transformations 3.2.4 Multiobject Density Functions 3.3 Set Integrals 3.4 Multiobject Differential Calculus 3.4.1 Gˆateaux Directional Derivatives 3.4.2 Volterra Functional Derivatives 3.4.3 Set Derivatives 3.5 Key Formulas of Multiobject Calculus 3.5.1 Fundamental Theorem of Multiobject Calculus 3.5.2 Change of Variables Formula for Set Integrals 3.5.3 Set Integrals on Joint Spaces 3.5.4 Constant Rule 3.5.5 Sum Rule 3.5.6 Linear Rule

59 59 60 60 60 61 62 62 64 65 66 67 69 70 71 71 73 73 73

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3.5.7 3.5.8 3.5.9 3.5.10 3.5.11 3.5.12 3.5.13 3.5.14

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Monomial Rule Power Rule Product Rules First Chain Rule Second Chain Rule Third Chain Rule Fourth Chain Rule Clark’s General Chain Rule

73 74 74 75 76 76 77 78

Chapter 4 Multiobject Statistics 4.1 Introduction 4.2 Basic Multiobject Statistical Descriptors 4.2.1 Belief-Mass Functions 4.2.2 Multiobject Probability Density Functions 4.2.3 Convolution and Deconvolution 4.2.4 Probability Generating Functionals (p.g.fl.’s) 4.2.5 Multivariate p.g.fl.’s 4.2.6 Cardinality Distributions 4.2.7 Probability Generating Functions (p.g.f.’s) 4.2.8 Probability Hypothesis Densities (PHDs) 4.2.9 Factorial Moment Density 4.2.10 Equivalence of the Fundamental Descriptors 4.2.11 Radon-Nikod´ym Formulas 4.2.12 Campbell’s Theorems 4.3 Important Multiobject Processes 4.3.1 Poisson RFSs 4.3.2 Identical, Independently Distributed Cluster (i.i.d.c.) RFSs 4.3.3 Bernoulli RFSs 4.3.4 Multi-Bernoulli RFSs 4.4 Basic Derived RFSs 4.4.1 Censored RFSs 4.4.2 Cluster RFSs

81 81 81 83 84 85 86 88 92 92 93 95 95 96 96 98 98 99 100 101 103 103 104

Chapter 5 Multiobject Modeling and Filtering 5.1 Introduction 5.2 The Multisensor-Multitarget Bayes Filter 5.3 Multitarget Bayes Optimality 5.4 RFS Multitarget Motion Models

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5.5 5.6 5.7 5.8

RFS Multitarget Measurement Models Multitarget Markov Densities Multisensor-Multitarget Likelihood Functions The Multitarget Bayes Filter in p.g.fl. Form 5.8.1 The p.g.fl. Time Update Equation 5.8.2 The p.g.fl. Measurement Update Equation 5.9 The Factored Multitarget Bayes Filter 5.10 Approximate Multitarget Filters 5.10.1 The p.g.fl. Time Update for Independent Targets 5.10.2 The p.g.fl. Measurement Update for Independent Measurements 5.10.3 A Principled Approximation Methodology 5.10.4 Poisson Approximation: PHD Filters 5.10.5 i.i.d.c. Approximation: CPHD Filters 5.10.6 Multi-Bernoulli Approximation: Multi-Bernoulli Filters 5.10.7 Bernoulli Approximation: Bernoulli Filters

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Chapter 6 Multiobject Metrology 6.1 Introduction 6.2 Multiobject Miss Distance 6.2.1 Multiobject Miss Distance: A History 6.2.2 The Optimal Sub-Pattern Assignment (OSPA) Metric 6.2.3 Extension of OSPA to Covariance (COSPA) 6.2.4 OSPA for Labeled Tracks (LOSPA) 6.2.5 Temporal OSPA (TOSPA) 6.3 Multiobject Information Functionals 6.3.1 Csisz´ar Information Functionals 6.3.2 Csisz´ar Functionals for Poisson Processes 6.3.3 Csisz´ar Functionals for i.i.d.c. Processes

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II

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RFS Filters: Standard Measurement Model

Chapter 7 Introduction to Part II 7.1 Summary of Major Lessons Learned 7.2 Standard Multitarget Measurement Model 7.2.1 Standard Multitarget Measurement Submodels

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7.2.2

7.3 7.4 7.5 7.6

Standard Multitarget Measurement Model: p.g.fl. and Likelihood 167 7.2.3 Standard Multitarget Measurement Model: Special Cases 168 7.2.4 Measurement-to-Track Association (MTA) 169 7.2.5 Relationship Between the MTA and RFS Approaches 173 An Approximate Standard Likelihood Function 173 Standard Multitarget Motion Model 174 Standard Motion Model with Target Spawning 178 Organization of Part II 178

Chapter 8 Classical PHD and CPHD Filters 8.1 Introduction 8.1.1 Summary of Major Lessons Learned 8.1.2 Organization of the Chapter 8.2 A General PHD Filter 8.2.1 General PHD Filter: Motion Modeling 8.2.2 General PHD Filter: Predictor 8.2.3 General PHD Filter: Measurement Modeling 8.2.4 General PHD Filter: Corrector 8.3 Arbitrary-Clutter PHD Filter 8.3.1 Time Update Equations for the Arbitrary-Clutter Classical PHD Filter 8.3.2 Measurement Modeling for the Arbitrary-Clutter Classical PHD Filter 8.3.3 Arbitrary-Clutter PHD Filter: Corrector 8.4 Classical PHD Filter 8.4.1 Classical PHD Filter: Predictor 8.4.2 Classical PHD Filter: Measurement Modeling 8.4.3 Classical PHD Filter: Corrector 8.4.4 Classical PHD Filter: State Estimation 8.4.5 Classical PHD Filter: Uncertainty Estimation 8.4.6 Classical PHD Filter: Characteristics 8.5 Classical Cardinalized PHD (CPHD) Filter 8.5.1 Classical CPHD Filter Motion Modeling 8.5.2 Classical CPHD Filter: Predictor 8.5.3 Classical CPHD Filter: Measurement Modeling 8.5.4 Classical CPHD Filter: Corrector

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8.6 8.7

8.5.5 Classical CPHD Filter: State Estimation 8.5.6 Classical CPHD Filter: Characteristics 8.5.7 Approximate Classical CPHD Filter Zero False Alarms (ZFA) CPHD Filter 8.6.1 Comparison of the PHD and ZFA-CPHD Filters PHD Filter for State-Dependent Poisson Clutter

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Chapter 9 Implementing Classical PHD/CPHD Filters 9.1 Introduction 9.1.1 Summary of Major Lessons Learned 9.1.2 Organization of the Chapter 9.2 “Spooky Action at a Distance” 9.3 Merging and Splitting for PHD Filters 9.3.1 Merging for PHD Filters 9.3.2 Splitting for PHD Filters 9.4 Merging and Splitting for CPHD Filters 9.4.1 Merging for CPHD Filters 9.4.2 Splitting for CPHD Filters 9.5 Gaussian Mixture (GM) Implementation 9.5.1 Standard GM Implementation 9.5.2 Pruning Gaussian Components 9.5.3 Merging Gaussian Components 9.5.4 GM-PHD Filter 9.5.5 GM-CPHD Filter 9.5.6 Implementation with Nonconstant pD 9.5.7 Implementation with Partially Uniform Target Births 9.5.8 Implementation with Target Identity 9.6 Sequential Monte Carlo (SMC) Implementation 9.6.1 SMC Approximation 9.6.2 SMC-PHD Filter 9.6.3 SMC-CPHD Filter 9.6.4 Using Measurements to Choose New Particles 9.6.5 Implementation with Target Identity

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Chapter 10Multisensor PHD and CPHD Filters 10.1 Introduction 10.1.1 Summary of Major Lessons Learned 10.1.2 Organization of the Chapter 10.2 The Multisensor-Multitarget Bayes Filter

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10.3 The General Multisensor PHD Filter 10.3.1 General Multisensor PHD Filter: Modeling 10.3.2 General Multisensor PHD Filter: Update 10.4 The Multisensor Classical PHD Filter 10.4.1 Implementations of the Exact Classical Multisensor PHD Filter 10.5 Iterated-Corrector Multisensor PHD/CPHD Filters 10.5.1 Limitations of the Iterated-Corrector Approach 10.6 Parallel Combination Multisensor PHD and CPHD Filters 10.6.1 Parallel Combination Multisensor CPHD Filter 10.6.2 Parallel Combination Multisensor PHD Filter 10.6.3 Simplified PCAM-PHD Filter 10.7 An Erroneous “Averaged” Multisensor PHD Filter 10.8 Performance Comparisons Chapter 11Jump-Markov PHD/CPHD Filters 11.1 Introduction 11.1.1 Summary of Major Lessons Learned 11.1.2 Organization of the Chapter 11.2 Jump-Markov Filters: A Review 11.2.1 The Jump-Markov Bayes Recursive Filter 11.2.2 State Estimation for Jump-Markov Filters 11.3 Multitarget Jump-Markov Systems 11.3.1 What Is a Multitarget Jump-Markov System? 11.3.2 The Multitarget Jump-Markov Filter 11.4 Jump-Markov PHD Filter 11.4.1 Jump-Markov PHD Filter: Models 11.4.2 Jump-Markov PHD Filter: Time Update 11.4.3 Jump–Markov PHD Filter: Measurement Update 11.4.4 Jump-Markov PHD Filter: State Estimation 11.5 Jump-Markov CPHD Filter 11.5.1 Jump-Markov CPHD Filter: Modeling 11.5.2 Jump-Markov CPHD Filter: Time Update 11.5.3 Jump-Markov CPHD Filter: Measurement Update 11.5.4 Jump-Markov CPHD Filter: State Estimation 11.6 Variable State Space Jump-Markov CPHD Filters 11.6.1 Variable State Space CPHD Filters: Modeling 11.6.2 Variable State Space CPHD Filters: Time Update

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11.6.3 Variable State Space CPHD Filters: Measurement Update 11.6.4 Variable State Space CPHD Filters: State Estimation 11.7 Implementing Jump-Markov PHD/CPHD Filters 11.7.1 Gaussian Mixture Jump-Markov PHD/CPHD Filters 11.7.2 Particle Implementation of Jump-Markov PHD and CPHD Filters 11.8 Implemented Jump-Markov PHD/CPHD Filters 11.8.1 Jump-Markov PHD Filter of Pasha et al. 11.8.2 IMM-Type JM-PHD Filter of Punithakumar et al. 11.8.3 Best-Fitting-Gaussian PHD Filter of Wenling Li and Yingmin Jia 11.8.4 JM-CPHD Filter of Georgescu et al. 11.8.5 Current Statistical Model (CSM) PHD Filter of Mengjun et al. 11.8.6 The Variable State Space CPHD Filter of Chen et al. Chapter 12Joint Tracking and Sensor-Bias Estimation 12.1 Introduction 12.1.1 Example: “Gridlocking” of Sensor Platforms 12.1.2 Gridlocking in General 12.1.3 Summary of Major Lessons Learned 12.1.4 Organization of the Chapter 12.2 Modeling Sensor Biases 12.3 Optimal Joint Tracking and Registration 12.3.1 Optimal BURT Filter: Single-Filter Version 12.3.2 Optimal BURT Filter: Two-Filter Version 12.3.3 Optimal BURT Procedure 12.4 The BURT-PHD Filter 12.4.1 BURT-PHD Filter: Single-Sensor Case 12.4.2 BURT-PHD Filter: Multisensor Case Using Iterated Corrector 12.4.3 BURT-PHD Filter: Multisensor Case Using Parallel Combination 12.5 Single-Filter BURT-PHD Filters 12.5.1 Single-Filter BURT-PHD Filter for Static Biases 12.5.2 A Heuristic Single-Filter BURT-PHD Filter 12.6 Implemented BURT-PHD Filters

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12.6.1 The BURT-PHD Filter of Ristic and Clark 12.6.2 The BURT-PHD Filter of Lian et al.

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Chapter 13Multi-Bernoulli Filters 13.1 Introduction 13.1.1 Summary of Major Lessons Learned 13.1.2 Organization of the Chapter 13.2 The Bernoulli Filter 13.2.1 Bernoulli Filter: Modeling 13.2.2 Bernoulli Filter: Time-Update 13.2.3 Bernoulli Filter: Measurement Update 13.2.4 Bernoulli Filter: State Estimation 13.2.5 Bernoulli Filter: Error Estimation 13.2.6 The Bernoulli Filter as an Exact PHD Filter 13.2.7 Bernoulli Filter: Practical Implementation 13.2.8 Bernoulli Filter: Implementations 13.3 The Multisensor Bernoulli Filter 13.4 The CBMeMBer Filter 13.4.1 CBMeMBer Filter: Modeling 13.4.2 CBMeMBer Filter: Predictor 13.4.3 CBMeMBer Filter: Corrector 13.4.4 CBMeMBer Filter: Merging and Pruning 13.4.5 CBMeMBer Filter: State and Error Estimation 13.4.6 CBMeMBer Filter: Track Management 13.4.7 CBMeMBer Filter: Gaussian-Mixture and Particle Implementation 13.4.8 CBMeMBer Filter: Performance 13.5 Jump-Markov CBMeMBer Filter 13.5.1 Jump-Markov CBMeMBer Filter: Modeling 13.5.2 Jump-Markov CBMeMBer Filter: Predictor 13.5.3 Jump-Markov CBMeMBer Filter: Corrector 13.5.4 Jump-Markov CBMeMBer Filter: Performance

379 379 380 381 382 383 384 384 385 386 386 387 388 388 390 392 392 393 395 395 396

Chapter 14RFS Multitarget Smoothers 14.1 Introduction 14.1.1 Summary of Major Lessons Learned 14.1.2 Organization of the Chapter 14.2 Single-Target Forward-Backward Smoother 14.2.1 Derivation of Forward-Backward Smoother

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14.3 14.4

14.5

14.6

14.2.2 Vo-Vo Alternative Form of the Forward-Backward Smoother 14.2.3 Vo-Vo Exact Closed-Form GM Forward-Backward Smoother General Multitarget Forward-Backward Smoother Bernoulli Forward-Backward Smoother 14.4.1 Bernoulli Forward-Backward Smoother: Modeling 14.4.2 Bernoulli Forward-Backward Smoother: Equations 14.4.3 Bernoulli Forward-Backward Smoother: Exact GM Implementation 14.4.4 Bernoulli Forward-Backward Smoother: Results PHD Forward-Backward Smoother 14.5.1 PHD Forward-Backward Smoother Equation 14.5.2 Derivation of the PHD Forward-Backward Smoother 14.5.3 Fast Particle-PHD Forward-Backward Smoother 14.5.4 Alternative PHD Forward-Backward Smoother 14.5.5 Gaussian-Mixture PHD Smoother 14.5.6 Implementations of the PHD Forward-Backward Smoother ZTA-CPHD Smoother

Chapter 15Exact Closed-Form Multitarget Filter 15.1 Introduction 15.1.1 Exact Closed-Form Solution of the Single-Target Bayes Filter 15.1.2 Exact Closed-Form Solution of the Multitarget Bayes Filter 15.1.3 Overview of the Vo-Vo Filter Approach 15.1.4 Summary of Major Lessons Learned 15.1.5 Organization of the Chapter 15.2 Labeled RFSs 15.2.1 Target Labels 15.2.2 Labeled Multitarget State Sets 15.2.3 Set Integrals for Labeled Multitarget States 15.3 Examples of Labeled RFSs 15.3.1 Labeled i.i.d.c. RFSs 15.3.2 Labeled Poisson RFSs 15.3.3 Labeled Multi-Bernoulli (LMB) RFSs

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15.3.4 Generalized Labeled Multi-Bernoulli (GLMB) RFSs 15.4 Modeling for the Vo-Vo Filter 15.4.1 Labeling Conventions 15.4.2 Overview of the Vo-Vo Filter 15.4.3 Basic Motion and Measurement Models 15.4.4 Motion and Measurement Models with Target ID 15.4.5 The Labeled Multitarget Likelihood Function 15.4.6 The Labeled Multitarget Markov Density—Standard Version 15.4.7 Labeled Multitarget Markov Density—Modified 15.5 Closure of Multitarget Bayes Filter 15.5.1 A “Road Map” for the Derivations 15.5.2 Closure Under Measurement Update with Respect to Vo-Vo Priors 15.5.3 Closure Under Time Update with Respect to Vo-Vo Priors 15.6 Implementation of the Vo-Vo Filter: Sketch 15.6.1 δ-GLMB Distributions 15.6.2 δ-GLMB Version of the Vo-Vo Filter 15.6.3 Characterization of Pruning 15.7 Performance Results 15.7.1 Gaussian Mixture Implementation of Vo-Vo Filter 15.7.2 Particle Implementation of the Vo-Vo Filter

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RFS Filters for Unknown Backgrounds

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Chapter 16Introduction to Part III 16.1 Introduction 16.2 Overview of the Approach 16.3 Models for Unknown Backgrounds 16.3.1 A Model for Unknown Detection Profile 16.3.2 A General Model for Unknown Clutter 16.3.3 Unknown-Clutter Models: Poisson-Mixture 16.3.4 Unknown-Clutter Models: General Bernoulli 16.3.5 Unknown-Clutter Models: Simplified Bernoulli 16.4 Organization of Part III

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Chapter 17RFS Filters for Unknown pD

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17.1 Introduction 17.1.1 Converting RFS Filters into pD -Agnostic Filters 17.1.2 A Motion Model for Probability of Detection 17.1.3 Summary of Major Lessons Learned 17.1.4 Organization of the Chapter 17.2 The pD -CPHD Filter 17.2.1 pD -CPHD Filter Models 17.2.2 pD -CPHD Filter Time Update 17.2.3 pD -CPHD Filter Measurement Update 17.2.4 pD -CPHD Filter Multitarget State Estimation 17.3 Beta-Gaussian Mixture (BGM) Approximation 17.3.1 Overview of the BGM Approach 17.3.2 Beta-Gaussian Mixtures (BGMs) 17.3.3 Pruning BGM Components 17.3.4 Merging BGM Components 17.4 BGM Implementation of the pD -PHD Filter 17.4.1 BGM pD -PHD Filter Modeling Assumptions 17.4.2 BGM pD -PHD Filter Time Update 17.4.3 BGM pD -PHD Filter Measurement Update 17.4.4 BGM pD -PHD Filter Multitarget State Estimation 17.5 BGM Implementation of the pD -CPHD Filter 17.5.1 BGM pD -CPHD Filter Modeling Assumptions 17.5.2 BGM pD -CPHD Filter Time Update 17.5.3 BGM pD -CPHD Filter Measurement Update 17.5.4 BGM pD -CPHD Filter Multitarget State Estimation 17.6 The pD -CBMeMBer Filter 17.7 Implementations of pD -Agnostic RFS Filters Chapter 18RFS Filters for Unknown Clutter 18.1 Introduction 18.1.1 Summary of Major Lessons Learned 18.1.2 Organization of the Chapter 18.2 A General Model for Unknown Bernoulli Clutter 18.2.1 The General Joint Target-Clutter Model 18.2.2 Phenomenology-Nonintermixing Motion Model 18.2.3 Phenomenology-Intermixing Motion Model 18.3 CPHD Filter for General Bernoulli Clutter

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18.4

18.5

18.6

18.7

18.8

18.3.1 General Bernoulli Clutter-Generator Model: CPHD Filter Time Update 18.3.2 General Bernoulli Clutter Model: CPHD Filter Measurement Update 18.3.3 General Bernoulli Clutter-Generator Model: PHD Filter Special Case 18.3.4 General Bernoulli Clutter Model: Multitarget State Estimation 18.3.5 General Bernoulli Clutter-Generator Model: Clutter Estimation The λ-CPHD Filter 18.4.1 λ-CPHD Filter: Models 18.4.2 λ-CPHD Filter: Time Update 18.4.3 λ-CPHD Filter: Measurement Update 18.4.4 λ-CPHD Filter: Multitarget State Estimation 18.4.5 λ-CPHD Filter: Clutter Estimation 18.4.6 Special Case: The λ-PHD Filter 18.4.7 λ-CPHD Filter Implementation: Gaussian Mixtures The κ-CPHD Filter 18.5.1 κ-CPHD Filter: Models 18.5.2 κ-CPHD Filter: Time Update 18.5.3 κ-CPHD Filter: Measurement Update 18.5.4 κ-CPHD Filter: Multitarget State Estimation 18.5.5 κ-CPHD Filter: Clutter Estimation 18.5.6 Special Case: The κ-PHD Filter 18.5.7 κ-CPHD Filter: Beta-Gaussian Mixtures 18.5.8 κ-CPHD Filter Implementation: Normal-Wishart Mixtures Multisensor κ-CPHD Filters 18.6.1 Iterated-Corrector κ-CPHD Filter 18.6.2 Parallel-Combination κ-CPHD Filter The κ-CBMeMBer Filter 18.7.1 κ-CBMeMBer Filter: Modeling 18.7.2 κ-CBMeMBer Filter: Time Update 18.7.3 κ-CBMeMBer Filter: Measurement Update 18.7.4 κ-CBMeMBer Filter: Multitarget State Estimation 18.7.5 κ-CBMeMBer Filter: Clutter Estimation Implemented Clutter-Agnostic RFS Filters

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18.8.1 Implemented λ-CPHD Filter 18.8.2 “Bootstrap” λ-CPHD Filter 18.8.3 Implemented λ-CBMeMBer Filter 18.8.4 Implemented NWM-PHD Filter 18.9 Clutter-Agnostic Pseudofilters 18.9.1 The λ-PHD Pseudofilter 18.9.2 Pathological Behavior of the λ-PHD Pseudofilter 18.10CPHD/PHD Filters with Poisson-Mixture Clutter 18.10.1 Poisson-Mixture Clutter-Agnostic CPHD Filter 18.10.2 Poisson-Mixture Clutter-Agnostic PHD Filter 18.11Related Work 18.11.1 Decoupled Target-Clutter PHD Filter 18.11.2 The “Dual PHD” Filter 18.11.3 The “iFilter ”

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RFS Filters for Nonstandard Measurement Models

628 629 630 631 631 632 635 636 638 640 641 642 643 644

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Chapter 19RFS Filters for Superpositional Sensors 19.1 Introduction 19.1.1 Examples of Superpositional Sensor Models 19.1.2 Summary of Major Lessons Learned 19.1.3 Organization of the Chapter 19.2 Exact Superpositional CPHD Filter 19.3 Hauschildt’s Approximation 19.3.1 Hauschildt Σ-CPHD Filter: Overview 19.3.2 Hauschildt Σ-CPHD Filter: Models 19.3.3 Hauschildt Σ-CPHD Filter: Measurement Update 19.3.4 Hauschildt Σ-CPHD Filter: Implementations 19.4 Thouin-Nannuru-Coates (TNC) Approximation 19.4.1 Generalized TNC Approximation: Overview 19.4.2 TNC Σ-CPHD Filter: Models 19.4.3 TNC Σ-CPHD Filter: Measurement Update 19.4.4 TNC Σ-CPHD Filter: Implementations

647 647 648 653 653 654 656 656 658 658 661 661 662 666 666 668

Chapter 20RFS Filters for Pixelized Images 20.1 Introduction 20.1.1 Summary of Major Lessons Learned 20.1.2 Organization of the Chapter

671 671 672 672

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20.2 20.3 20.4 20.5

The IO Multitarget Measurement Model IO Motion Model IO-CPHD Filter IO-MeMBer Filter 20.5.1 IO-MeMBer Filter: Measurement Update 20.5.2 IO-MeMBer Filter: Track Merging 20.5.3 IO-MeMBer Filter: Multitarget State Estimation 20.5.4 IO-MeMBer Filter: Track Management 20.6 Implementations of IO-MeMBer Filters 20.6.1 Track-Before-Detect (TBD) in Image Data 20.6.2 Tracking in Color Videos 20.6.3 Tracking Road-Constrained Targets Chapter 21RFS Filters for Cluster-Type Targets 21.1 Introduction 21.1.1 Summary of Major Lessons Learned 21.1.2 Organization of the Chapter 21.2 Extended-Target Measurement Models 21.2.1 The Statistics of Extended Targets 21.2.2 Exact Rigid-Body (ERB) Model 21.2.3 Approximate Rigid-Body (ARB) Model 21.2.4 Approximate Poisson-Body (APB) Model 21.3 Extended-Target Bernoulli Filters 21.3.1 Extended-Target Bernoulli Filters: Performance 21.4 Extended-Target PHD/CPHD Filters 21.4.1 General Extended-Target PHD Filter 21.4.2 PHD Filter for Extended Targets: ERB Model 21.4.3 PHD Filter for Extended Targets: APB Model 21.5 Extended-Target CPHD Filter: APB Model 21.5.1 APB-CPHD Filter: Theory 21.5.2 Gaussian Mixture APB-CPHD Filter: Performance 21.5.3 Gamma Gaussian Inverse-Wishart APB-CPHD Filter: Performance 21.5.4 APB-CPHD Filter of Lian et al.: Performance 21.6 Cluster-Target Measurement Model 21.6.1 Likelihood Function for Cluster Targets 21.6.2 Estimation of Soft Clusters 21.7 Cluster-Target PHD and CPHD Filters

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21.7.1 Cluster-Target CPHD Filter 21.7.2 Cluster-Target PHD Filter 21.8 Measurement Models for Level-1 Group Targets 21.8.1 “Natural” State Representation of Single Level-1 Group Targets 21.8.2 “Natural” State Representation of Multiple Level-1 Group Targets 21.8.3 Simplified State Representation of Multiple Level-1 Group Targets 21.8.4 Multiple Level-1 Group Targets with the Standard Measurement Model 21.9 PHD/CPHD Filters for Level-1 Group Targets 21.9.1 PHD Filter for Level-1 Group Targets: Standard Model 21.9.2 CPHD Filter for Level-1 Group Targets: Standard Model 21.9.3 PHD Filter for Single Level-1 Group Targets: Standard Measurement Model 21.9.4 CPHD Filter for Single Level-1 Group Targets: Standard Model 21.10Measurement Models for General Group Targets 21.10.1 Simplified State Representation of Level-ℓ Group Targets 21.10.2 Standard Measurement Model for Level-ℓ Group Targets 21.11PHD/CPHD Filters for Level-ℓ Group Targets 21.12A Model for Unresolved Targets 21.13Motion Model for Unresolved Targets 21.14The Unresolved-Target PHD Filter 21.15Approximate Unresolved-Target PHD Filter 21.16Approximate Unresolved-Target CPHD Filter Chapter 22RFS Filters for Ambiguous Measurements 22.1 Introduction 22.1.1 Motivation: Quantized Measurements 22.1.2 Generalized Measurements, Measurement Models, and Likelihoods 22.1.3 Summary of Major Lessons Learned

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22.1.4 Organization of the Chapter 22.2 Random Set Models of Ambiguous Measurements 22.2.1 Imprecise Measurements 22.2.2 Vague Measurements 22.2.3 Uncertain Measurements 22.2.4 Contingent Measurements (Inference Rules) 22.2.5 Generalized Fuzzy Measurements 22.3 Generalized Likelihood Functions (GLFs) 22.3.1 GLFs for Nonnoisy Nontraditional Measurements 22.3.2 GLFs for Noisy Nontraditional Measurements 22.3.3 Bayesian Processing of Generalized Measurements 22.3.4 Bayes Optimality of the GLF Approach 22.4 Unification of Expert-System Theories 22.4.1 Bayesian Unification of Measurement Fusion 22.4.2 Dempster’s Rule Arises as a Particular Instance of Bayes’ Rule 22.4.3 Bayes-Optimal Measurement Conversion 22.5 GLFs for Imperfectly Characterized Targets 22.5.1 Example: Imperfectly Characterized Target Types 22.5.2 Example: Received Signal Strength (RSS) 22.5.3 Modeling Imperfectly Characterized Targets 22.5.4 GLFs for Imperfectly Characterized Targets 22.5.5 Bayes Filtering with Imperfectly Characterized Targets 22.6 GLFs for Unknown Target Types 22.6.1 Unmodeled Target Type 22.6.2 Unmodeled Target Types—Imperfectly Characterized Measurement Function 22.7 GLFs for Information with Unknown Correlations 22.8 GLFs for Unreliable Information Sources 22.9 Using GLFs in Multitarget Filters 22.10GLFs in RFS Multitarget Filters 22.10.1 Using GLFs in PHD Filters 22.10.2 Using GLFs in CPHD Filters 22.10.3 Using GLFs in CBMeMBer Filters 22.10.4 Using GLFs in Bernoulli Filters 22.10.5 Implementations of RFS Filters for Nontraditional Measurements

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22.11Using GLFs with Conventional Multitarget Filters 22.11.1 Measurement-to-Track Association (MTA) with Nontraditional Measurements 22.11.2 A Closed-Form Example: Fuzzy Measurements 22.11.3 MTA with Joint Kinematic and Nonkinematic Measurements

V

Sensor, Platform, and Weapons Management

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Chapter 23Introduction to Part V 23.1 Basic Issues in Sensor Management 23.1.1 Top-Down or Bottom-Up? 23.1.2 Single-Step or Multistep? 23.1.3 Information-Theoretic or Mission-Oriented? 23.2 Information Theory and Intuition: An Example 23.2.1 PENT for “Cookie Cutter” Sensor Fields of View (FoVs) 23.2.2 PENT for General Sensor Fields of View 23.2.3 Characteristics of PENT 23.2.4 The Cardinality-Covariance Objective Function 23.2.5 The Cauchy-Schwartz Objective Function 23.3 Summary of RFS Sensor Control 23.3.1 RFS Control Summary: General Approach (SingleStep) 23.3.2 RFS Control Summary: Ideal Sensor Dynamics 23.3.3 RFS Control Summary: Simplified Nonideal Sensor Dynamics 23.3.4 RFS Control Summary: Control with PHD and CPHD Filters 23.3.5 RFS Control Summary: “Pseudosensor” Approximation for Multisensor Control 23.3.6 RFS Control Summary: General Approach (Multistep) 23.4 Organization of Part V

827 830 831 831 832 834

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Chapter 24Single-Target Sensor Management 24.1 Introduction 24.1.1 Summary of Major Lessons Learned

861 861 861

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Contents

24.1.2 Organization of the Chapter 24.2 Example: Missile-Tracking Cameras 24.2.1 Single-Camera Missile Tracking 24.2.2 Two-Camera Missile Tracking 24.3 Single-Sensor, Single-Target Control: Modeling 24.4 Single-Sensor, Single-Target Control: Single-Step 24.5 Single-Sensor, Single-Target Control: Objective Functions 24.5.1 Kullback-Leibler Information Gain 24.5.2 Csisz´ar Information Gain 24.5.3 Cauchy-Schwartz Information Gain 24.6 Single-Sensor, Single-Target Control: Hedging 24.6.1 Expected-Value Hedging 24.6.2 Minimum-Value Hedging 24.6.3 Multisample Approximate Hedging 24.6.4 Single-Sample Approximate Hedging 24.6.5 Mixed Expected-Value and PM Hedging 24.7 Single–Sensor, Single-Target Control: Optimization 24.8 Special Case 1: Ideal Sensor Dynamics 24.9 Simple Example: Linear-Gaussian Case 24.10Special Case 2: Simplified Nonideal Dynamics 24.10.1 Simplified Nonideal Single-Sensor Dynamics: Modeling 24.10.2 Simplified Nonideal Single-Sensor Dynamics: Filtering Equations 24.10.3 Simplified Nonideal Single-Sensor Dynamics: Optimization Chapter 25Multitarget Sensor Management 25.1 Introduction 25.1.1 Summary of Major Lessons Learned 25.1.2 Organization of the Chapter 25.2 Multitarget Control: Target and Sensor State Spaces 25.2.1 Target State Spaces 25.2.2 Sensor State Spaces 25.2.3 Joint Multisensor-Multitarget State Space 25.2.4 Integrals and Set Integrals on State Spaces 25.2.5 p.g.fl.’s on Target/Sensor State Spaces

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25.3 Multitarget Control: Control Spaces 25.4 Multitarget Control: Measurement Spaces 25.4.1 Sensor Measurements 25.4.2 Actuator-Sensor Measurements 25.4.3 Joint Multisensor-Multitarget Measurements 25.4.4 Integrals and Set Integrals on Measurement Spaces 25.4.5 p.g.fl.’s on Measurement Spaces 25.5 Multitarget Control: Motion Models 25.5.1 Single-Target and Multitarget Motion Models 25.5.2 Single-Sensor Motion and Multisensor Motion with Sensor Controls 25.5.3 Joint Multisensor-Multitarget Motion 25.6 Multitarget Control: Measurement Models 25.6.1 Measurements: Assumptions 25.6.2 Measurements: Sensor Noise 25.6.3 Measurements: Fields of View (FoVs) and Clutter 25.6.4 Measurements: Actuator Sensors and Transmission Failure 25.6.5 Measurements: Multitarget Likelihood Functions 25.6.6 Measurements: Joint Multitarget Likelihood Functions 25.7 Multitarget Control: Summary of Notation 25.7.1 Notation for Spaces of Interest 25.7.2 Notation for Motion Models 25.7.3 Notation for Measurement Models 25.8 Multitarget Control: Single Step 25.9 Multitarget Control: Objective Functions 25.9.1 Information-Theoretic Objective Functions 25.9.2 The PENT Objective Function 25.9.3 The Cardinality-Variance Objective Function 25.9.4 PENT as an Approximate Information-Theoretic Objective Function 25.10Multisensor-Multitarget Control: Hedging 25.10.1 Hedging Using Predicted Measurement Set (PMS)? 25.10.2 Predicted Ideal Measurement Set (PIMS): A General Approach 25.10.3 Predicted Ideal Measurement Set (PIMS): Special Cases

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25.10.4 Predicted Ideal Measurement Set (PIMS): Derivation of General Approach 25.11Multisensor-Multitarget Control: Optimization 25.12Sensor Management with Ideal Sensor Dynamics 25.13Simplified Nonideal Multisensor Dynamics 25.13.1 Simplified Nonideal Multisensor Dynamics: Assumptions 25.13.2 Simplified Nonideal Multisensor Dynamics: Filtering Equations 25.13.3 Simplified Nonideal Single-Sensor Dynamics: Hedgingand Optimization 25.14Target Prioritization 25.14.1 The Concept of Tactical Significance 25.14.2 Tactical Importance Functions (TIFs) and HigherLevel Fusion 25.14.3 Characteristics of TIFs 25.14.4 The Multitarget Statistics of TIFs 25.14.5 Posterior Expected Number of Targets of Interest (PENTI) 25.14.6 Biasing the Cardinality Variance to Targets of Interest (ToIs) Chapter 26Approximate Sensor Management 26.1 Introduction 26.1.1 Summary of Major Lessons Learned 26.1.2 Organization of the Chapter 26.2 Sensor Management with Bernoulli Filters 26.2.1 Sensor Management with Bernoulli Filters: Filtering Equations 26.2.2 Sensor Management with Bernoulli Filters: Objective Functions 26.2.3 Bernoulli Filter Control: Hedging 26.2.4 Bernoulli Filter Control: Multisensor 26.3 Sensor Management with PHD Filters 26.3.1 Single-Sensor, Single-Step PHD Filter Control 26.3.2 PHD Filter Sensor Management: Multisensor SingleStep

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26.4 Sensor Management with CPHD Filters 989 26.4.1 Single-Sensor, Single-Step CPHD Filter Control 990 26.4.2 Multisensor, Single-Step CPHD Filter Control 1001 26.5 Sensor Management with CBMeMBer Filters 1008 26.5.1 Single-Sensor, Single-Step CBMeMBer Filter Control1008 26.5.2 Multisensor, Single-Step CBMeMBer Control 1015 26.6 RFS Sensor Management Implementations 1021 26.6.1 RFS Control Implementations: Multitarget Bayes Filter 1021 26.6.2 RFS Control Implementations: Bernoulli Filters 1024 26.6.3 RFS Control Implementations: PHD Filters 1025 26.6.4 RFS Control Implementations: CBMeMBer Filters 1030 Appendix A Glossary of Notation and Terminology A.1 Transparent Notational System A.2 General Mathematics A.3 Set Theory A.4 Fuzzy Logic and Dempster-Shafer Theory A.5 Probability and Statistics A.6 Random Sets A.7 Multitarget Calculus A.8 Finite-Set Statistics A.9 Generalized Measurements

1033 1033 1034 1035 1036 1036 1038 1038 1039 1040

Appendix B Bayesian Analysis of Dynamic Systems B.1 Formal Bayes Modeling in General B.2 The Bayes Filter in General

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Appendix C Rigorous Functional Derivatives C.1 Nonconstructive Definition of the Functional Derivative C.2 The Constructive Radon-Nikod´ym Derivative C.3 Constructive Definition of the Functional Derivative

1045 1045 1047 1048

Appendix D Partitions of Finite Sets D.1 Counting Partitions D.2 Recursive Construction of Partitions

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Appendix E Beta Distributions

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Appendix F Markov Time Update of Beta Distributions

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Appendix G Normal-Wishart Distributions G.1 Proof of (G.8) G.2 Proof of (G.22) G.3 Proof of (G.23) G.4 Proof of (G.29)

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Appendix H Complex-Number Gaussian Distributions

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Appendix I Statistics of Level-1 Group Targets

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Appendix J FISST Calculus and Moyal’s Calculus J.1 A “Point Process ”Functional Calculus J.2 Volterra Functional Derivatives J.3 Moyal’s Functional Calculus of p.g.fl.’s J.3.1 Moyal’s p.g.fl. J.3.2 Moyal’s Functional Calculus

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Appendix K Mathematical Derivations

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References

1089

About the Author

1109

Index

1110

Preface This book is a sequel to my 2007 book, Statistical Multisource-Multitarget Information Fusion [179]. That earlier book was a textbook-style introduction to finiteset statistics (also known as random set information fusion), a fundamentally new, seamlessly unified, and fully probabilistic approach to multisource-multitarget detection, tracking, classification, and information fusion. This sequel provides a comprehensive description of the state of the art in random set information fusion since 2007—a description not otherwise available. Its intended audience is signal processing graduate students, researchers, and engineers, as well as mathematicians and statisticians interested in tracking, information fusion, robotics, and related subjects. Finite-set statistics has five major elements: • A general theory of measurements, based on a stochastic-geometry formulation of random set theory. • A general theory of stochastic multiobject systems, based on a stochasticgeometry formulation of point process theory or, equivalently, random finite set theory. • A general approach to multisource-multitarget modeling based on multiobject integro-differential calculus. • A general optimal approach to multisource-multitarget processing based on these models and Bayesian filter theory. • A general approach to multisource-multitarget algorithmic approximation, also based on multiobject integro-differential calculus.

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Tutorial introductions to the subject can be found in [198], [181], and [311]; and extended summaries in the book chapters [177] and [311]. An authorized Chinese-language edition of Statistical Multisource-Multitarget Information Fusion is also available [180]. Other information sources are: • The “Random Finite Set Filtering” web site, with U.K. and Australian mirrors: – http://randomsets.eps.hw.ac.UK/index.html. – http://randomsets.ee.unimelb.edu.au/index.html. • Professor Ba-Tuong Vo’s web sites: – http://ba-tuong-vo-au.com. – http://ba-tuong-vo-au.com/codes/html. – Various MATLAB algorithm codes are available at the second link: probability hypothesis density (PHD) filter; cardinalized PHD (CPHD) filter; cardinality-balanced multi-Bernoulli (CBMeMBer) filter; the singletarget RFS filter described in [309]; and (eventually) the λ-CPHD filter (described in Chapter 18) and the track-before-detect multi-Bernoulli filter (described in Chapter 20). Since 2007, the approach has inspired a considerable amount of research, conducted by many dozens of researchers in at least a dozen nations, reported in many hundreds of research publications. As a result, progress in random set information fusion has been rapid and has proceeded in diverse and sometimes unexpected directions, propelled by many clever new ideas. Indeed, the rapidity and extent of progress has itself been somewhat unexpected, especially given my cautious disclaimer in Statistical Multisource-Multitarget Information Fusion ([179], p. 566): “...preliminary research has suggested that the PHD and CPHD filters may be more effective than [multihypothesis correlator-type] filters in some conventional multitarget detection and tracking problems. Whether such claims hold up is for future research to determine. Here we emphasize that the PHD and CPHD approaches were originally devised to address non-traditional tracking problems such as those just described [that is, tracking of target clusters].”

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A summary of advances in the field will be given in Section 1.2. In brief, the progression of research emphasis has been roughly as follows: from PHD filter to CPHD filter; from CPHD filter to multi-Bernoulli filters and, in particular, to the CBMeMBer filter; and most recently, to “background-agnosic” CPHD and CBMeMBer filters and the Vo-Vo exact closed-form multitarget detection and tracking filter. Ancillary advances have occurred in regard to joint tracking and sensor registration; superpositional sensors and track-before-detect (TBD); distributed fusion; sensor management; and robotics. As time progressed, it became increasingly clear to me that the most intriguing aspects of the new research should be aggregated and systematized, in a single place, into a coherent and integrated picture. That is the purpose of this book. Thus one of my primary goals is to provide a deep-dive overview of the state of the art in the field. However, this overview is not intended to be comprehensive. Many rather intricate implementation strategies for PHD and CPHD filters have been neglected, for example. I have placed a greater emphasis on exact closed-form implementation approaches, rather than particle implementation approaches. I have not included research whose veracity I have been unable to verify in detail. In particular I have, albeit with a few exceptions, excluded mention of research that is mathematically erroneous or that I have been unable to understand. Also, as the book neared finalization at the end of 2013, it became impossible to include later-breaking developments. Certain advances are not described here, or are described at only a broad conceptual level, because they are already addressed in book-length or book-chapter form elsewhere. The reader’s attention is directed to the following publications: • B. Ristic, Particle Filters for Random Set Models [250]. – As the title indicates, this book emphasizes techniques for the sequential Monte Carlo (SMC) implementation of RFS multitarget detection and tracking filters. – An unusual aspect of the book is its application of particle-RFS filters to nontraditional measurements, such as natural-language statements and inference rules. • J. Mullane, B.-N. Vo, M. Adams, and B.-T. Vo, Random Finite Sets in Robotic Map Building and SLAM [210].

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– This is a concise introduction to the new random finite set approach to the robotics field known as Simultaneous Localization and Mapping (SLAM). Its topics include an introduction to RFSs, estimation with RFSs, a “brute force” PHD filter approach to SLAM, and a RaoBlackwellized PHD filter approach to SLAM. – The RFS-SLAM techniques reported in this book provide the first provably Bayes-optimal approach to SLAM. – In high-clutter environments, PHD filter-based SLAM algorithms described in the book have been shown to significantly outperform conventional approaches such as EKF-SLAM and MH-FastSLAM. – A tutorial overview of the approach can be found in [1]. – Another source is Professor Martin Adams’ web site: ∗ http://www.cec.uchile.cl/˜martin/Martin research 18 8 11.html. – The RFS-SLAM paper [141] by Lee, Clark, and Salvi is also pertinent in this regard, as is the June 2014 special issue of the IEEE Robotics & Automation Magazine on “Applications of Stochastic Geometry.” • R. Mahler, “Toward a Theoretical Foundation for Distributed Fusion” [183]. – This book chapter is essentially a summary and elaboration of work by D. Clark and his associates. This work—for example, the paper [294] by Uney, Clark, and Julier—leverages the multitarget covarianceintersection concept of [172], which is immune to unknown doublecounting of measurements. – Of special interest in this regard is the potentially breakthrough-level paper [15] by Battistelli, Chisci, Fantacci, Farina, and Graziano. – Additional papers of interest are [242], [10]. For those who may wish to employ this publication as a university textbook, I suggest the following. It is divided into five parts, devoted to increasingly more specialized and increasingly more research-oriented topics. Part I provides a fairly condensed summary of the basic elements of finite-set statistics. When used in conjunction with Statistical Multisource-Multitarget Information Fusion, it would be suitable for the first part of an introductory one-semester course. The beginning of Part II is devoted to topics that have become standard in finite-set statistics (PHD, CPHD, Bernoulli, and multi-Bernoulli filters). This material is

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suitable for the second part of an introductory one-semester course. Chapter 22 is appropriate for a course oriented more towards expert systems and higher-level information fusion. The remaining chapters of Part II, as well as Parts III, IV, and V, address more specialized topics and leading-edge research. When selected with a particular research focus in mind, a subset of these chapters could be the basis of a graduate seminar or advanced one-semester course, with the purpose of leading students to the threshold of dissertation-level research. For pedagogical reasons and ease of reference, many well-known concepts are reviewed—for example, unscented Kalman filters, jump-Markov filters, complex Gaussian distributions, beta distributions, and Wishart and inverse-Wishart distributions. As with Statistical Multisource-Multitarget Information Fusion, it is my sincere hope that the reader will find the book informative, useful, stimulating, thought-provoking, occasionally provocative, and possibly even a bit exciting.

Acknowledgments The contents of this book do not necessarily reflect the position or policy of Lockheed Martin Corporation. No official endorsement should be inferred. I gratefully acknowledge the ongoing original research and correspondences of Professors Ba-Ngu Vo and Ba-Tuong Vo (Curtin University, Perth, Australia); Professor Daniel Clark (Heriot-Watt University, Edinburgh, U.K.); and Dr. Branko Ristic (Defence Science and Technology Organization, Melbourne, Australia). Readers will also quickly discover the importance of the research of the following individuals and their students: Professor Thia Kirubarajan (McMaster University, Hamilton, Ontario, Canada); Professor Peter Willett (University of Connecticut, Storrs, Connecticut, United States of America); Professor Lennart Svensson (Chalmers University of Technology, G¨oteborg, Sweden); and Professor Mark Coates (McGill University, Montreal, Canada). I also wish to express my gratitude for the long-standing assistance and researches of my collaborators at Scientific Systems Company, Inc.—especially, Dr. Adel El-Fallah and Dr. Alexsander Zatezalo. I am particularly grateful to Dr. El-Fallah for his invaluable help in preparing the final camera-ready copy of the book. The work of dozens of researchers is described in this book. I have endeavored to describe their work as accurately and as clearly as possible. In many cases, I found it necessary to directly contact them for clarification of various technical issues. I wish to thank them for their time, assistance, and patience. Finally, and as I did in Statistical Multisource-Multitarget Information Fusion, I acknowledge my profound debt to the groundbreaking research of Dr. I.R. Goodman (U.S. Navy SPAWAR Systems Center, retired). It was Dr. Goodman who

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first made me aware of the potentially revolutionary implications of random set techniques in information fusion. The manuscript for this book was produced using Version 5.50 of MacKichan Software’s Scientific WorkPlace. The task of writing it would have been vastly more difficult without it. Figures were prepared using Scientific WorkPlace and Microsoft PowerPoint.

Chapter 1 Introduction to the Book The subject of the earlier book, Statistical Multisource-Multitarget Information Fusion [179], was finite-set statistics (FISST), a form of random finite set (RFS) theory specialized for application to information fusion Finite-set statistics unifies much of information fusion under a single probabilistic—in fact, Bayesian—paradigm. It does so by directly generalizing, to the multisource-multitarget realm, the “statistics 101” formalism that most signal processing practitioners learn as undergraduates. Increasingly more detailed tutorial descriptions of the approach can be found in the following publications: • R. Mahler, “‘Statistics 101’ for multisensor, multitarget data fusion” [198] (a tutorial introduction at a very elementary level); • R. Mahler, “‘Statistics 102’ for multisensor-multitarget tracking” [181] (a more detailed tutorial); • R. Mahler, “Random set theory for target tracking and identification” [177] (an extended summary); • B.-N.Vo, B.-T. Vo, and D. Clark, “Bayesian multiple target filtering using random finite sets” [311] (a tutorial with emphasis on implementation of the PHD, CPHD, and multi-Bernoulli filters). Finite-set statistics was introduced in 1994 in its basic form (set integrals, set derivatives, RFS motion and measurement models, multitarget Bayes filter) [162], [94]; and, in its current refined form (probability generating functionals, functional derivatives) in 2001 [168]. From the beginning it has been conceived as a systematic

1

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Figure 1.1 The finite-set statistics research program. Unification of expert systems, and of generalized measurements, sets the stage for a unification of multisource integration. This sets the stage for a unification of sensor and platform management. This is intended to set the stage for a unification of all of the information fusion levels.

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research program, as indicated in Figure 1.1. In the United States at least, information fusion has been conceptualized in terms of a taxonomy of “fusion levels” [25]. In the original and simplest form of this taxonomy, there are four fusion levels. Level 1 fusion (also known as “multisource integration”) algorithms address basic issues: the detection, tracking, localization, and identification of one or more targets using one or more information sources. Levels 2 and 3 fusion (“higher-level fusion,” “threat assessment,” “situation assessment”) algorithms address more complex and amorphous issues such as degree of threat and adversarial intent. Level 4 fusion (“information refinement” or “resource allocation”) algorithms enable sensing and other assets to collect better information about poorly understood targets of interest. Statistical Multisource-Multitarget Information Fusion was primarily concerned with the top two layers of Figure 1.1: unification of expert-systems theory, and unification of Level 1 fusion. The purpose of this book is therefore threefold: • To provide detailed and integrated descriptions of many of the most interesting and significant advances in Level 1 information fusion. • To place these advances in the context of the finite-set statistics research program, and with respect to each other. • To systematically address, for the first time, the bottom two layers in Figure 1.1—unification of sensor and platform management (see Part V), and the elements of a foundation for Levels 2 and 3 information fusion (Section 25.14 ). In consequence of the first goal, the reader will discover that much of the book is devoted to the innovations of other researchers. While I have endeavored to describe these as accurately as possible, any errors in their description should be attributed to me. The remainder of this chapter is structured as follows: 1. Section 1.1: An overview of the philosophy, methods, and techniques of finite-set statistics. 2. Section 1.2: A summary of recent advances in finite-set statistics. 3. Section 1.3: The organization of the book.

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1.1

OVERVIEW OF FINITE-SET STATISTICS

The purpose of this section is to provide a brief introduction to finite-set statistics and random finite set (RFS) methods for those readers who may not yet be familiar with them. An extended and detailed summary will be provided in Chapters 2 through 6 and Chapter 22. The section is organized as follows: 1. Section 1.1.1: The philosophy of finite-set statistics. 2. Section 1.1.2: Misconceptions about finite-set statistics. 3. Section 1.1.3: Measurement-to-track association (MTA)—the conventional approach to multisensor-multitarget information fusion. 4. Section 1.1.4: The random finite-set (RFS) approach, compared to MTA. 5. Section 1.1.5: Extension of the RFS approach to nontraditional measurements, using general random set theory. 1.1.1

The Philosophy of Finite-Set Statistics

Finite-set statistics has attracted a considerable amount of interest in a relatively short period of time. The lesson that a few seem to have drawn is that renown will follow if one skims a few insights from finite-set statistics while changing its notation and terminology; strips off some or all of the mathematical tools that make these insights rigorous, general, and useful; and then proclaims the resulting halfcopy to be an advance over finite-set statistics. All such imitator-critics have not only completely missed the point, but have embraced the same fallacy: the belief that mere changes of notation and terminology add technical substance. The point of finite-set statistics is not that multitarget problems can be formulated in terms of random finite sets, random counting measures, or anything else. The choice of a particular mathematical formalism is of limited practical interest in and of itself. The point is, rather, that random set techniques provide a carefully constructed practitioner’s toolbox of explicit, rigorous, systematic, and general procedures. This toolbox cannot be supplanted by extemporized, ad hoc reasoning that (as we shall see) actually facilitates the commission of error. The purpose of this section is to summarize the finite-set statistics toolbox and the philosophy that underlies it. Finite-set statistics is based on Bayesian probability and filtering theory, which is reviewed in Appendix B. Challa, Evans, and Musicki have succinctly

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summarized one of the primary viewpoints that motivate finite-set statistics [31, p. 437]: In practice, an intelligent combination of probability theory using Bayes’ theorem and ad hoc logic is used to solve tracking problems. The ad hoc aspects of practical tracking algorithms are there to limit the complexity of the probability theory based solutions. One of the key things that is worth noting is that the success of Bayesian solution depends on the models used. Hence, before applying Bayes’ theorem to practical problems, a significant effort must be spent on modeling. In pursuit of this goal, finite-set statistics addresses multisource-multitarget information fusion problems using the following systematic methodology: • Step 1: Approach information fusion problems in a unified, statistically topdown fashion, by constructing comprehensive statistically accurate models of multitarget-multisensor-multiplatform systems, including: – Top-down, comprehensive statistically accurate (as opposed to extemporized, ad hoc) models of multitarget sensing—encompassing phenomena such as sensor-platform dynamics, sensor slew rates, sensor noise, sensor fields of view (FoVs), missed-detection processes, clutter processes, obscurations, and transmission dropouts. – Top-down, comprehensive statistically accurate (as opposed to extemporized, ad hoc) models of multitarget motion—encompassing phenomena such as individual target motion, target disappearance, and target appearance. – Top-down, comprehensive statistically accurate models of “nontraditional measurements”—encompassing attributes, features, naturallanguage statements, and inference rules, as well as certain expertsystem uncertainty representations such as fuzzy sets, the DempsterShafer theory, and rule-based inference. • Step 2: Use these statistically accurate models to construct the optimal solution to the problem at hand—typically, some kind of multitarget recursive Bayes filter. This necessitates the explicit construction of: – “True” multitarget Markov transition densities—meaning that: ∗ These densities faithfully reflect the underlying multitarget motion model and thus are not heuristic or ad hoc.

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∗ No extraneous information has inadvertently been introduced. – “True” multitarget likelihood functions—meaning that ∗ These functions faithfully reflect the underlying multitarget measurement model and thus are not heuristic or ad hoc. ∗ No extraneous information has inadvertently been introduced. • Step 3: Since the optimal solution will usually be computationally intractable in general, use principled approximation techniques to “trim down” the optimal solution to an approximate one that is tractable and yet preserves, as faithfully as possible, the underlying models and their interrelationships. In particular, it is presumed that: – A principled approximation must be statistically top-down, in the sense that it has been directly constructed from the optimal multitarget solution. However, a statistically top-down approach forces us into unfamiliar theoretical territory: • Multisensor, multitarget systems are comprised of randomly varying numbers of randomly varying objects of various kinds: varying numbers of targets; varying numbers of sensors with varying number of sensor measurements collected by each sensor; and varying numbers of sensor-carrying platforms. A rigorous mathematical foundation for stochastic multiobject problems— point process theory [55], [278]—has been in existence for a half-century. However, this theory has traditionally been formulated with the requirements of mathematicians rather than tracking and information fusion practitioners in mind. The formulation usually preferred by mathematicians, random counting-measures, is inherently abstract and complex (especially in regard to probabilistic foundations) and not easily assimilable with practical physical intuition (see Section 2.3.1). The fundamental motivation underlying the finite-set statistics treatment of point process theory is this: • Tracking and information fusion R&D researchers and practitioners should not have to be virtuoso experts in point process theory to produce meaningful practical innovations. As was emphasized in [198], engineering statistics is a tool and not an end in itself. It must have two qualities:

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• Trustworthiness: Constructed upon a systematic, reliable mathematical foundation, to which we can appeal when the going gets rough. • Fire and forget: This foundation can be safely neglected in most situations, leaving a serviceable mathematical machinery in its place. These two qualities are inherently in conflict. If foundations are so mathematically complex that they cannot be taken for granted in most practical situations, then they are shackles and not foundations. If they are so simple that they repeatedly result in practical blunders, then they are simplistic rather than simple. This inherent gap between mathematical trustworthiness and practical pragmatism is what finite-set statistics attempts to bridge. Four objectives are paramount: • Directly generalize familiar single-sensor, single-target Bayesian “Statistics 101” concepts to the multisource-multitarget realm. • Avoid all avoidable abstractions. • As much as possible, replace theorem-proving with “mechanical,” “turn-thecrank,” purely algebraic procedures. • Nevertheless retain all mathematical power necessary for effective practical problem-solving. The following are specific illustrations of the second point. 1.1.1.1

Illustration 1: Avoid Avoidable Concepts

Consider the concepts of “thinning” and “marking.” These are basic to purely mathematical treatments of point process theory. But in multitarget detection and tracking, they appear only in a few, concrete contexts that can be adequately addressed at a practitioner level of complexity. Missed detections and disappearing targets can both be described as forms of thinning; and target identity as a form of marking. But does the imposition of such concepts represent an increase of practically actionable understanding—or of pedantry? 1.1.1.2

Illustration 2: Avoid Abstract Point Process Theory

Finite-set statistics is based on a specific formulation of point process theory—the stochastic-geometry version of the theory of random finite sets (RFSs) [134], [278]. The reason for this is that the stochastic geometry formulation offers the following advantages:

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1. It is more “practitioner friendly” than the random counting measure and other formulations of point processes. A finite set {x1 , ..., xn } is easily visualizable as a point pattern—for example, in the plane or in three dimensions. Similarly, an RFS is easily visualizable as a random point pattern. The following is an everyday example of an RFS regarded as a random point pattern: the stars in a night sky, with many stars winking in and out of visibility, and/or slightly varying in their apparent position ([179], pp. 349-356). 2. It usually permits us to avoid abstractions such as topologies, measurable mappings, and the “randomness” of point processes in the formal mathematical sense. The topology presumed in the stochastic-geometry formulation of RFS theory—the Fell-Matheron topology—permits a major theoretical simplification. Abstract probability measures, defined on “hyperspaces” of finite sets, can be equivalently replaced by belief-mass functions (b.m.f.’s, also known as belief measures) defined on ordinary (that is, nonhyper) spaces. Just as the probability measure pk|k (S) = Pr(Xk|k ∈ S)

(1.1)

completely characterizes the probability law of the random target-state Xk|k , so the b.m.f. βk|k (S) = Pr(Ξk|k ⊆ S) (1.2) completely characterizes the probability law of a random finite set Ξk|k . Formulas for the probability distribution fΞk|k (X) of Ξk|k can be derived from βk|k (S) via set differentiation and “turn-the-crank” differentiation rules. 3. It results in a multitarget mathematical formalism that is nearly identical to the single-target “Statistics 101” formalism with which tracking practitioners are already familiar. 4. It provides a systematic probabilistic foundation for expert systems theory (fuzzy logic, Dempster-Shafer theory, rule-based inference) in addition to multisensor-multitarget estimation and filtering. This permits a systematic and mathematically rigorous unification of these two quite different aspects of information fusion (see Chapter 22).

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Illustration 3: Avoid Abstract Measure Theory

In finite-set statistics density functions are systematically used in place of measures, except when this is not possible. Thus the Dirac delta function is employed even though it produces practitioner-heuristic abbreviations of rigorous measuretheoretic expressions. In particular, in finite-set statistics probability functionals and their functional derivatives are addressed using a constructive, rather than an abstract measuretheoretic, approach. The reasons for this choice, explained in greater detail in Appendix J, are as follows. 1. The purely measure-theoretic treatments preferred by pure mathematicians, such as Moyal [207], are ill-suited for practical purposes because: (a) Measure-theoretic definitions of probability functionals are based on abstract probability measures on abstract hyperspaces—concrete formulas for which are extremely cumbersome and difficult to determine. (b) Measure-theoretic treatments of functional differentiation are nonconstructive. They lead to formulas for abstract measures on product spaces (or, equivalently, multilinear functionals) rather than ordinary density functions. Even if we succeed in proving that these very abstract measures are absolutely continuous, the Radon-Nikod´ym theorem tells us only that their densities exist. It does not provide what is required for practical application: concrete formulas for these densities themselves. 2. By way of contrast, finite-set statistics has, from its inception in 1996, been based on belief-mass functions on (nonhyper) spaces and their set derivatives. In particular: (a) Probability functionals, such as the probability generating functional (p.g.fl.), can be defined in terms of belief-mass functions on ordinary (nonhyper) spaces. (b) A functional derivative is a particular kind of set derivative, and set derivatives are constructive. That is, they produce concrete formulas for density functions—as opposed to abstract formulas for abstract measures on product spaces (equivalently, abstract multilinear functionals).

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1.1.2

Misconceptions About Finite-Set Statistics

As might be expected to occur with any new technical specialty, a few misconceptions about finite-set statistics have arisen and sometimes dogmatically asserted as fact. The purpose of this section is to address the more common of these. • Misconception 1: PHD filters, like all RFS filters, are defined only for Euclidean state and measurement spaces. – RFS filters are based on the stochastic-geometry formulation of random set theory—in particular, on the Fell-Matheron topology on the hyperspace of closed subsets of an underlying space Y. The topology on the subhyperspace of finite sets is the restriction of the Fell-Matheron topology to that subhyperspace. Consequently, a state space or measurement space can be any topological space that is Hausdorff, locally compact, and completely separable [278]. This space is further assumed to be endowed with a measure-theoretic integration concept. – In applications, the underlying space Y commonly has the “hybrid” continuous-discrete form L × S, where S ⊆ RN is some region of a Euclidean space and L is a finite set. • Misconception 2: Because finite sets are order-independent, RFS filters are inherently incapable of constructing time sequences of labeled tracks—and therefore are not true tracking filters. – This misconception is a consequence of the first one. Target states can have the non-Euclidean form x = (ℓ, u),

(1.3)

where ℓ ∈ L is an identifying label unique to each track and where u ∈ S is the kinematic part of the state. Given this, the multitarget Bayes filter—as well as any RFS approximation of it, including PHD and CPHD filters—are in principle inherently capable of maintaining temporally-connected tracks. (See pp. 505-508 of [179].) They are capable of track maintenance also in practice, as much recent research has shown. (The fact that the first such algorithms did not appear until 2004 reflects the fact that implementers did not begin addressing the track labeling issue until then.)

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– In particular and as is explained in detail in Chapter 15, Vo and Vo have devised an exact, closed-form, computationally tractable solution of the multitarget recursive Bayes filter. This results in what appears to be the first provably Bayes-optimal, implementable, multitarget detection and tracking algorithm. The track management approach inherent to this filter is, therefore, also provably Bayes-optimal. • Misconception 3: Target identifiability is lost in RFS models. – This misconception is a second consequence of the first one. An identifying label ℓ can include target-identity information as well as track label information—meaning that target identifiability is not lost in RFS models. • Misconception 4: Because target identifiability is lost in RFS models, PHD and other RFS filters require that motion models and likelihood functions are the same for all targets. – This misconception is a third consequence of the first one. Since singletarget likelihood functions are allowed to have the form fk+1 (z|ℓ, u),

(1.4)

one can specify a different measurement model for each choice of ℓ. Similarly, because single-target Markov densities are allowed to have the form fk+1|k (ℓ, u|ℓ′ , u′ ) = fk+1|k (ℓ|ℓ′ , u′ ) · fk+1|k (u|ℓ, ℓ′ , u′ ),

(1.5)

one can specify a different single-target Markov density fk+1|k (u|ℓ, ℓ′ , u′ )

(1.6)

for each choice of ℓ, ℓ′ . • Misconception 5: The CPHD filter cannot address target-spawning because— unlike the PHD filter—it does not have a target-spawning model. – Consider the following analogy. A jump-Markov motion model is not necessarily required for a single-target Bayes filter to successfully track maneuvering targets. In the same manner, a spawning model is not

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necessarily required for an RFS filter to track spawning targets. The CPHD filter can potentially detect and track spawning targets (though with a time-delay, as the filter accounts for measurements generated by the spawned targets). – Furthermore, the CPHD filter’s model for (nonspawned) target appearance can be used to address spawning targets. – Moreover: a spawning model can actually cause tracking performance to deteriorate. Consider, for example, the staging events of a multistage missile. If the missile type is unknown, these events will occur at unpredictable times. If the spawning model is active at all times, it will almost always mismodel the missile’s actual spawning behavior—since spawning occurs only at one, two, or possibly three isolated instants. Because the spawning model is thereby usually quite inaccurate, a tracking filter with such a model will be forced to “waste” measurements to overcome the ongoing mismatch between reality (no spawning is occurring) and the spawning model (spawning is occurring). • Misconception 6: The previous misconception derives from a deeper misconception—that models are always applicable in all situations. As the target-spawning example illustrates, this is not true: – Detailed statistical models can be counterproductive if inappropriately applied. – Some models are appropriate in some circumstances and not in others. • Misconception 7: Both the time-update Dk|k (x|Z (k) ) → Dk+1|k (x|Z (k) ) and the measurement-update Dk+1|k (x|Z (k) ) → Dk+1|k+1 (x|Z (k+1) ) for the (classical) PHD filter require the following assumptions: for the timeupdate, fk|k (X|Z (k) ) is approximately Poisson; and for the measurementupdate, fk+1|k (X|Z (k) ) is approximately Poisson. – In actuality, only the second assumption is required. The formula for the time-update does not require the Poisson approximation and, in this sense, is exact (given the underlying models and independence assumptions). An exact derivation is possible because of the product and chain rules of the multitarget differential calculus.

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• Misconception 8: The (classical) PHD filter is obsolete, because it has been superseded by better RFS filters such as the (classical) CPHD filter and the CBMeMBer filter. – Practical real-time application usually involves a trade-off between algorithm performance and algorithm computational complexity. While the (classical) CPHD filter has significantly better performance than the (classical) PHD filter, it is also significantly more computationally intensive. Consequently, there will be applications in which the CPHD filter cannot be employed, but the PHD filter can. Likewise, the CBMeMBer filter has different limitations that sometimes hinder its use. – In any case, the (classical) PHD filter often exhibits surprisingly good performance, especially in dense-clutter situations that tax the capabilities of conventional combinatorial algorithms. • Misconception 9: The RFS model of the multiple target state is an approximation, because the Bayes posterior RFS is not exact, but is an approximation based on the earlier invocations of the PHD approximation used to close the Bayesian recursion. The Bayes posterior RFS is an approximation even before the PHD approximation is invoked. – This misconception appears to be due to an extremely superficial reading of the finite-set statistics literature. In this reading, the entire RFS approach is misunderstood to be synonymous with one particular aspect of the RFS approach: the PHD filter. That is, it is mistakenly believed that the multitarget RFS is always presumed to be Poisson (“PHD approximation”). – In actuality, in the RFS approach a random multitarget state at time tk is mathematically represented as an RFS Ξk|k , and the statistical behavior of Ξk|k is represented by its RFS probability distribution fk|k (X|Z (k) ). This distribution is statistically exact—not approximate. In particular, the RFS approach includes a “multitarget calculus” methodology for explicitly constructing fk|k (X|Z (k) ) from explicitlydefined RFS multitarget motion and measurement models. – For purposes of computation, in the RFS approach multitarget distributions are approximated in various ways. To arrive at the (classical) PHD filter, the predicted multitarget distribution fk+1|k (X|Z (k) ) is

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approximated as a Poisson multitarget distribution: fk+1|k (X|Z (k) ) ∼ = e−Nk+1|k

∏

Dk+1|k (x|Z (k) ).

(1.7)

x∈X

However, this is not the only approximation used in the RFS approach. For example, fk+1|k (X|Z (k) ) can also be approximated as an i.i.d.c. process, a multi-Bernoulli process, or a generalized labeled multi-Bernoulli process—none of which are necessarily Poisson. – The Poisson approximation is the simplest and least accurate RFS approximation of the random multitarget state. It is not a “representation” of it. • Misconception 10: The right model of the multitarget state is that used in the multihypothesis tracker (MHT) paradigm, not the RFS paradigm. – The MHT “representation” of the multitarget state is—no less than the Poisson, i.i.d.c., multi-Bernoulli, or generalized labeled multi-Bernoulli “representations” used in the RFS paradigm—an approximation of fk|k (X|Z (k) ). – Moreover, if the MHT “representation” is the “right” one, then it should be—as is the case with the RFS representation—provably Bayesoptimal. But it is apparently the case that no proof exists showing that MHT is Bayes-optimal—or even approximately Bayes-optimal. – An algorithm is not Bayes-optimal simply because it employs Bayes’ rule in some fashion. The term “Bayes optimal” has a specific mathematical meaning. In the multitarget case, it requires the minimization of the multitarget Bayes risk corresponding to some multitarget cost function (see Section 5.3). – By way of contrast, the approximation of fk|k (X|Z (k) ) as a generalized labeled multi-Bernoulli distribution is provably Bayes-optimal— because it leads to a exact closed-form solution of the multitarget Bayes filter (see Chapter 15). • Misconception 11: The RFS approach is questionable because it is computationally intractable, and requires extreme approximations to make it tractable.

Introduction to the Book

15

– This statement appears to reflect the existence of a double standard. The “ideal” MHT is inherently computationally intractable. Its combinatorics can be “beat down” only by resort to rather extreme approximations—approximations that can severely degrade its performance in heavy-clutter and other scenarios. • Misconception 12: The RFS approach has not “panned out”—that is, RFS algorithms, such as PHD and CPHD filters, have failed to demonstrate significant improvement over more conventional approaches. – Provided that we eschew less magnanimous interpretations of this statement, this appears to be another misconception that is attributable to a superficial reading of the finite-set statistics literature. While this issue will be more fully addressed in Section 1.2, the following examples will suffice here: – Vo, Vo, and Cantoni have reported a single-target RFS filter that significantly outperforms traditional approaches such as the probabilistic data association (PDA) filter [309]. (This RFS filter can also address state-dependent clutter, such as multipath returns.) – In conventional multitarget detection and tracking, CPHD filters can successfully perform in clutter with clutter rates of 70 measurements per frame and higher. In such circumstances, conventional algorithms such as MHTs tend to suffer combinatorial breakdown. This is especially the case when there are newly appearing targets, in which case such targets tend to be overlooked because of measurement-gating. – Another example is the TNC Σ-CPHD filter for superpositional sensors (Chapter 19). It has been shown to significantly outperform conventional Markov Chain Monte Carlo (MCMC) methods, while also being 30 to 87 times faster (depending on the specific application). – Still another example is the IO-MeMBer track-before-detect (TBD) filter for tracking in images (Chapter 20). It has been shown to significantly outperform the previously best TBD algorithm, the histogramPMHT. – As a final example, consider the RFS-SLAM algorithms [210], [208], [1]. In high-clutter environments, these have been shown to significantly outperform conventional simultaneous localization and mapping (SLAM) algorithms such as MH-FastSLAM.

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Advances in Statistical Multisource-Multitarget Information Fusion

1.1.3

The Measurement-to-Track Association Approach

The purpose of this and the following subsections is to provide an overview of the finite-set statistics approach by contrasting it with the ubiquitous conventional approach, measurement-to-track association (MTA). A more complete and precise discussion will be provided in Sections 7.2.2, 7.2.4, and 7.2.5. It should also be pointed out that the following discussion is conceptual. It is not intended to be a description of the internal logic of any particular conventional multitarget tracking algorithm. (For an encyclopedic treatment of conventional tracking methods, see the book by Blackman and Popoli [24].) The most familiar tracking algorithms presume the following measurement model—hereafter referred to as the “standard multitarget measurement model.” A detection process (for example, a threshold) is applied to a sensor signature (for example, a radar scan or an image). Signal-to-noise ratio (SNR) is relatively small, meaning that the number m of measurements will typically outnumber the number n of targets. The result is a set Z = {z1 , ...., zm } of measurements (for example, measured positions). For each zi there are three possibilities: • zi was generated by a target (a “target detection”). • zi was caused by sensor noise (a “false detection”). • zi was generated by some real entity in the background environment that is momentarily target-like but not an actual target (a “clutter detection”). There is also a fourth possibility: • A target was present but did not generate a measurement (a “missed detection”). For mathematical convenience, false and clutter detections are usually grouped together as a single statistical process, which is referred to as “clutter,” and which is usually assumed to be Poisson. For the target-generated detections, the “small-target” model is presumed. That is: • Targets are distant enough (relative to the sensor’s resolution capability) that a single target generates at most a single detection. • They are also near enough that any given detection is generated by at most a single target.

Introduction to the Book

17

Because of the small-target assumption, a bottom-up, “divide and conquer” strategy can be applied to the multitarget detection and tracking problem ([179]. pp. 321-335). Because of this strategy, the multitarget tracking problem can be decomposed into multiple single-target tracking problems. Suppose that, at time tk , a multitarget tracking algorithm has produced n hypothesized targets, called “tracks.” These tracks have the form k|k

k|k

k|k

k|k k|k (ℓ1 , x1 , P1 ), ..., (ℓk|k n , xn , Pn )

where xi is a state-vector (for example, position, velocity), Pi a error covariance k|k matrix (modeling the uncertainty in xi ), and ℓi is a “track label” (to clearly distinguish the tracks from one another as they evolve through time). The Gaussian k|k distribution fi (x) = NP k|k (x − xi ) is called the “track density” or “track i

distribution” or “spatial density” of the ith track. Next, suppose that at time tk+1 the sensor collects m detections Zk+1 = {z1 , ..., zm } with |Zk+1 | = m where, in more difficult tracking scenarios, m > n because of clutter. The prediction step of some single-target filter—most typically, an extended Kalman filter (EKF)—is used to construct predicted tracks k+1|k

(ℓ1

k+1|k

, x1

k+1|k

, P1

), ..., (ℓk+1|k , xnk+1|k , Pnk+1|k ). n

Let I be a (possibly empty) subset of {1, ..., n}. From it, construct the following hypothesis HI,τ about how the predicted tracks are related to the new measurements at time tk+1 : k+1|k

k+1|k

k+1|k

k+1|k

k+1|k

k+1|k

• The tracks (ℓi , xi , Pi ) with i ∈ I generated the detections zτ (i) for some selection τ (i) ∈ {1, ..., m} of indices, so that ZI,τ = {zτ (i) | i ∈ I} is the set of target-generated measurements. • The tracks (ℓi

, xi

, Pi

) with i ∈ / I were not detected.

• The excess measurements Zk+1 − ZI,τ were generated by clutter.1 The hypothesis HI,τ is a measurement-to-track association (MTA) or k+1|k+1 association hypothesis. We end up with a list HI,τ of MTAs, one for each k+1|k+1

I ⊆ {1, ..., n} and τ . For each HI,τ , we can apply the corrector step of the single-target filter—most typically, an EKF—to use the measurements in ZI,τ to 1

The possibility of newly appearing targets has been ignored for the sake of conceptual clarity.

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Advances in Statistical Multisource-Multitarget Information Fusion

construct measurement-updated tracks for i ∈ I: k+1|k+1

(ℓi,τ (i)

k+1|k+1

, xi,τ (i)

k+1|k+1

, Pi,τ (i)

).

This process is repeated indefinitely. Multihypothesis trackers (MHTs) are currently the dominant MTA-based tracking algorithms [245], [23]. 1.1.4

The Random Finite Set (RFS) Approach

In contrast to MTA, finite-set statistics employs a top-down paradigm grounded in a particular version of point process theory—the stochastic-geometry formulation of the theory of random finite sets. The basic ideas are as follows. 1.1.4.1

Multitarget Density Functions k+1|k+1

In the place of the list {HI,τ }I,τ of measurement-updated association hypotheses, the RFS approach employs a multitarget probability density function (m.p.d.f.) fk+1|k+1 (X|Z (k+1) ) defined on the finite-set variable X = {x1 , ..., xn } with n ≥ 0, where Z (k+1) : Z1 , ..., Zk+1 is the time history (sample path) of measurement sets at time tk+1 . The quantity fk+1|k+1 (X|Z (k+1) ) is the probability (density) that the targets have state set X = {x1 , ..., xn } with n ≥ 0, given the measurement history Z (k+1) . Because the number n of targets can be variable, we can have:     

∅ {x1 } X= {x1 , x2 }    ..  .

if if if .. .

no targets are present one target with state x1 is present two targets with states x1 ̸= x2 are present . .. .

(1.8)

The units of measurement of fk+1|k+1 (X|Z (k+1) ) are u−|X| , where u denotes the units of measurement of the single-target state x.

Introduction to the Book

1.1.4.2

19

RFS Measurement Models

In the place of the standard multitarget measurement model, one constructs from this model an RFS measurement model of the form all measurements

measurements, 1st target

measurements, nth target

clutter

Σk+1 = Υk+1 (x1 ) ∪ ... ∪ Υk+1 (xn ) ∪ Ck+1

(1.9)

where Υk+1 (x) is the RFS of the measurement generated by a target with state x; where Ck+1 is the RFS describing the clutter-measurement process; and where Σk+1 is the RFS of all measurements. For the standard multitarget measurement model, Ck+1 is assumed Poisson, and either Υk+1 (x) = ∅ (the target x was not detected) or Υk+1 (x) = {Zk+1 } (the target was detected and the random measurement Zk+1 was collected). For a nonlinear-additive single-target measurement model, for example, if Υk+1 (x) ̸= ∅ then Υk+1 (x) = {ηk+1 (x) + Vk+1 }. (1.10) 1.1.4.3

Multitarget Likelihood Functions

The RFS measurement model is transformed into an equivalent multitarget likelihood function LZ (X) abbr. = fk+1 (Z|X). (1.11) The value LZ (X) is the likelihood that, at time tk+1 , the measurement set Z = {z1 , ..., zm } with m ≥ 0 will be generated, if targets with state set X = {x1 , ..., xn } with n ≥ 0 are present. The transformation of the RFS model to a multitarget likelihood function is accomplished using multiobject differential calculus: [ ] δ fk+1 (Z|X) = βk+1 (T |X) (1.12) δZ T =∅ where βk+1 (T |X) = Pr(Σk+1 ⊆ T |Ξk+1|k = X)

(1.13)

is the belief-mass function (belief measure) of the multitarget measurement model Σk+1 ; and where δ/δZ denotes a set derivative. This belief-mass function is the multiobject analog of the probability-mass function pk+1 (T |x) = Pr(Zk+1 ∈ T |Xk+1|k = x) of a single-target measurement model Zk+1 = ηk+1 (x) + Vk+1 .

(1.14)

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Advances in Statistical Multisource-Multitarget Information Fusion

1.1.4.4

Multitarget Bayes’ Rule

Instead of constructing the list of measurement-updated hypotheses, we instead apply the multitarget analog of Bayes’ rule: fk+1|k+1 (X|Z (k+1) ) = ∫

fk+1 (Z|X) · fk+1|k (X|Z (k) ) . fk+1 (Z|Y ) · fk+1|k (Y |Z (k) )δY

(1.15)

∫ Here, ·δY is a set integral that accounts for the fact that both the elements and the number of elements of the finite-set variable Y are variable. The set integral operation is inverse to the set derivative operation. 1.1.4.5

RFS Multitarget Motion Models

The standard multitarget motion model is an analog of the standard multitarget measurement model. It is based on the following presumptions: • A target may disappear from the scene (this is the motion-model analog of a missed detection). • New targets may appear in the scene (this is the motion-model analog of clutter). • Target motions are independent of each other and of those of newly appearing targets. From this standard model, one constructs an RFS motion model, all targets

transition for target 1

transition for target n′

new targets

Ξk+1|k = Tk+1|k (x′1 ) ∪ ... ∪ Tk+1|k (x′n′ ) ∪ Bk+1|k

(1.16)

where Tk+1|k (x′ ) is the RFS at time tk+1 of a target that had state x′ at time tk ; Bk+1|k is the RFS of all appearing targets; and Ξk+1|k is the RFS of all predicted tracks at time tk+1 . For the standard multitarget motion model, Bk+1|k is assumed Poisson, and either Tk+1|k (x′ ) = ∅ (the target x′ did not persist) or Tk+1|k (x′ ) = {Xk+1|k } (the target persisted and transitioned to Xk+1|k ). For a nonlinear-additive singletarget motion model, for example, if Tk+1|k (x′ ) ̸= ∅ then Tk+1|k (x′ ) = {φk (x′ ) + Wk }.

(1.17)

Introduction to the Book

1.1.4.6

21

Multitarget Markov Densities

The RFS motion model is transformed into a multitarget Markov transition density MX (X ′ ) abbr. = fk+1|k (X|X ′ ). The value MX (X ′ ) is the likelihood that, at time tk+1 , the targets will have state set X = {x1 , ..., xn } with n ≥ 0 if, at time tk , the state set of the targets was X ′ = {x′1 , ..., x′n′ } with n′ ≥ 0. The transformation of an RFS model to a multitarget Markov density is accomplished using, once again, a set derivative: [ ] δ ′ ′ fk+1|k (X|X ) = βk+1|k (S|X ) (1.18) δX S=∅ where βk+1|k (S|X ′ ) = Pr(Ξk+1|k ⊆ S|Ξk|k = X ′ )

(1.19)

is the belief-mass function of the RFS motion model Ξk+1|k . This belief-mass function is the multiobject analog of the probability-mass function pk+1 (S|x′ ) = Pr(Xk+1|k ∈ S|Xk|k = x′ )

(1.20)

of a single-target motion model Xk+1|k = φk (x′ ) + Wk . 1.1.4.7

Multitarget Prediction Integral

Instead of constructing the predicted hypothesis list, in finite-set statistics one instead applies a multitarget analog of the prediction integral: ∫ (k) fk+1|k (X|Z ) = fk+1|k (X|X ′ ) · fk|k (X ′ |Z (k) )δX ′ (1.21) where 1.1.4.8

∫

·X ′ is a set integral. The Multitarget Recursive Bayes Filter

The time-update and measurement-update steps just summarized result in the multitarget recursive Bayes filter: ... →

fk|k (X|Z (k) )

→

fk+1|k (X|Z (k) )

→

fk+1|k+1 (X|Z (k+1) )

→ ...

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Advances in Statistical Multisource-Multitarget Information Fusion

This is the optimal solution to the problem of detecting, tracking, and classifying an unknown number of unknown targets. It is optimal because of the existence of Bayes-optimal multitarget state estimators—for example, the joint multitarget (JoM) estimator ([179], p. 500): JOM Xk+1|k+1 = arg sup X

c|X| · fk+1|k+1 (X|Z (k+1) ). |X|!

(1.22)

Here, c is a constant that has the same units of measurement as the single-target state x; and its magnitude should be the size of the desired localization accuracy ([179], pp. 498-499). 1.1.4.9

Approximate RFS Filters

The multitarget Bayes filter is computationally intractable in most situations of practical interest. Consequently, approximations are necessary. These approximations should be principled, in the sense that they are statistically top-down in form, and thus preserve the essential nature of the original motion and measurements models, as well as the relationships between these models. The type of principled approximation employed in finite-set statistics is that of assuming that the multitarget distributions fk|k (X|Z (k) ) and/or fk+1|k (X|Z (k) ) have some simplified form—one that permits approximate closed-form solution of the multitarget Bayes filter. This is accomplished using probability generating functionals (p.g.fl.’s) and multiobject calculus. The p.g.fl. of a multitarget distribution is defined as ∫ (k) Gk|k [X|Z ] = hX · fk|k (X|Z (k) )δX (1.23) ∫ Gk+1|k [X|Z (k) ] = hX · fk+1|k (X|Z (k) )δX (1.24) where the power functional is defined as hX =

{

∏

1 x∈X h(x)

if if

X=∅ X ̸= ∅

(1.25)

and where h(x) is a “test function” such that 0 ≤ h(x) ≤ 1 identically. Three types of approximation have been extensively investigated in the literature thus far, and are most simply expressed using p.g.fl.’s:

Introduction to the Book

23

1. fk+1|k (X|Z (k) ) and/or fk|k (X|Z (k) ) are the distributions of Poisson RFSs. The various probability hypothesis density (PHD) filters are the result: ... → Dk|k (x|Z (k) ) → Dk+1|k (x|Z (k) ) → Dk+1|k+1 (x|Z (k+1) ) → ... where Dk|k (x|Z (k) ) is a probability hypothesis density (PHD). 2. fk|k (X|Z (k) ) and/or fk+1|k (X|Z (k) ) are the distributions of independent identically distributed cluster (i.i.d.c.) processes. The various cardinalized PHD (CPHD) filters are the result: ... →

pk|k (n|Z (k) )

→

pk+1|k (n|Z (k) )

... →

sk|k (x|Z (k) )

→

sk+1|k (x|Z (k) )

→ ↑↓ →

pk+1|k+1 (n|Z (k+1) )

→ ...

sk+1|k+1 (x|Z (k+1) )

→ ...

where sk|k (x|Z (k) ) is a spatial distribution (that is, the normalized PHD) and where pk|k (n|Z (k) ) is the probability distribution on the number n of targets (also known as the “cardinality distribution”). 3. fk|k (X|Z (k) ) and/or fk+1|k (X|Z (k) ) are the distributions of multiBernoulli processes. The various multi-Bernoulli filters are the result: k|k v

k+1|k vk+1|k }i=1

k|k {qi }i=1 → {qi

... → k|k

v

k+1|k

k|k ... → {si (x)}i=1 → {si

→

v

k+1|k (x)}i=1

→

k+1|k+1 vk+1|k+1 }i=1

{qi

→ ... ↑↓ vk+1|k+1 k+1|k+1 {si (x)}i=1 → ...

k|k

k|k

where s1 (x), ..., sνk|k (x) are the track distributions (spatial densities) of k|k

k|k

νk|k target tracks; and where q1 , ..., qνk|k are the probabilities that these tracks are actual targets. 1.1.4.10

Derivation of Approximate RFS Filters

These filters are derived using the following methodology. It can be shown that the multitarget Bayes filter can be equivalently expressed as a filter on p.g.fl.’s: ... →

Gk|k [h|Z (k) ]

→

Gk+1|k [h|Z (k) ]

→

Gk+1|k+1 [h|Z (k+1) ]

→ ...

Given one of the above approximations, the formulas for Gk+1|k [h|Z (k) ] and Gk+1|k+1 [h|Z (k+1) ] become algebraically simplified. The formulas for items such

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Advances in Statistical Multisource-Multitarget Information Fusion

k+1|k+1

k+1|k+1

as pk+1|k+1 (n|Z (k+1) ), Dk+1|k+1 (x|Z (k+1) ), qi , and si then be derived using multiobject calculus. For example, [ ] δ (k+1) (k+1) Dk+1|k+1 (x|Z )= Gk+1|k+1 [h|Z ] . δx h=1

(x) can

(1.26)

Here, δG/δx is a generalization of a set derivative called a functional derivative. It is defined by δ G[h + ε · δx ] − G[h] G[h] = lim (1.27) ε→0 δx ε where δx denotes the Dirac delta function concentrated at x. (Note that this is a heuristic, intuitive definition since δx is not a valid test function. See Appendix C or [181] for a more rigorous definition.) 1.1.5

Extension to Nontraditional Measurements

In this book, the terminology “traditional measurement” refers to a measurement that is produced by a conventional sensor—whether this measurement be a signature or a detection extracted from a signature. Other forms of information, referred to as “nontraditional,” usually (but not always) involve human mediation. These include: • Attribute—an identifying characteristic of a target. An example is an identifying characteristic of a target inferred by a human operator while examining an image. • Feature—typically, an identifying characteristic extracted from a signature by a digital signal processing (DSP) algorithm. Examples include intuitively understandable features such as “blobs” extracted from images; but also mathematically abstract features such as principal components or wavelet coefficients. • Natural-language statement. These consist of verbal or written texts generated by a human information source. • Inference rule. This is contingent information, in which a consequent statement is held to be true in the event that an antecedent statement is true. Besides its employment of RFS theory to model multiobject systems, finiteset statistics includes: • A general theory of measurements—called “generalized measurements”— based on the concept of a random (but not necessarily finite) set.

Introduction to the Book

25

• Single-target and multitarget Bayes-optimal processing of general measurements via generalized likelihood functions (GLFs), which are employed in Bayes’ rule in the same way as conventional likelihood functions.

1.2

RECENT ADVANCES IN FINITE-SET STATISTICS

As has already been stated, this book is a consequence of the many new developments that have arisen in RFS-based multisource-multitarget information fusion. The purpose of this section is to briefly describe some of these advances. Although most will be described in technical detail in later chapters, in some cases the reader will be directed to other publications. The section is organized as follows: 1. Section 1.2.1: Advances in conventional PHD and CPHD filters: multisensor PHD and CPHD filters, and multiple-model PHD and CPHD filters. 2. Section 1.2.2: PHD multitarget smoothers. 3. Section 1.2.3: PHD and CPHD filters for unknown detection profiles and unknown clutter backgrounds. 4. Section 1.2.4: PHD filters for extended targets, cluster targets, group targets, and unresolved targets. 5. Section 1.2.5: Advances in conventional multi-Bernoulli filters. 6. Section 1.2.6: RFS filters for “raw data” sensors. 7. Section 1.2.7: Theoretical advances: Clark’s general chain rule for functional derivatives; the general PHD filter; and an exact closed-form solution of the multitarget Bayes filter. 8. Section 1.2.8: Fusing nontraditional information sources. 9. Section 1.2.9: Advances in the development of unified processing systems: unified simultaneous localization and mapping (SLAM); unified sensor/platform management; unified multisensor-multitarget tracking and sensor registration; and unified track-to-track fusion.

26

1.2.1

Advances in Statistical Multisource-Multitarget Information Fusion

Advances in Conventional PHD and CPHD Filters

Chapters 10 and 11 report advances in, respectively, multisensor PHD/CPHD filters and jump-Markov PHD/CPHD filters. 1.2.1.1

Principled Approximate Classical Multisensor PHD/CPHD Filters

The classical PHD and CPHD filters are single-sensor filters. The corresponding multisensor filters are computationally intractable in general. The most common approximate approach for implementing multisensor PHD and CPHD filters, the heuristic “iterated corrector” approach, involves repeating the single-sensor corrector formulas, once for each sensor. However, this approach produces different answers, depending on the order of the sensors. In particular, sensors with larger probabilities of detection should be processed first. Section 10.6 reports a significant advance: multisensor PHD and CPHD filters that are principled, computationally tractable, and do not depend on sensor order. 1.2.1.2

Multiple Motion Model PHD and CPHD Filters

The PHD and CPHD filters presume the existence of an a priori single-target motion model, in the form of a single-target Markov density fk+1|k (x|x′ ). Since in general x = (u, c) where u is the kinematic state and c is a target-ID variable, this model can have different motion models for targets of different type: fk+1|k (x|x′ )

= =

fk+1|k (u, c|u′ , c′ ) ′

′

(1.28) ′

′

fk+1|k (c|u , c ) · fk+1|k (u|u , c , c)

(1.29)

where the final equation is a consequence of Bayes’ rule. However, because the motion model fk+1|k (u|u′ , c′ , c) is specified a priori, the filter may fail to adequately track evasive, rapidly maneuvering targets. Later chapters report significant advances, due to several researchers, that extend jump-Markov (multiple motion model) techniques to PHD and CPHD filters (see Chapter 11) and multi-Bernoulli filters (see Section 13.5). 1.2.2

Multitarget Smoothers

The single-target Bayes filter is an online algorithm. That is, it propagates a track density fk|k (x|Z k ) that describes the state x of the target at time tk , given

Introduction to the Book

27

the measurement-stream Z k : z1 , ..., zk up through time tk . If we are willing to employ time-late “batch” processing, it is possible—for purposes such as track reconstruction—to produce more accurate tracking results. This can be done by using the data in the entire time-window Z k of measurements to estimate the target states at time tℓ for all 1 ≤ ℓ ≤ k. An algorithm that computes the track distributions fℓ|k (x|Z k ) for 1 ≤ ℓ ≤ k is called a smoother. Single-target smoothers can be directly extended to multitarget smoothers, but these smoothers will in general be computationally intractable. Chapter 14 reports four significant advances. The first is a forward-backward Bernoulli smoother that provides an optimal approach when at most a single target is present. The second is a principled and computationally tractable PHD smoother. The third is a closed-form Gaussian-mixture implementation of this smoother. The fourth, generated as a byproduct, is what appears to be the first-ever closed-form Gaussian mixture solution to the conventional single-sensor, single-target smoother (Section 14.2.3). The PHD smoother “works,” but its performance turns out to be less satisfactory than one might hope—only about a 30% improvement. However, the PHD smoother may point the way to more effective approaches in the future. 1.2.3

PHD and CPHD Filters for Unknown Backgrounds

The conventional PHD and CPHD filters require two a priori sensing models. First, a state-dependent probability of detection pD (x) that describes the detection profile of the sensor. Second, a model of the clutter process, in the form of an independent, identically distributed cluster (i.i.d.c.) process model. This consists of a clutter spatial distribution ck+1 (z) and a distribution pκk+1 (m) on the number m of clutter measurements. (For the PHD filter, pκk+1 (m) = e−λk+1 λm k+1 /m! is assumed to be Poisson.) In practical application, one or both of these models is typically unknown. Chapters 17 and 18 report significant advances: PHD and CPHD filters that do not require these a priori models. These can be summarized as follows. 1.2.3.1

PHD/CPHD Filters for Unknown Detection Profiles

As is reported in Chapter 17, any RFS filter can be converted to a filter that does not require a priori knowledge of pD (x). The approach is simple: append a new state variable 0 ≤ a ≤ 1 to the single-target state x, which represents the unknown probability of detection. These RFS filters are capable in principle of estimating the probability of detection at any given track. The closed-form implementation

28

Advances in Statistical Multisource-Multitarget Information Fusion

of such filters—using “beta-Gaussian mixtures (BGMs)”—is reported in Section 17.3.2. 1.2.3.2

PHD/CPHD Filters for Unknown Clutter

As is reported in Chapter 18, it is possible to derive PHD and CPHD filters that do not require the i.i.d.c. clutter model ck+1 (z), pκk+1 (m). This is accomplished by assuming a multi-Bernoulli clutter model instead of an i.i.d.c. clutter model. That is, clutter generators are assumed to be target-like, in that they generate single measurements rather than a cluster of measurements. The usual single-target state space X is replaced by a joint target-clutter state space X ⊎ C where C is the space of clutter generators. Mahler, Vo, and Vo have shown that the resulting PHD and CPHD filters can be implemented in exact closed form using BGMs (see Section 18.5.7). Chen Xin, Tharmarasa, Kirubarajan, and Pelletier have shown that they can be implemented in exact closed form using normal-Wishart mixtures (NWMs, see Section 18.5.8). 1.2.4

PHD Filters for Nonpoint Targets

The small-target model, which was discussed in Section 1.1.3, is based on the assumption that targets are neither too near nor too far from the sensor. This model can be violated in two ways. First, if a target is sufficiently near the sensor, it will generate multiple detections rather than a single detection. Such a target is called an extended target. Second, some unknown phenomenon may generate time-evolving measurement-clusters. Such a target is called a cluster target. Third, measurementclusters may be produced by a coordinated system of conventional targets. Such a target is called a group target. Finally, if a group of targets are sufficiently distant then they will generate a single detection rather than multiple detections. Such targets are called unresolved targets. Several advances have been made in the use of RFS filters to address these kinds of targets—especially, for extended and unresolved targets. 1.2.4.1

PHD and CPHD Filters for Extended Targets

As is reported in Chapter 21, it is possible to derive filtering equations for a PHD filter designed to track multiple extended targets. The measurement-update equation for this filter is combinatorial and thus computationally impractical in general. However, recent research by Lundquist, Granstr¨om, and Orguner has

Introduction to the Book

29

shown how certain approximations can be employed to render the filter potentially practical. 1.2.4.2

PHD and CPHD Filters for Unresolved Targets

As is reported in Chapter 21, it is possible to derive filtering equations for a PHD filter designed to track multiple unresolved targets. The basic approach is to represent an unresolved target as a point target cluster—that is, multiple targets that have the same (single-target) state, but with a continuously variable target number. Given this, it is possible to derive filtering equations for an unresolved-target PHD filter. As with the extended-target PHD filter, the measurement-update equation is combinatorial. However, approximations similar to those being used for the extended-target case can potentially be applied here. 1.2.5

Advances in Classical Multi-Bernoulli Filters

Three significant advances in the theory and application multi-Bernoulli filters have been proposed since their introduction in 2007. These are the cardinality balanced multi-Bernoulli (CBMeMBer) filter; multiple-model versions of the CBMeMBer filter; and “background agnostic” versions of the CBMeMBer filter. 1.2.5.1

Cardinality-Balanced Multi-Bernoulli (CBMeMBer) Filters

The first multi-Bernoulli filter, the “multitarget multi-Bernoulli (MeMBer)” filter, was proposed in Chapter 17 of [179]. Because of an ill-conceived linear approximation, it was subsequently shown to exhibit a pronounced bias in the estimate of target number. Vo and Vo corrected this deficiency with their “cardinality balanced” MeMBer (CBMeMBer) filter. This advance is reported in Chapter 13. 1.2.5.2

Multiple Motion Model CBMeMBer Filters

As with PHD and CPHD filters, the CBMeMBer filter employs a single-target motion model, in the form of a single-target Markov density fk+1|k (x|x′ ) that is specified a priori. Dunne and Kirubarajan have devised a jump-Markov version of the CBMeMBer filter that more easily tracks evasive, maneuvering targets. This advance is reported in Section 13.5.

30

1.2.5.3

Advances in Statistical Multisource-Multitarget Information Fusion

CBMeMBer Filters with Unknown Backgrounds

Again like the PHD and CPHD filters, the CBMeMBer filter employs a priori models of the detection profile and the clutter background. Vo and Vo have generalized the CBMeMBer filter so that these models are no longer necessary. This advance is reported in Section 18.7. 1.2.6

RFS Filters for “Raw-Data” Sensors

Until recently, all RFS filters (including the ones just described) have been based on the presumption that measurements are detections. That is, the measurements are points extracted from a sensor signature using some detection technique. However, detection-based approaches inherently discard information, since the extracted detections only approximate the original signatures. This is especially the case when signal-to-noise ratio (SNR) is small. It is therefore desirable to devise PHD and CPHD filters that can be used directly with the original “raw” signature measurement. Such filters are often called track-before-detect (TBD) filters. 1.2.6.1

Exact Multi-Bernoulli Filters for Pixelized Image Data

In modern optical, infrared, and other imaging systems, the measurement is usually an image-signature in the form of a two-dimensional matrix of gray-scale pixels or red-green-blue (RGB) color pixels. As is reported in Chapter 20, Vo and Vo have shown that it is possible to derive exact closed-form multi-Bernoulli filters for the optimal TBD processing of pixelized images. In this approach, targets are assumed to have physical extent, and thus cannot occlude each other. These filters have been applied successfully to real video data, and have been shown to outperform the previously best TBD algorithm for image data, the histogram-PMHT filter. 1.2.6.2

PHD/CPHD Filters for Superpositional Sensors

The most familiar sensors are based on electromagnetic-wave or acoustic-wave signatures. Such sensors are superpositional, in that the received measurements are summations of the (usually complex-valued) signatures generated by each target. Mahler has shown that it is possible to derive exact RFS filters for such signatures (Section 19.2) , but these filters are computationally intractable in general. As is reported in Chapter 19, approximation techniques make it possible to derive computationally tractable RFS filters for superpositional signatures. The

Introduction to the Book

31

Hauschildt technique (Section 19.3) employs a direct approximation of the general approach. A second approach, due to Thouin, Nannuru, and Coates and subsequently generalized by Mahler (Section 19.4), employs Campbell’s theorem. A CPHD filter based on this approximation has been shown to outperform a conventional Markov Chain Monte Carlo (MCMC) algorithm for multitarget tracking in both radio-frequency (RF) tomography data and passive-acoustic data. 1.2.7

Theoretical Advances

Three advances of a more theoretical nature have been achieved. The first two are a general chain rule for functional derivatives, due to D. Clark; and one of its consequences, a general PHD filter based on this chain rule. The third is an exact closed-form solution of the multitarget Bayes filter. 1.2.7.1

Clark’s General Chain Rule for Functional Derivatives

The derivation of approximate RFS filters, as summarized in Section 1.1.4.10, frequently results in expressions of the general form G[T [h]] where G[h] is a functional and where T : h ?→ T [h] is a functional transformation—that is, a function that transforms functions into functions. Consequently, these derivations require the determination of functional derivatives of the form δ G[T [h]]. δX That is, they require a chain rule. Various chain rules were given in [179], but only for special cases. As a result, the derivation of new results typically required a complicated induction proof. Thanks to a powerful general chain rule due to D. Clark, this is no longer required. Intuitively speaking, the previously required case-by-case induction proof is now built into the general chain rule. This advance is reported in Section 3.5.14. 1.2.7.2

The Generalized Classical PHD Filter

An immediate consequence of Clark’s general chain rule is the derivation of the measurement-update formula for a generalization of the classical PHD filter. This filter is general in that both the clutter process and the target measurementgeneration process can be general. This advance is reported in Section 8.2.

32

1.2.7.3

Advances in Statistical Multisource-Multitarget Information Fusion

Exact Closed-Form Solution of the Multitarget Bayes Filter

PHD, CPHD, and multi-Bernoulli filters are based on approximations that attempt to approximate the actual multitarget density functions fk|k (X|Z (k) ) with increasingly greater accuracy. The filtering steps for these filters are approximate (with the exception of the PHD filter and CBMeMBer filter time-updates). Chapter 15 reports a major theoretical advance, due to B.-T. Vo and B.-N. Vo: an exact closed-form solution of the multitarget Bayes filter. This filter is based on the assumption that the targets in a single continuous target-track all share a distinctive identifying track label. Given this, Vo and Vo construct a class of multitarget distributions—“generalized labeled multi-Bernoulli” distributions. They then show that the class of these distributions is algebraically closed with respect to the multitarget prediction integral, and with respect to the multitarget version of Bayes’ rule. One consequence is that this new filter appears to be the first provably Bayes-optimal, implementable multitarget detection and tracking algorithm. A second consequence is that it apparently has the first provably Bayes-optimal trackmanagement scheme. A Gaussian-mixture implementation of this filter is computationally tractable, and resembles track-oriented multihypothesis tracker (track-oriented MHT) algorithms. 1.2.8

Advances in Fusing Nontraditional Measurements

Two advances in the fusion of nontraditional measurements are reported in Chapter 22: the Bayes optimality of the random set “generalized likelihood function” approach in the single-target case (when the measurement function is precisely known); and the Bayes-optimal extension of this approach to both RFS and conventional multitarget filters. 1.2.8.1

Bayes Optimality of the Generalized Likelihood Function (GLF) Approach

GLFs, as introduced in Section 1.1.5, differ from conventional likelihood functions in that they are unitless probabilities rather than probability densities. It has been shown that, despite this difference, GLFs are provably Bayes-optimal in the singletarget case. (This result is true, however, only when the underlying measurement

Introduction to the Book

33

function ηk+1 (x) is precisely known—that is, for “UGA measurements.”) This advance is reported in Section 22.3.4. 1.2.8.2

Extension of the GLF Approach to Multitarget Filtering

It has also been shown that: • The GLF approach can be naturally generalized for use with both RFS and conventional multitarget filters. • The generalization to RFS multitarget filters is Bayes-optimal (given that the measurement function is precisely known)—though the RFS filters themselves may not be Bayes-optimal. The Bayes-optimal generalization of the GLF approach to RFS filters—the PHD filter, CPHD filter, and multi-Bernoulli filter—is reported in Section 22.10. A paper by Bishop and Ristic is particularly interesting in this regard—see [22] and Section 22.10.5.3. The extension of GLF techniques to conventional measurement-to-track association (MTA), and thereby to conventional multitarget filters, is described in Section 22.11. 1.2.9

Advances Toward Fully Unified Systems

Existing information fusion systems are typically patched together in an ad hoc fashion from various subsystems, many of which are themselves based on ad hoc heuristics. The following are some of the information fusion functionalities that must be integrated after being addressed separately: sensor registration; target detection; target tracking; target classification; measurement-to-track fusion; trackto-track fusion; sensor management; platform management; and so on. ad hoc design and integration reduces algorithmic efficiency by introducing unknown errors. Every heuristic, whether in the integration approach or in the individual components, introduces hidden assumptions. Not only does each such assumption potentially introduce a hidden error and/or statistical bias, but the inefficiencies due to such errors and/or biases are typically magnified as they are propagated through a heuristically designed system. A recent book and book chapter, and Chapters 12, 24, 25, and 26, report recent advances towards the goal of constructing fully unified information fusion systems, based on principled RFS statistical reasoning. These advances include: unified simultaneous localization and mapping (SLAM) for robotics; unified sensor

34

Advances in Statistical Multisource-Multitarget Information Fusion

and platform management; unified multisensor-multitarget tracking and sensor registration; and unified multisensor-multitarget track-to-track fusion. 1.2.9.1

Unified Simultaneous Localization and Mapping (RFS-SLAM)

In SLAM, one or more robots are inserted into an unknown environment. They must construct a map of the environment based on detected landmarks, and then situate themselves within this map. The book Random Finite Sets in Robotic Map Building and SLAM, by Mullane, Vo, Adams, and Vo, describes an RFS approach to SLAM—indeed, the first provably Bayes-optimal approach to SLAM [210]. An approximate SLAM algorithm, based on the PHD filter, has been shown to significantly outperform traditional methods such as EKF-SLAM and MH-FastSLAM in high-clutter environments. See also the papers [208], [209], [1]. 1.2.9.2

Unified Sensor and Platform Management

The performance of multitarget detection and tracking algorithms can be greatly improved if allocatable sensor resources (including the platforms that carry them) can be efficiently and adaptively redirected to preferentially collect measurements from under-collected targets. This process is also known as sensor/platform management or Level 4 information fusion. Chapters 24 through 26 report a significant advance in the development of a unified but potentially tractable RFS-based, control-theoretic and informationtheoretic approach to multisensor-multitarget sensor and platform management. However, the use of purely abstract measures of information, such as KullbackLeibler cross-entropy or R´enyi α-divergence, involves a certain amount of risk: one is “flying blind” insofar as practical intuition is concerned. Thus the emphasis of this work is on the development of information-theoretic objective functions that are not only computationally tractable but physically intuitive (rather than purely abstract). Examples of such objective functions are the posterior expected number of targets (PENT), the cardinality variance (that is, the variance of PENT), and the Cauchy-Schwartz divergence. The work regarding the cardinality variance and Cauchy-Schwartz divergence is quite new but promising—see Section 26.6.4.3. 1.2.9.3

Unified Multisensor-Multitarget Tracking and Sensor Registration

Existing RFS algorithms, like essentially all current multitarget tracking algorithms, are based on the presumption that the sensors are temporally and spatially registered

Introduction to the Book

35

with perfect accuracy. In practice, sensor measurements are often contaminated by one or more unknown biases (also known as misregistrations). One example is a terrain map that has an unknown translational offset. Another is drift error in the inertial navigation system (INS) of a sensor-carrying platform. In such circumstances, the performance of multitarget detection and tracking algorithms will typically be seriously degraded unless such sensor biases can be estimated and removed. Conventional approaches attempt to estimate registration errors prior to target detection and tracking. However, recent research has demonstrated that, at least in certain circumstances, sensor registration and multitarget tracking can be accomplished simultaneously. These advances are reported in Chapter 12. 1.2.9.4

Unified Multisensor-Multitarget Track-to-Track Fusion

Conventional information fusion is based on the concept of collecting measurements from sensors in order to detect and track targets. Such an approach is often not viable for distributed information fusion systems. This is because raw measurements are often too large to be effectively transmitted through bandwidth-limited communications channels. This difficulty is typically addressed by transmitting track estimates rather than measurements. However, this approach introduces two new difficulties: temporal correlation and double-counting (spatial correlation). Measurements can usually be assumed to be statistically independent. This is the not the case with tracks which, being the output of some filtering process, are time-correlated. They therefore cannot be fused in the same manner as measurements. The other difficulty arises from the fact that, in ad hoc networks, seemingly independent data arriving at a fusion node may have actually originated with the same information source. Consider a simple network in which node X communicates data D about a target to the nodes A and B, each of which then transmits D to node Y as data D ′ and D ′′ . If Y fuses D ′ and D ′′ presuming that they are independent, the result will be a spuriously accurate localization of the target. This phenomenon is known as double-counting. A unified RFS approach to track-to-track fusion, much of it based on the work of Clark and his associates, has been described in Chapter 8 of the book Distributed Data Fusion for Network-Centric Operations [183].

36

Advances in Statistical Multisource-Multitarget Information Fusion

1.2.9.5

Unified Situation Assessment

The term situation assessment (also known as Levels 2 and 3 information fusion) refers to the process of determining the current and/or predicted levels of threat in a given environment. Section 25.14.2 describes the elements of a unified approach to the problem, based on the concept of a tactical importance function (TIF).

1.3

ORGANIZATION OF THE BOOK

The purpose of this section is to summarize the contents of the book. The emphasis of the book is on the derivation of concrete techniques and formulas that can be applied to concrete problems. Consequently, the introduction to every chapter includes a “Major Lessons Learned” section detailing the most significant concepts, formulas, and results of that chapter. Mathematical proofs and other extended mathematical derivations have been relegated to the appendices, when it has not been possible or desirable to direct the more theoretically engaged reader to other publications. Because of a “transparent” system of notation (see Appendix A.1), the reader will usually be able to infer the meaning of mathematical symbols at a glance. A crawl-walk-run style of exposition is also employed. We will begin with more familiar concepts and techniques and then build upon them to introduce more complex ones. The book is organized as follows and as indicated in Figure 1.2: Part I: Elements of Finite-Set Statistics 1. Chapter 2: Basic concepts of random finite sets. 2. Chapter 3: Basic concepts of multiobject calculus. 3. Chapter 4: Basic concepts of multiobject statistics. 4. Chapter 5: Basic concepts of RFS modeling and multitarget filtering. 5. Chapter 6: Basic concepts of multitarget metrology: multitarget missdistances, and measures of multitarget information. Part II: RFS Filters for Standard Measurement Models 1. Chapter 7: Introduction to Part II.

Introduction to the Book

37

Figure 1.2 The structure of the book at a glance. It is divided into five major parts: Elements of FISST, Standard Measurement Models, Unknown Backgrounds, Nonstandard Measurement Models, and Sensor/Platform Management.

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Advances in Statistical Multisource-Multitarget Information Fusion

2. Chapter 8: The “classical” PHD and CPHD filters; the zero false-alarm (ZFA) CPHD filter; the state-dependent-clutter PHD filter; and the generalized classical PHD filter. 3. Chapter 9: Implementation of the classical PHD and CPHD filters. 4. Chapter 10: Extension of the classical PHD and CPHD filters to multiple sensors. 5. Chapter 11: Jump-Markov versions of the classical PHD and CPHD filters (for rapidly maneuvering, “noncooperative” targets). 6. Chapter 12: Joint sensor registration and multitarget detection and tracking. 7. Chapter 13: Multi-Bernoulli filters and the CBMeMBer filter. 8. Chapter 14: PHD smoothers and the ZTA-CPHD smoother. 9. Chapter 15: The Vo-Vo exact closed-form solution of the multitarget Bayes filter. Part III: RFS Filters for Unknown Backgrounds 1. Chapter 16: Introduction to Part III. 2. Chapter 17: Unknown probability of detection. 3. Chapter 18: Unknown clutter backgrounds. Part IV: RFS Filters for Nonstandard Measurement Models 1. Chapter 19: PHD and CPHD filters for superpositional sensors. 2. Chapter 20: Track-before-detect multi-Bernoulli filters for pixelized images. 3. Chapter 21: PHD filters for extended, cluster, group, and unresolved targets. 4. Chapter 22: The theory and RFS filtering of nontraditional measurements. Part V: RFS Sensor and Platform Management 1. Chapter 23: Introduction to Part V. 2. Chapter 24: Sensor management for single-sensor, single-target systems. 3. Chapter 25: RFS sensor management for multisensor-multitarget systems.

Introduction to the Book

39

4. Chapter 26: Sensor management using approximate RFS filters: Bernoulli, PHD, CPHD, and CBMeMBer. Appendices Various mathematical details and tangential matters have been relegated to the appendices, as follows: • Appendix A: A glossary of notation. • Appendix B: The core Bayesian approach. • Appendix C: Functional derivatives. • Appendix D: The theory of partitions. • Appendix E: Beta distributions. • Appendix F: Markov time-update of beta distributions. • Appendix G: Wishart and normal-Wishart distributions. • Appendix H: Complex-valued Gaussian random variables. • Appendix I: The statistics of level-1 group targets. • Appendix J: Comparing the functional calculi of FISST and Moyal. • Appendix K: Mathematical derivations: (This appendix can be found online. See the citations in the text.)

Part I

Elements of Finite-Set Statistics

Chapter 2 Random Finite Sets 2.1

INTRODUCTION

The purpose of this chapter is to provide a summary of the basic concepts, formulas, and methodologies of finite-set statistics. It begins with an introduction to conventional single-sensor, single-target statistics and ends with a conceptual sketch of multisource-multitarget statistics. The latter has three basic aspects: • A formulation of multisource-multitarget detection, tracking, identification,1 and information fusion in terms of the theory of random finite sets (RFSs). • A general theory of measurements, which encompasses many forms of nontraditional, human-mediated information. • A set of practitioner-oriented mathematical tools, based on multitarget integral and differential calculus, that facilitates problem-solving. 2.1.1

Organization of the Chapter

The chapter is organized as follows: 1. Section 2.2: A review of single-sensor, single-target Bayes statistics. 2. Section 2.3: An introduction to the theory of random finite sets (RFSs). 1

In this book the term “target identification” will be used flexibly. Depending on application, it can refer to determining (1) the broad class or type of a target (for example, jet fighter versus commercial jet); (2) the narrow class or type of a target (for example, F-16 versus F-35); or even (3) a specific identity (for example, an aircraft tail number).

43

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Advances in Statistical Multisource-Multitarget Information Fusion

3. Section 2.4: Multitarget statistics in a nutshell.

2.2

SINGLE-SENSOR, SINGLE-TARGET STATISTICS

For the sake of conceptual and notational clarity, let us begin with a summary of the basic concepts of single-sensor, single-target Bayesian statistics. The following topics are covered: 1. Section 2.2.1: Basic notation. 2. Section 2.2.2: Single-target state spaces and single-sensor measurement spaces. 3. Section 2.2.3: Random vectors, probability-mass functions, and probability density functions. 4. Section 2.2.4: Target motion models and Markov state transition densities. 5. Section 2.2.5: Sensor measurement models and likelihood functions. 6. Section 2.2.6: Generalized measurement models and generalized likelihood functions for nontraditional information sources. 7. Section 2.2.7: The single-sensor, single-target recursive Bayes filter. 2.2.1

Basic Notation

A glossary of notation can be found in Appendix A. In addition, the following notation will be employed throughout the book: • Binomial (also known as combinatorial) coefficient: Cn,i =

{

n! i!·(n−i)!

0

if if

0≤i≤n . otherwise

(2.1)

• Multidimensional Gaussian distribution: Let x be an N -dimensional column vector and C be an N × N covariance matrix. Then this is defined by ( ) 1 1 NC (x) = √ · exp − xT C −1 x . (2.2) 2 det 2πC

Random Finite Sets

45

• Fundamental identity for multidimensional Gaussian distributions: Let x be an N -dimensional column vector, z be an M -dimensional column vector with M ≤ N , P be an N × N covariance matrix, C be an M × M covariance matrix, and H be an M × N matrix. Then NR (z − Hx) · NP (x − x0 ) = NR+HP H T (z − Hx0 ) · NC (x − c) (2.3) where C −1 C −1 c

= =

P −1 + H T R−1 H P −1 x0 + H T R−1 z

(2.4) (2.5)

or, equivalently, where

2.2.2

c C

= =

K

=

x0 + K(z − Hx0 ) (I − KH)P ( )−1 P H T HP H T + R .

(2.6) (2.7) (2.8)

State Spaces and Measurement Spaces

The state of a single-target contains the primary information about the target that we want to know. It is, most commonly, a column vector x = (x1 , ..., xN )T

(2.9)

in some Euclidean space X = RN or in some region S ⊆ RN . Less commonly, it has the hybrid discrete-continuous form x = (c, x1 , ..., xN )T ∈ X = C × S

(2.10)

where C is a finite set of discrete state-variables—most typically, a set of track labels and/or target-identity classes. But, as indicated in Appendix B, in general X can be any Hausdorff, locally compact, and completely separable topological space. For example—and as will be the case in Chapter 18—it could have the form X = RN1 ⊎ RN2 where ‘⊎’ indicates a disjoint union. The measurement generated by a target, and collected by a single-sensor, is the information from the target that is actually observable. It is, most commonly, a column vector z = (z1 , ..., zM )T (2.11)

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Advances in Statistical Multisource-Multitarget Information Fusion

in some Euclidean space Z = RM or in some region T ⊆ RM . Less commonly, it is hybrid discrete-continuous: z = (b, z1 , ..., zM )T ∈ Z = B × T

(2.12)

where B is a finite set of discrete state-variables—most typically, a set of attributes associated with target-identity classes. In general, a measurement space can be any Hausdorff, locally compact, and completely separable topological space. State and measurement spaces are both presumed to come equipped with a measure-theoretic integration concept. For X = RN and Z = RM this is typically the Lebesgue integral. For X = C × S and Z = B × T it is the product discrete-Lebesgue integral: ∫ ∑∫ h(x)dx = f (c, x1 , ..., xN )dx1 · · · dxN (2.13) c∈C

∫

g(z)dz

=

b∈B

2.2.3

S

∑∫

g(b, z1 , ..., zM )dz1 · · · dzM .

(2.14)

T

Random States and Measurements, Probability-Mass Functions, and Probability Densities

A random state X is a random element of the state space X, and a random measurement is a random element Z of the measurement space Z. They both have corresponding probability-mass functions (also known as probability measures), defined as pX (S) = Pr(X ∈ S), pZ (T ) = Pr(Z ∈ T ). (2.15) If pX (S) = 0 whenever S is of measure zero (with respect to the baseline measure), then the Radon-Nikod´ym theorem tells us that ∫ pX (S) = fX (x)dx (2.16) S

where fX (x) is an almost-everywhere unique function called the probability density function (p.d.f.) of X. Similarly, ∫ pZ (T ) = fZ (z)dz. (2.17) T

Random Finite Sets

2.2.4

47

Target Motion Models and Markov Densities

A single-target target motion model has the general form Xk+1|k = φk (x′ , Wk )

(2.18)

where Wk is a random noise vector and x′ is the target state at time tk , and φk is the state transition function. More commonly, it has the additive form Xk+1|k = φk (x′ ) + Wk

(2.19)

where Wk is zero-mean and φk (x) is the deterministic motion model (state transition function). For a linear-Gaussian sensor, Wk is Gaussian and φk (x′ ) = Fk x′ , where Fk is the state transition matrix. The probability-mass function corresponding to Xk+1|k , conditioned on the event Xk|k = x′ , is pk+1|k (S|x′ ) = Pr(Xk+1|k ∈ S|x′ ) =

∫

fk+1|k (x|x′ )dx

(2.20)

S

where Mx (x′ ) = fk+1|k (x|x′ ) is the Markov state transition density function. It is the probability (density) that the target will have state x at time tk+1 , if it had state x′ at time tk . 2.2.5

Measurement Models and Likelihood Functions

A single-sensor, single-target measurement model has the general form Zk+1 = ηk+1 (x, Vk+1 )

(2.21)

where Vk+1 is a random noise vector and ηk+1 is the measurement function. More commonly, it has the additive form Zk+1 = ηk+1 (x) + Vk+1

(2.22)

where Vk+1 is zero-mean and ηk+1 (x) is the deterministic measurement model (measurement function). For a linear-Gaussian sensor, Vk+1 is Gaussian and ηk+1 (x) = Hk+1 x, where Hk+1 is the measurement matrix.

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The probability-mass function corresponding to Zk+1 , conditioned on the event Xk+1|k = x, is pk+1 (T |x) = Pr(Zk+1 ∈ T |x) =

∫

fk+1 (z|x)dz

(2.23)

T

where Lz (x) abbr. = fk+1 (z|x)

(2.24)

is the sensor likelihood function. It is the probability (density) that the measurement z will be collected, given that there is a target with state x in the scene at time tk+1 . 2.2.6

Nontraditional Measurements

“Traditional measurements” are those produced by conventional sensor information sources, such as radars. However, much information originates with intermediary sources such as digital signal processors and human observers. As was indicated in Section 1.1.5, such “nontraditional” information includes attributes, features, natural-language statements, and inference rules. A fundamental feature of finite-set statistics is that information of this sort can, from a rigorous Bayesian point of view, be processed in the same way as traditional information. The basic concepts here are the generalized measurement, the generalized measurement model, and the generalized likelihood function (GLF). Detailed discussion of this aspect of finite-set statistics is deferred until Chapter 22. 2.2.7

The Single-Sensor, Single-Target Bayes Filter

The single-sensor, single-target recursive Bayes filter is the theoretical foundation for single-sensor, single-target tracking and identification ([179], Chapter 2). Given a time sequence Z k : z1 , ..., zk of measurements collected by the sensor through time tk , this filter propagates a posterior distribution fk|k (x|Z k )—commonly

Random Finite Sets

49

known as the “track distribution”—through time:2 ... →

fk|k (x|Z k )

→

fk+1|k (x|Z k )

→

fk+1|k+1 (x|Z k+1 )

It is defined by the time-update and measurement-update equations ∫ fk+1|k (x|Z k ) = fk+1|k (x|x′ ) · fk|k (x′ |Z k )dx′ fk+1|k+1 (x|Z k+1 )

=

fk+1 (zk+1 |x) · fk+1|k (x|Z k ) fk+1 (zk+1 |Z k )

where the second equation is Bayes’ rule and where ∫ k fk+1 (zk+1 |Z ) = fk+1 (zk+1 |x) · fk+1|k (x|Z k )dx

→ ...

(2.25) (2.26)

(2.27)

is the Bayes normalization factor. When motion and measurement models are linear-Gaussian and the initial distribution f0|0 (x|Z 0 ) = f0|0 (x) is linearGaussian, then the Bayes filter reduces to the Kalman filter. Information of interest—target position, velocity, type, and so on—can be extracted from fk|k (x|Z k ) using a Bayes-optimal state estimator, such as the maximum a posteriori (MAP) estimator3 AP k+1 x ˆM ) k+1|k+1 = arg sup fk+1|k+1 (x|Z

(2.28)

x∈X

or the expected a posteriori (EAP) estimator: ∫ x ˆEAP = x · fk+1|k+1 (x|Z k+1 )dx. k+1|k+1 2

(2.29)

Note: In two other common systems of notation, the probability distribution fk+1|k (x|Z k ) is written as fk+1|k (x|Z k ) = f (xk+1 |z1:k ) or as fk+1|k (x|Z (k) ) = fXk+1|k |Z1 ,....,Zk (x|z1 , ..., zk )

3

where Xk+1|k is the predicted-target process at time tk+1 and Zj is the measurement process at time tj . Caution: It is commonly asserted that the MAP estimator is Bayes-optimal. In actuality, when the state space is continuously infinite it is only approximately Bayes-optimal (although it is Bayesoptimal to within an arbitrarily small degree of accuracy). This is due to the fact that the associated cost function is a “finite notch” C(x, y) = 1E (x − y), where E is some arbitrarily small neighborhood of 0.

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RANDOM FINITE SETS (RFSs)

Let Y be an underlying space, such as a state space X or a measurement space Z. Then Y∞ denotes the hyperspace of all finite subsets of Y, the empty set included.4 A random finite set (RFS) is a random variable Ψ on Y∞ .5 In this book, we will deal primarily with two types of RFSs: • Y = X is the space of single-target states, Y∞ = X∞ is the hyperspace of all finite subsets of X, and Ψ = Ξ is a random target state set. • Y = Z is the space of single-target, single-sensor measurements, Y∞ = Z∞ is the hyperspace of all finite subsets of Z, and Ψ = Σ is a random measurement set. Thus a random state set Ξ will have the instantiations X X

= =

∅ {x1 }

X

=

{x1 , x2 } .. .

(no targets are present) (a single target with state x1 is present) (two targets with states x1 ̸= x2 are present)

(2.30) (2.31) (2.32)

Similarly, a random measurement set Σ will have the instantiations Z Z Z

4 5

= = =

∅ (no measurement has been collected) (2.33) {z1 } (a single measurement z1 has been collected) (2.34) {z1 , z2 } (two measurements z1 ̸= z2 have been collected) (2.35) .. .

In the mathematical literature, a “hyperspace” is any space whose points are subsets of another space. Formally, an RFS Ψ is a measurable mapping Ψ : Ω → Y∞ from an underlying probability space Ω to Y∞ . In turn, the definition of a measurable mapping requires us to first define a topology on Y∞ . In finite-set statistics, this topology is the restriction to Y of the Fell-Matheron “hit and miss” topology (defined on the class of all closed subsets of Y0 ). For more details, see [179], Appendix F, pp. 711-716.

Random Finite Sets

2.3.1

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RFSs and Point Processes

As was noted earlier in Section 1.1.1, RFS theory is the mathematically simplest version of point process theory. Kingman’s book on Poisson point processes is a classic introduction [134].6 However, point process theory was introduced fifty years ago by Moyal. He formulated two equivalent versions, one based on random unordered finite sequences ([207], pp. 2-3, 5) and the other—the one that is now dominant among pure mathematicians—based on counting measures ([207], p. 6). ˜∞ It will be instructive to briefly examine Moyal’s first formulation. Let Y 7 be the set of unordered finite sequences θ = [y1 ...yn ] for any n. Then a point ˜ ∞ . If it process is a random variable P whose instantiations are elements of Y is always the case that the y1 , ..., yn are distinct, then the point process is called simple. A finite sequence of distinct, unordered elements is the same thing as a finite set: [y1 ...yn ] = {y1 ...yn }. Therefore, a simple point process is the same thing as an RFS. The statistics of P are described by its probability measure pP (O) = ˜ ∞ . A basic result of Pr(P ∈ O), where O is a measurable subset of Y point process theory is the following: the probability density function fP (θ) corresponding to pP (O) exists (that is, is finite-valued and integrable) if and only if P is simple—that is, if it is an RFS ([55], p. 138, Prop. 5.4.V). In this case the functions n! · jn,P (x1 , ..., xn ) = fP ([y1 ...yn ]) = fP ({y1 , ..., yn })

(2.36)

are known as “Janossy densities”—or, as described in Section 4.2.2, as the multiobject probability distribution of the RFS P. The intuitive meaning of Prop. 5.4.V of [55] is this: if there are repeated elements in [y1 ...yn ] (i..e, if yi = yj for some i ̸= j) then fP ([y1 ...yn ]) = ∞ which cannot be used in formulas of practical interest. Stated differently: • To be of practical interest, a point process must be an RFS. 6 7

More precisely, RFSs in Kingman are local RFSs—that is, their intersection with any bounded closed set is finite. Caution: Contrary to modern notation, Moyal used the notation {y1 , ..., yn } to denote an ˜ ∞ = ⊎n≥0 (Yn /Rn ) unordered finite sequence rather than a finite set. Technically speaking, Y where (Yn /Rn ) denotes the space of equivalence classes of Yn with respect to to equivalence Rn

relation (y1 , ..., yn ) ≡ (y1 , ..., yn ) if and only if (y1 , ..., yn ) = (yπ1 , ..., yπn ) for some permutation π on 1, ..., n.

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As was argued in [179], Appendix E.3, pp. 708-710, for the purposes of information fusion and multitarget detection and tracking, the non-RFS formulations of point process theory: • Unnecessarily increase notational and theoretical complexity. • Add no new substance, especially from a practitioner’s point of view. • Lose the simple and geometrically intuitive tools of ordinary set theory. • Are the same thing as RFS theory, the minute that one applies them to practical problems. Despite this fact, the increasing notoriety of finite-set statistics has inspired the creation of supposed “point process” alternatives to RFS-based information fusion. As a typical example, the authors of [281] cite Kingman as an authority on Poisson point processes.8 Rather than adopting Kingman’s RFS formulation, however, they define an instantiation of a point process on Y as follows. Rather than a finite set {y1 , ..., yn }, it is an (n + 1)-tuple ξ = (n, y1 , ..., yn ) for any n ≥ 0.9 In this case the probability distribution f (ξ) of ξ must have the form f (n, y1 , ..., yn ) with f (n, y1 , ..., yn ) = f (n, yπ1 , ..., yπn ) for any permutation π on 1, ..., n. To be of practical use, f (ξ) must be finite-valued—in which case the point process must be an RFS. Thus: what in [281] is called a “point process” is actually an RFS with different notation and terminology. Another difficulty with all such “point process” alternatives is that they are technically vacuous. They imitate a single insight from finite-set statistics, while stripping away the mathematical tools and systematic methodologies that give it its problem-solving power. The most visible of these tools, all introduced in 2001 [168], are the probability generating functional (p.g.fl.), the functional derivative, and their employment to derive PHD and CPHD filters. The vacuousness of the original “point process” imitations appears to have inspired the discovery—a full decade after the fact—of a supposed alternative “point process” formulation of this aspect of finite-set statistics [279], [282]. The following question must be raised every time that a newly-minted “point process” alternative arrives on the scene: notation and terminology aside, is there any actual difference between it and finite-set statistics? As will be shown in Appendix J: 8 9

To wit: “For further background on PPP’s from a multidimensional perspective, see Kingman [6]” [281], first paragraph of Section 2. See [281], discussion following Eq. (1).

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• Except for changes in notation and terminology, the “point process” formulation in [279], [282] is identical to a truncated version of finite-set statistics. • Its treatment of functional derivatives is mathematically erroneous in two different respects—but is identical to the engineering-heuristic version of the functional derivative of finite-set statistics. As was pointed out in [198] and the beginning of Section 1.1.1, the mere fact that multitarget tracking can be formulated in terms of RFSs—or any other mathematical formalism—is of little practical interest in and of itself. A mere change of notation or terminology does not add new substance. In particular: • “Point process” or “PP” means the same thing as “RFS.” • “Poisson point process” or “PPP” means the same thing as “Poisson RFS.” • “Superposition of point processes” means the same thing as “set-theoretic union of RFSs.” • “Intensity” or “intensity function” means the same thing as “first-moment density” or “probability hypothesis density” or “PHD.” • “Intensity filters,” in the general sense of the term, means the same thing as “PHD filters.” • “Cardinal number density”10 is a rather transparent renaming of “cardinality distribution” (as well as being incorrect terminology, since the term “density” is reserved for continuously infinite spaces). ”Canonical number distribution” is yet another transparent renaming. There are many more such examples. It is the tools of finite-set statistics— as summarized in this and the following chapters—that render the RFS formalism useful for information fusion from a practical point of view. 2.3.2

Examples of RFSs

Many specific examples of RFSs are presented in Section 11.2, pp. 348-356, of [179]. Here are a few simple instances. • Random singleton: Ψ = {Y} 10 [282], p. 45, paragraph following Eq. (14).

(2.37)

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where Y is a random element of Y. • Union of random singletons: Ψ = {Y1 , ..., Yn }

(2.38)

where n > 1 is a fixed integer and Y1 , ..., Yn are random elements of Y. If Y1 , ..., Yn are independent and Y is continuously infinite, then |Ψ| = n with probability 1. • “Twinkling” random singleton: Ψ = {Y} ∩ ∅q

(2.39)

where ∅q is the discrete random subset of Y defined by Pr(∅q = T ) =

 

q 1−q  0

if if if

T =Y T =∅ . otherwise

(2.40)

That is, Ψ ̸= ∅ with probability q; and in this case it is a singleton. As an example, the RFS Ψ could be employed as a simple model of a twinkling star in the night sky. 2.3.3

Algebraic Properties of RFSs

• Union: Suppose that Ψ1 , ..., Ψn ⊆ Y are RFSs. Then their set-theoretic union—also called “superposition”—is an RFS: Ψ = Ψ1 ∪ ... ∪ Ψn .

(2.41)

• Intersection: Suppose that Ψ1 , Ψ2 ⊆ Y are RFSs. Then their set-theoretic intersection Ψ = Ψ1 ∩ Ψ2 (2.42) is an RFS. If Ψ1 , Ψ2 are independent and Y is continuously infinite, then they are almost always disjoint: Pr( Ψ1 ∩ Ψ2 = ∅) = 1.

(2.43)

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55

As a special case, let Ψ ⊆ Y be an RFS and y ∈ Y. If {y} is regarded as a constant RFS, then Ψ, {y} are independent and so Pr(y ∈ Ψ) = Pr({y} ∩ Ψ ̸= ∅) = 0.

(2.44)

That is, y ∈ Ψ is a zero-probability event. As we shall see, the PHD DΨ (y) of Ψ provides a mathematically rigorous description of the event y ∈ Ψ, in the same way that a conventional probability density function fY (y) provides a rigorous description of the zero-probability event Y = y. More generally, let Θ ⊆ Y be any random closed subset and Ψ ⊆ Y an RFS. Then Ψ′ = Ψ ∩ Θ (2.45) is an RFS. In particular, let Θ = T be a constant subset. Then Ψ′ = Ψ ∩ T

(2.46)

is an RFS, called a censored RFS. It will be further considered in Section 4.4.1, and will play an important role throughout the book.

2.4

MULTIOBJECT STATISTICS IN A NUTSHELL

Finite-set statistics draws on a general, Bayesian formulation of the dynamic state space model. This general formulation is described in detail in Appendix B. The finite-set statistics special case of this formulation consists of the following basic elements: 1. A practitioner-oriented multisource-multitarget integral and differential calculus ([179], Chapter 11). Just as “turn-the-crank” rules exist for ordinary differential and integral calculus, so similar “turn-the-crank” rules exist for the multitarget differential and integral calculus—specifically, for the set integral, set derivative, and functional derivative. 2. Formal statistical multisouce-multitarget measurement models ([179], Chapter 12). Just as single-sensor, single-target data can be modeled using a measurement model Zk+1 = ηk+1 (x, Vk+1 ), (2.47) so multitarget, multisensor data can be modeled using a multisensor and multitarget measurement model—a randomly varying finite set Σk+1 = Υk+1 (X) ∪ Ck+1 (X)

(2.48)

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of measurements, where Υk+1 (X) and Ck+1 (X) are RFSs of targetgenerated and (possibly state-dependent) clutter measurements, respectively. 3. “True” multisource-multitarget likelihood functions, derived via multitarget calculus from the measurement models ([179], Chapter 12). Just as the true single-sensor, single-target likelihood function fk+1 (z|x) can be derived from the probability mass function pk+1 (T |x) = Pr(Zk+1 ∈ T |Xk+1|k = x)

(2.49)

of the measurement model Σk+1 via differentiation, so the true multitarget likelihood function fk (Z|Xk ) can be derived from the belief-mass function (also known as belief measure) βk+1 (T |X) = Pr(Σk+1 ⊆ T |Ξk+1|k = X)

(2.50)

of the multisensor-multitarget measurement model via the set derivative. Here, “true” means that fk+1 (z|x) contains exactly the same information as the measurement model Zk+1 = ηk+1 (x, Vk+1 ) That is, no modeled information has been left out or replaced with heuristics; and no information extraneous to the model has inadvertently been inserted. 4. Formal statistical multitarget motion models ([179], Chapter 13). Just as single-target motion can be modeled using a motion model Xk+1|k = φk (x, Wk ),

(2.51)

so the motion of multitarget systems can be modeled using a multitarget motion model—a randomly varying finite set Ξk+1|k = Tk (X) ∪ Bk (X)

(2.52)

of predicted targets, where Tk (X) and Bk (X) are RFSs of persisting and newly appearing targets, respectively. 5. “True” multitarget Markov transition densities, derived via multitarget calculus from the motion models ([179], Chapter 14). Just as the true Markov transition density fk+1|k (x|x′ ) can be derived from the probability mass function pk+1|k (S|x′ ) = Pr(Xk+1|k ∈ S|Xk|k = x′ ) (2.53)

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of the motion model via differentiation, so the true multitarget Markov transition density fk+1|k (X|X ′ ) can be derived from the belief-mass function βk+1|k (S|X ′ ) = Pr(Ξk+1|k ⊆ S|Ξk|k = X ′ ),

(2.54)

via set differentiation. Here, “true” means that fk+1|k (x|x′ ) contains exactly the same information as the motion model Xk+1|k = φk (x, Wk ). No modeled information has been left out or replaced with heuristics; and no information extraneous to the model has inadvertently been inserted. 6. Principled statistical approximation ([179], Chapters 16, 17). Just as the Kalman filter can be derived as a principled approximation of the singlesensor, single-target Bayes filter, so various multitarget filters—for example, PHD, CPHD, and multi-Bernoulli filters—can be derived as principled approximations of the multisensor-multitarget Bayes filter. Here, “principled” means that the single-target distributions fk|k (x|Z k ) or the multitarget distributions fk|k (X|Z (k) ) are assumed to have statistically approximate forms that result in closed-form formulas. Examples of such approximations include linear-Gaussian in the case of fk|k (x|Z k ) and Poisson in the case of fk|k (X|Z (k) ). 7. Exact derivation of approximate filters using multiobject differential calculus. Using the multisensor-multitarget measurement model and the multitarget motion model, the recursive Bayes filter is reformulated in terms of probability generating functionals (p.g.fl.’s). Then functional derivatives are used to derive the approximate multitarget filters from this p.g.fl. form of the Bayes filter. Additional aspects of finite-set statistics will be addressed in Chapter 6 (multiobject metrology) and Chapter 22 (nontraditional measurements): 1. Multiobject miss distances ([179], pp. 510-512). The optimal subpattern assignment (OSPA) metric (Section 6.2.2) and its generalizations permit the measurement of distances between finite sets Y = {y1 , ..., yn } of points. 2. Multiobject information-theoretic functionals ([179], pp. 513-513). Multiobject generalizations of the Csisz´ar family of information-theoretic divergences permit the comparison of one multiobject probability distribution f1 (Y ) with another multiobject probability distribution f0 (Y ). 3. Formal statistical models for nontraditional measurements ([179], Chapter 4). Just as conventional measurements are modeled as random vectors

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Zk+1 ∈ Z of a measurement space Z, so nontraditional measurements— attributes, features, natural-language statements, and inference rules—are modeled as “generalized measurements”—that is, random closed subsets Θk+1 ⊆ Z of a measurement space Z. 4. Formal measurement-generation models for nontraditional measurements ([179], Chapters 5 and 6). Just as the generation of single-sensor, singletarget data can be modeled using a measurement model ηk+1 (x) + Vk+1 = Zk+1 ,

(2.55)

so the generation of nontraditional measurements can be modeled using generalized measurement models. These have the form ηk+1 (x) + Vk+1 ∈ Θk+1

(2.56)

if the measurement function ηk+1 (x) is known precisely; and Θk+1 ∩ Ξx,k+1 ̸= ∅

(2.57)

if otherwise, where a target-related RFS Ξx,k+1 replaces ηk+1 (x). 5. Likelihood functions for nontraditional measurements ([179], Chapters 5 and 6). Just as the measurement models for conventional measurements can be transformed into likelihood functions fk+1 (z|x), so the measurement models for nontraditional measurements can be transformed into “generalized likelihood functions”: ρk+1 (Θ|x) = Pr(ηk+1 (x) + Vk+1 ∈ Θ)

(2.58)

ρk+1 (Θ|x) = Pr(Θ ∩ Ξx ̸= ∅).

(2.59)

or, alternatively, Generalized likelihood functions are not conventional likelihood functions. This is, for example, because ρk+1 (Θ|x) is a probability rather than a probability density. Nevertheless, it has been shown that they are mathematically rigorous from a strict Bayesian point of view (see [191] and Section 22.3.4).

Chapter 3 Multiobject Calculus 3.1

INTRODUCTION

The multiobject (also known as multitarget) integro-differential calculus, as summarized in this chapter, is the core “mathematical machine” of finite-set statistics. This calculus and its associated concepts—random finite sets, belief-mass functions, multiobject probability density functions, and probability generating functionals— permit: • The mathematically rigorous derivation of multiobject Markov density functions from mathematically rigorous RFS multitarget motion models. • The mathematically rigorous derivation of multisensor-multitarget likelihood functions from mathematically rigorous RFS multisensor-multitarget measurement models. • The mathematically rigorous derivation of approximate multisensor and multitarget detection, tracking, and identification algorithms such as PHD filters, CPHD filters, and multi-Bernoulli filters. The remainder of the chapter is organized as follows: 1. Section 3.2: Basic concepts—set functions, functionals, functional transformations and multiobject density functions. 2. Section 3.3: Set integrals—the fundamental integration concept on hyperspaces of finite sets.

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3. Section 3.4: Multiobject differential calculus—gradient derivatives, functional derivatives, and set derivatives. 4. Section 3.5: Key formulas of multiobject calculus—the fundamental theorem of multiobject calculus; change of variables formula for set integrals; set integrals on joint spaces; and various “turn-the-crank” rules for functional derivatives (constant rule, sum rule, linear rule, monomial rule, power rule, product rules, and the general chain rule of Clark and its special cases). (More complex identities will be described in Chapter 4.)

3.2

BASIC CONCEPTS

The purpose of this section is to introduce several basic concepts: set functions, functionals, functional transformations, and multiobject density functions. 3.2.1

Set Functions

A set function is a real-valued function ϕ(T ) defined on the measurable subsets T of Y. Some simple examples: • Probability-mass function (also known as probability measure): pY (T ) = Pr(Y ∈ T )

(3.1)

where Y is a random element of Y. • Possibility measure: µg (T ) = sup g(y)

(3.2)

y∈T

where g(y) is a fuzzy membership function on Y. • Belief-mass function or belief measure (see Section 4.2.1): βΨ (T ) = Pr(Ψ ⊆ T )

(3.3)

where Ψ ⊆ Y is an RFS. 3.2.2

Functionals

A functional on Y is a real-valued (and typically unitless) function F [h] in the variable h(y), where h(y) is an ordinary real-valued (and typically unitless)

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function on y ∈ Y.1 The following are examples of functionals that will be important in what follows. • The linear functional induced by a fixed density function f (y) on Y, defined by ∫ s[h] =

h(y) · f (y)dy.

• The power functional. Let Y ⊆ Y be a finite set. Then { if Y = ∅ ∏ 1 hY = . y∈Y h(y) if Y ̸= ∅

(3.4)

(3.5)

The following identity involving the power functional—a generalization of the binomial theorem—is sometimes useful:2 ∑ (h + h0 )X = hW · hX−W (3.6) 0 W ⊆X

or, equivalently,

∑

X

hW = (1 + h) .

(3.7)

W ⊆X

3.2.3

Functional Transformations

A functional transformation is similar to a functional, except that its value (and not just its argument) is a function. Thus a functional transformation T : h ?→ T [h] transforms the function h(y) in the variable y ∈ Y into another function T [h](w) in the variable w ∈ W. A simple example is the transformation T [h] = 1 − γ + γ · h

(3.8)

T [h](y) = 1 − γ(y) + γ(y) · h(y),

(3.9)

defined pointwise by

for all y ∈ Y and where 0 ≤ γ(y) ≤ 1 is a unitless function. 1

2

The bracket notation ‘[·]’ has been borrowed from quantum physics [264], which in turn borrowed it from Volterra [317]. Its purpose is to clearly distinguish functionals F [h] from conventional functions F (y). This equation is a consequence of the fact that the superposition of two Poisson RFSs with PHDs D1 (x), D2 (x) is a Poisson RFS with PHD D1 (x) + D2 (x).

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3.2.4

Multiobject Density Functions

A real-valued function f (Y ) of the finite-set variable Y ⊆ Y is a multiobject density function if, for each Y , the units of measurement of f (Y ) are u−|Y | , where u denotes the units of y. Thus if Y = R and the units of R are meters (u = m), • f (∅) is unitless. • The units of f ({y}) are m−1 . • The units of f ({y1 , y2 }) with y1 ̸= y2 are m−2 . • Similarly for larger values of |Y |. Multiobject density functions can be written in vector notation:3 { 1 i! · f ({y1 , ..., yi }) if |{y1 , ..., yi }| = i . fi (y1 , ..., yi ) = 0 if otherwise

(3.10)

That is, fi (y1 , ..., yi ) vanishes if any two of the y1 , ..., yi are equal.

3.3

SET INTEGRALS

Let f (Y ) be a multiobject density function (as defined in Section 3.2.1). Then its set integral is defined as: ∫

f (Y )δY

=

= def.

=

∫ ∞ ∑ 1 f ({y1 , ..., yi })dy1 · · · dyi i! i=0 ∫ ∞ ∑ 1 f (∅) + f ({y1 , ..., yi })dy1 · · · dyi i! i=1 ∞ ∫ ∑ f (∅) + fi (y1 , ..., yi )dy1 · · · dyi

(3.11)

(3.12)

(3.13)

i=1

3

Equation (3.10) involves more than just a change of notation. The finite-set variable Y , as an instantiation of an RFS Ψ, is defined in terms of the Fell-Matheron topology. However, for each i, (y1 , ..., yi ) is defined in terms of the product topology on Y × ... × Y (Cartesian product taken i times). It is necessary to verify that the two notations are topologically consistent—see [94], p. 133, Proposition 2.

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where fi (y1 , ..., yi ) was defined in (3.10). Note that by the definition of a multiobject density function, the products fi (y1 , ..., yi )dy1 · · · dyi are unitless for every i ≥ 0. Thus the set integral is mathematically well defined. Let T ⊆ Y be a measurable subset. Then the set integral concentrated in T is4 ∫ ∫ f (Y )δY = 1YT · f (Y )δY (3.14) T

=

∫ ∞ ∑ 1 f (∅) + f ({y1 , ..., yi })dy1 · · · dyi i! T × ... × T i=1 ? ?? ?

(3.15)

i times

where the power functional hY was defined in (3.5). Like conventional integrals, the set integral is linear in f : ∫ ∫ ∫ (a1 f1 (Y ) + a2 f2 (Y )) δY = a1 f1 (Y )δY + a2 f2 (Y )δY. T

T

(3.16)

T

Unlike conventional integrals, it is usually not additive in T . That is, if T1 ∩T2 = ∅ then it is usually the case that ∫ ∫ ∫ f (Y )δY ̸= f (Y )δY + f (Y )δY. (3.17) T1 ∪T2

T1

T2

Remark 1 (Caution: Units of measurement) When forming a set integral, care should be taken with respect to units of measurement in the integrand. For example, the following multiobject analog of the L2 distance, √∫ 2

∥f1 − f2 ∥2 =

(f1 (Y ) − f2 (Y )) δY ,

(3.18)

is not well defined because of incompatibility of units in the set integral. 4

As is more fully explained in [179], Appendix F.3, pp. 714-715, set integrals are not, strictly speaking, measure-theoretic integrals. It is possible to define them as measure-theoretic integrals with respect to a certain extension of the measure on Y to a measure on Y∞ . However, the definition of this measure requires the introduction of an arbitrary constant c. Conventional practice in point process theory is to set c = 1 · u, where u denotes the units of measurement in Y. This measure-theoretic definition of the set integral is potentially problematic for multitarget state estimation, however. As is explained more fully in [179], pp. 499-500, the size of c should be approximately equal to the accuracy to which any single-target state y is to be estimated.

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As an example, suppose that    

(1 − q)2 q (1 − q) · (f1 (y) + f2 (y)) f (Y ) = 2 q · (f (y1 ) · f2 (y2 ) + f1 (y2 ) · f2 (y1 ))  1   0

if if if if

Y =∅ Y = {y} Y = {y1 , y2 }, |Y | = 2 |Y | > 2 (3.19) where f1 (y), f2 (y) are probability density functions on Y and where 0 < q ≤ 1. Then ∫ f (Y )δY (3.20) ∫ ∫ 1 1 = f (∅) + f ({y})dy + f ({y1 , y2 })dy1 dy2 1! 2! ∫ = (1 − q)2 + q (1 − q) (f1 (y) + f2 (y)) dy (3.21) ∫ q2 + (f1 (y1 ) · f2 (y2 ) + f1 (y2 ) · f2 (y1 )) dy1 dy2 2 = (1 − q)2 + 2q (1 − q) + q 2 (3.22) =

3.4

1.

(3.23)

MULTIOBJECT DIFFERENTIAL CALCULUS

The purpose of this section is to introduce the basic concepts of multiobject differential calculus: 1. Section 3.4.1: Gˆateaux directional derivatives—the theoretical basis of multiobject differential calculus. 2. Section 3.4.2: Volterra functional derivatives—important for the derivation of approximate multisensor-multitarget filters such as the PHD filter and CPHD filter. 3. Section 3.4.3: Set derivatives—important for the derivation of multitarget Markov transition densities and multisensor-multitarget likelihood functions.

Multiobject Calculus

3.4.1

65

Gˆateaux Directional Derivatives

Let F [h] be a functional defined on unitless real-valued functions h(y) with argument y ∈ Y. Let g(y) be another unitless real-valued function with argument y ∈ Y. Then, if it exists, the Gˆateaux directional derivative of F [h] in the direction of g satisfies the following two properties ([179], Appendix C, pp. 695697): 1. Its value is given by the differential quotient ∂F F [h + ε · g] − F [h] [h] = lim . ε→0 ∂g ε

(3.24)

2. For each fixed h, the function in g defined by g ?−→

∂G [h] ∂g

(3.25)

exists for all g is both linear and continuous.5 Note that if we instead allow g(y) to be a density function, then the units of ε must be the same as the units of y—so that ε · g is unitless. It follows that, in this case, (∂F/∂g)[h] has the units of a density function. The iterated Gˆateaux derivative is defined inductively as ∂nF ∂ n+1 F ∂ n+1 [h] = [h]. ∂gn+1 ∂gn · · · ∂g1 ∂gn+1 δgn · · · ∂g1 5

(3.26)

If one does not insist on linearity or continuity, (3.24) defines a “Gˆateaux differential.” Mathematicians prefer a special case of the Gˆateaux derivative called the Frech´et derivative (also known as gradient derivative), which will not concern us in this book.

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3.4.2

Volterra Functional Derivatives

The Volterra functional derivative of F [h] with respect to a finite subset Y ⊆ Y is defined to be ([179], Section 11.4.1): 6 δF [h] = F [h] δY

(3.27)

if Y = ∅ and, if Y = {y1 , ..., yn } ⊆ Y with |Y | = n > 0,7 δF ∂nF [h] = [h] δY ∂δy1 · · · ∂δyn

(3.28)

where δy (w) denotes the Dirac delta function concentrated at y. In particular, when Y = {y},8 δF δF F [h + εδy ] − F [h] [h] abbr. = [h] = lim . ε→0 δy δ{y} ε

(3.29)

Note that δy is not an actual function and therefore the expression F [δy ] is not mathematically well defined. Consequently, (3.28) and (3.29) are intuitive practitioner heuristics rather than mathematically rigorous definitions. Rigorous definitions can be found in Appendix C. In this case y ?→(δF/δy)[h] turns out to be the density function corresponding to the probability measure defined by 6

7

8

The concept of a functional derivative in the sense of (3.28) and (3.29) goes back to Volterra in 1927 ([317], p. 24, Eq. (3)), where the left side of (3.29) is notated as F ′ [h(x); y]. It should be distinguished from a more casual and careless usage of the term “functional derivative” —that is, application of the Frech´et derivative, Gˆateaux derivative, or Gˆateaux differential to a functional, as in (3.24). Both the notation and the definitions in (3.28) and (3.29) are borrowed from quantum physics (see [264]; [79], p. 406, Eq. (A.15); or [179], Remark 14, p. 376, and Remark 16, p. 382). As is explained more fully in [179], Appendix F.4, pp. 715-716, general functional derivatives δF/δY are not quite Radon-Nikod´ym derivatives of some measure, despite the fact that first-order functional derivatives can be defined in terms of Radon-Nikod´ym derivatives. The Volterra functional derivative (δF/δy)[h] of a functional F [h] is analogous to the partial derivative (∂f /∂xi )(x) of a function f (x) of a Euclidean vector variable x, with y being a continuous analog of the finite-valued index i. That is, (3.29) is analogous to ∂f f (x + εˆ ei ) − f (x) (x) = lim ε→0 ∂xi ε where ˆ e1 , ..., ˆ eN is an orthonormal basis. For this reason, the notation (δF/δh(y))[h] is often employed. The notation (δF/δy)[h] is an abbreviation of this.

Multiobject Calculus

T ?→ (∂F/∂1T )[h]. That is, the defining equation for (δF/δy)[h] is ∫ ∂F δF [h] = [h]dy. ∂1T T δy

67

(3.30)

In other words, the first-order functional derivative δF/δy for all y is defined as the Radon-Nikod´ym derivative of the probability measure ∂F/1T . This definition is problematic from a practical point of view, since it asserts only that the first-order functional derivatives exist. It does not tell us how to actually construct them. It turns out that functional derivatives can be constructively defined in terms of set derivatives—see Appendix C.3 and Remark 2 in Section 3.4.3. Just as the functional derivative can be regarded as a special kind of Gˆateaux derivative, the Gˆateaux derivative can be expressed in terms of functional derivatives: ∫ ∂F δF [h] = g(y) · [h]dy. (3.31) ∂g δy This equation, which generalizes (3.30), is proved in Appendix C. Several examples of functional derivatives are presented in pp. 377-380 of [179]. For the sake of illustration, only one is presented here. Let ∫ f [h] = h(y) · f (y)dy (3.32) be the linear functional defined in (3.4). Then δ f [h] δy

= = =

3.4.3

f [h + εδy ] − f [h] ε→0 ε ∫ ε · f [δy ] lim = δy (w) · f (w)dw ε→0 ε f (y). lim

(3.33) (3.34) (3.35)

Set Derivatives

Let F [h] be a functional defined on unitless functions h(y) on y ∈ Y with 0 ≤ h(x) ≤ 1. Define the set function ϕF (T ) for closed T ⊆ Y by ϕF (T ) = F [1T ]. Then the set derivative of ϕF (T ) with respect to the finite subset Y ⊆ Y is, if it exists ([179], Section 11.4.2), δϕF δF (T ) = [1T ]. δY δY

(3.36)

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The set derivative can alternatively be defined independently of functionals. Let ϕ(T ) be a set function defined on closed sets T . Then the set derivative with respect to y is (see Eq. (11.229) of [179]): δϕ ϕ(T ∪ Ey ) − ϕ(T ) (T ) = lim δy |Ey | |Ey |↘0

(3.37)

where Ey is an arbitrarily small closed neighborhood of the point y that is disjoint from T . (For T that are not disjoint from Ey , a slightly more complex definition is required—see Appendix C.2 or [94], pp. 145-146, Definition 13.) Remark 2 (Constructive functional derivatives) As noted earlier, (3.30) defines the functional derivative only implicitly. However, as is explained in Appendix C.3, the set derivative can be used to construct functional derivatives: [ ] δF δ [h] = Fh (T ) δy δy T =∅

(3.38)

where the set function Fh (T ) is defined by Fh (T ) =

∂F [h]. ∂1T

(3.39)

That is: by (3.30), if the functional derivatives (δF/δx)[h] of F [h] exist for all x and if the function x ?→(δF/δx)[h] is integrable for each fixed h, then Fh (T ) is a probability measure absolutely continuous with respect to the base measure (see also Appendix C). Its set derivative must equal the Radon-Nikod´ym derivative of the set function Fh (S). The general set derivative with respect to a finite subset Y = {y1 , ..., yn } ⊆ Y with |Y | = n is defined iteratively: δϕ δnϕ δ δ n−1 ϕ (T ) = (T ) = (T ). δY δyn · · · δy1 δyn δyn−1 · · · δy1

(3.40)

Remark 3 (Set derivative as an inverse M¨obius transform) The set derivative is a continuous-space generalization of the inverse M¨obius transform of DempsterShafer theory (see [94], p. 149, Proposition 9, and [179], p. 383, Remark 17).

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Remark 4 (Relationship with Moyal’s calculus) Fifty years ago in [207], Moyal introduced a differential calculus of probability functionals. It is not only different than the one just described, it is very ill-suited for practical application. This is for two reasons: (1) it is highly abstract and measure-theoretic; and (2) it provides only existence proofs for the probability distributions of RFSs, which are defined only implicitly. It does not provide any explicit means of actually constructing them. The finite-set statistics calculus is based on set derivatives—which in turn are based on constructive Radon-Nikod´ym derivatives. See Appendix J for details.

3.5

KEY FORMULAS OF MULTIOBJECT CALCULUS

The purpose of this section is to summarize the most important mathematical identities associated with multiobject differential and integral calculus. Additional useful identities can be found in Chapter 4: convolution and deconvolution formulas (Section 4.2.3), Campbell’s theorems (Section 4.2.12), and Radon-Nikod´ym formulas (Section 4.2.11). The section is organized as follows: 1. Section 3.5.1: Fundamental theorem of multiobject calculus—states that set integrals and functional/set derivatives are inverse operations. 2. Section 3.5.2: Change of variables formula for set integrals—transforms set integrals into ordinary integrals. 3. Section 3.5.3: Set integrals on disjoint unions of spaces—showing that they can be transformed into multiple set integrals on the individual spaces. 4. Section 3.5.4: Constant rule—the functional derivatives of a constant functional are zero. 5. Section 3.5.5: Sum rule—showing that the functional derivative is a linear operator. 6. Section 3.5.6: Linear rule—formula for the functional derivatives of a linear functional. 7. Section 3.5.7: Monomial rule—formula for the functional derivatives of the integer powers of a linear functional. 8. Section 3.5.8: Power rule—formula for the first functional derivative of the power of a functional.

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9. Section 3.5.9: Product rules—formula for the functional derivatives of the product of a finite number of functionals. 10. Section 3.5.10: First chain rule—formula for the first functional derivative of a functional of the form F˜ [h] = f (F1 [h], ..., Fn [h]) where f (x1 , ..., xn ) is an ordinary function and F1 [h], ..., Fn [h] are functionals. 11. Section 3.5.11: Second chain rule—formula for the general functional derivatives of a functional of the form F˜ [h] = F [T −1 h] where F [h] is a functional and T is a nonsingular vector transformation. 12. Section 3.5.12: Third chain rule—formula for the first functional derivative of a functional of the form F˜ [h] = F [fh ] where f (x) is an ordinary function and fh (y) = f (h(y)). 13. Section 3.5.13: Fourth chain rule—formula for the first functional derivative of a functional of the form F˜ [h] = F [T [h]] where F [h] is a functional and T [h] is a functional transformation. 14. Section 3.5.14: Clark’s general chain rule—formula for the general functional derivatives of a functional of the form F˜ [h] = F [T [h]], where F [h] is a functional and T [h] is a functional transformation. 3.5.1

Fundamental Theorem of Multiobject Calculus

The set integral and functional derivative are inverse operations ([179], Eqs. (11.246-11.247)): ∫ δF F [h] = hY · [0]δY (3.41) δY [ ] ∫ δ f (Y ) = hW · f (W )δW (3.42) δY h=0 where the power functional hY was defined in (3.5). The corresponding formulas for set derivatives are ([179], Eqs. (11.24411.245)): ∫ δϕ ϕ(T ) = (∅)δY (3.43) δY T [ ] ∫ δ f (Y ) = f (W )δW . (3.44) δY T T =∅

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Change of Variables Formula for Set Integrals

Let Ψ ⊆ Y be an RFS and fΨ (X) its probability distribution. Let η be a transformation from finite sets Y ⊆ Y to the elements η(Y ) ∈ W of some other space W, and let T : W → V be a transformation from the space W to the elements of some other space V.9 Let PW (w) = PΨ,η (w) be the probability distribution of the random variable (3.45)

W = η(Ψ).

Then the change of variables formula for set integrals is ([94], p. 180, Prop. 4): ∫ ∫ T (η(Y )) · fΨ (Y )δY = T (w) · PΨ,η (w)dw. (3.46) That is, the set integral on the left side can be transformed into the ordinary integral on the right side, via the change of variables w = η(Y ). 3.5.3 1

Set Integrals on Joint Spaces s

1

s

Let Y, ..., Y be s spaces, each of which is endowed with an integral. Let Ψ, ..., Ψ be respective RFSs on these spaces. Then the superposition (set theoretic union) of these RFSs is an RFS 1 s ˘ = Ψ ⊎ ... ⊎ Ψ Ψ (3.47) on their “joint space,” that is, the disjoint union or “topological sum” 1

s

˘ = Y ⊎ ... ⊎ Y. Y

(3.48)

˘ have the form Functions with arguments in Y j ˘ y) = h( ˘ y) h(˘

The integral

∫

y ˘=y .

˘ is defined as ·d˘ y on Y ∫ ∫ ∫ 1 1 s s ˘ y)d˘ ˘ y)d ˘ y)d y. h(˘ y = 1 h( y + ... + s h( Y

9

j

if

(3.49)

(3.50)

Y

Note: In this and the following sections, the symbol ‘T ’ will be used to denote a vector or functional transformation, as well as a subset of the space Y. The proper notational meaning will be clear from context.

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˘ is Thus the set integral on Y ∫ ∑ 1 ∫ f˘(Y˘ )δ Y˘ = f˘({˘ y1 , ..., y ˘n˘ })d˘ y1 · · · d˘ yn˘ . n ˘!

(3.51)

n ˘ ≥0

Equation (3.51) can be written in a more mathematically convenient form. Abbreviate 1 s 1 s ˘˘ (Y ⊎ ... ⊎ Y ) abbr. ˘ f˘Ψ = f 1 s (Y , ..., Y ). (3.52) ˘ (Y ) = f Ψ Ψ,...,Ψ

Then the single set integral on the joint space can be written as a multiple set integral on the disjoint subspaces: ∫ ∫ 1 s 1 s ˘ ˘ ˘ fΨ f 1 s (Y , ..., Y )δ Y · · · δ Y . (3.53) ˘ (Y )δ Y = Ψ,...,Ψ

To demonstrate (3.53), it is sufficient to prove it for s = 2. Abbreviate 1

2

1

f (Y , Y ) = f 1

2

2

(Y , Y ). First note that an induction proof on n ˘ (see Section K.1)

Ψ,Ψ

shows that ∫ =

f˘({˘ y1 , ..., y ˘n˘ })d˘ y1 · · · d˘ yn˘ ∫ ∑ 1 1 2 2 Cn+n′ ,n f˘({y1 , ..., yn , y1 , ..., yn′ })

(3.54)

n+n′ =˘ n 1

1

2

2

·dy1 · · · dyn dy1 · · · dyn′ where Cn+n′ ,n is the binomial coefficient as defined in (2.1). Thus ∫ ∑ 1 ∫ f˘(Y˘ )δ Y˘ = f˘({˘ y1 , ..., y ˘n˘ })d˘ y1 · · · d˘ yn˘ n ˘! n ˘ ≥0 ∑ ∑ Cn+n′ ,n ∫ 1 1 2 2 = f ({y1 , ..., yn }, {y1 , ..., yn′ }) ′ )! (n + n ′

(3.55)

(3.56)

n ˘ ≥0 n+n =˘ n 1

=

1

2

2

·dy1 · · · dyn dy1 · · · dyn′ ∫ ∑ 1 1 1 2 2 f ({y1 , ..., yn }, {y1 , ..., yn′ }) ′! n! · n ∗

(3.57)

n,n≥0 1

=

1

2

2

·dy1 · · · dyn dy1 · · · dyn′ ∫ 1 2 1 2 f (Y , Y )δ Y δ Y .

(3.58)

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73

Constant Rule

Let K be a constant functional or constant set function. Then

3.5.5

δ K = 0. δY

(3.59)

δ δF1 δF2 (a1 F1 [h] + a2 F2 [h]) = a1 [h] + a2 [h]. δY δY δY

(3.60)

Sum Rule

• Functional form:

• Set-function form: δ δϕ1 δϕ2 (a1 ϕ1 (T ) + a2 ϕ2 (T )) = a1 (T ) + a2 (T ). δY δY δY 3.5.6

Linear Rule

• Functional form: Let f [h] = ([179], Eq. (11.261)):

∫

h(y) · f (y)dy be a linear functional. Then

  f [h] δ f (y) f [h] =  δY 0 • Set-function form: Let p(T ) =

∫

T

if if if

Y =∅ Y = {y} . |Y | ≥ 2

(3.62)

f (y)dy. Then ([179], Eq. (11.260)):

  p(T ) δ f (y) p(T ) =  δY 0 3.5.7

(3.61)

if if if

Y =∅ Y = {y} . |Y | ≥ 2

(3.63)

Monomial Rule

∫ Let f [h] = h(y) · f (y)dy be a linear functional and let N ≥ 0 be a nonnegative integer. Then ([179], Eq. (11.11.263)):

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• Functional form:  f [h]N  δ N f [h] = |Y |! · CN,|Y | · f [h]N −|Y | · f Y  δY 0

if if if

Y =∅ 0 < |Y | ≤ N . |Y | > N (3.64) was defined in (3.5) and where

where the power-functional notation f Y CN,n was defined in (2.1). ∫ • Set-function form: Let p(T ) = T f (y)dy and N ≥ 0 a nonnegative integer. Then ([179], Eq. (11.11.262)):  p(T )N if Y =∅  δ N p(T ) = |Y |! · CN,|Y | · p(T )N −|Y | · f Y if 0 < |Y | ≤ N .  δY 0 if |Y | > N (3.65) 3.5.8

Power Rule

• Functional form: Let F [h] be a functional and a be a real number. Then ([179], Eq. (11.265)): δ δF F [h]a = a · F [h]a−1 · [h]. δy δy

(3.66)

• Set function form: Let ϕ(T ) be a set function and a be a real number. Then ([179], Eq. (11.264)): δ δϕ ϕ(T )a = a · ϕ(T )a−1 · (T ). δy δy 3.5.9

(3.67)

Product Rules

• Functional form: Let F [h] = F1 [h] · · · Fn [h] be a product of functionals and Y ⊆ Y be a finite subset. Then ([179], Eq. (11.274)): δF [h] = δY

∑

δF1 δFn [h] · · · [h] δW1 δWn

(3.68)

W1 ⊎...⊎Wn =Y

where the summation is taken over all mutually disjoint subsets W1 , ..., Wn of Y (the empty subset included) such that W1 ∪ ... ∪ Wn = Y .

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• Set function form: Let ϕ(T ) = ϕ1 (T ) · · · ϕn (T ) be a product of set functions and Y ⊆ Y be a finite subset. Then ([179], Eq. (11.273)): ∑

δϕ (T ) = δY

δϕ1 δϕn (T ) · · · (T ). δW1 δWn

(3.69)

W1 ⊎...⊎Wn =Y

When n = 2 the product rule becomes ∑ δF1 δF δFn [h] = [h] · [h]. δY δW δ(Y − W )

(3.70)

W ⊆Y

If Y = {y} then (3.68) can be rewritten as n ∑ δF 1 δFi [h] = F [h] · [h]. δy F [h] δy i=1 i

(3.71)

δF δF1 δF2 [h] = [h] · F2 [h] + F1 [h] · [h]. δy δy δy

(3.72)

If in addition n = 2,

3.5.10

First Chain Rule

• Functional form: Let F1 [h], ..., Fn [h] be functionals and f (y1 , ..., yn ) a real-valued function of real arguments y1 , ..., yn . Then ([179], Eq. (11.279)): n

∑ ∂f δ δFj f (F1 [h], ..., Fn [h]) = (F1 [h], ..., Fn [h]) · [h]. δy ∂y δy j j=1

(3.73)

• Set function form: Let ϕ1 (T ), ...., ϕn (T ) be set functions and f (y1 , ..., yn ) a real-valued function of real arguments y1 , ..., yn . Then ([179], Eq. (11.276)): n

∑ ∂f δ δϕj f (ϕ1 (T ), ..., ϕn (T )) = (ϕ1 (T ), ..., ϕn (T )) · (T ). δy ∂y δy j j=1

(3.74)

When n = 1 the first chain rule becomes δ df δF f (F [h]) = (F [h]) · [h]. δy dy δy

(3.75)

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3.5.11

Second Chain Rule

• Functional form: Let Y be a vector space, F [h] be a functional, and T : Y → Y be a nonsingular transformation. For any test function h(y) define (T −1 h)(y) T

=

h(T (y))

(3.76)

Y

=

−1

(3.77)

JTY

=

−1

{T y| y ∈ Y } ∏ JT (y)

(3.78)

y∈Y

where JT (y) is the Jacobian determinant of T . Then ([179], Eq. (11.282)): δ 1 δF F [T −1 h] = Y · [T −1 h]. −1 Y ) δY δ(T JT

(3.79)

• Set function form: Let Y be a vector space, ϕ(S) a set function, and T : Y → Y a nonsingular transformation. Further define T −1 S = {T −1 y| y ∈ S} Then ([179], Eq. (11.281)): 1 δϕ δ ϕ(T −1 S) = Y · (T −1 S). δY JT δ(T −1 Y ) 3.5.12

(3.80)

Third Chain Rule

Functional form only: Let f (y) be a real-valued function of the real variable y and let F [h] be a functional. Define fh (y) = f (h(y))

(3.81)

for all y. Then ([179], Eq. (11.283)): δ δF df F [fh ] = [fh ] · (h(y)). δy δy dy

(3.82)

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Fourth Chain Rule

The second and third chain rules are actually special cases of this more general chain rule. Let h ?→ T [h] be a functional transformation (that is, it transforms test functions h to test functions T [h]). Define the functional derivative of T [h] to be the pointwise functional derivative. That is, for all y, w ∈ Y, define δT δ T [h + εδy ](w) − T [h](w) [h](w) = T [h](w) = lim . ε→0 δy δy ε

(3.83)

Let F [h] be a functional. Then the following chain rule is true for functional derivatives ([179], Eq. (11.285)): ∫ δ δT δF F [T [h]] = [h](w) · [T [h]]dw. (3.84) δy δy δw Because of the linearity and continuity properties of Gˆateaux derivatives, (3.31), this can be rewritten as a Gˆateaux derivative: δ ∂F ) [T [h]]. F [T [h]] = ( δy ∂ δT δy [h] The notation on the right side of this equation means this: [ ] ∂F ∂F ( ) [h] = [h] . ∂g g= δT [h] ∂ δT [h] δy δy

(3.85)

(3.86)

That is, first compute the gradient derivative (∂F/∂g)[h] of F [h] in the direction of an arbitrary g(y). Then substitute the functional derivative (δT /δy)[h] of T [h] at y in place of g. As a simple example of (3.84), let T [h](x) = 1 − τ (x) + τ (x) · h(x) with 0 ≤ τ (x) ≤ 1 identically and let f (x) = F [1 − τ + τ · x]. Then f ′ (x)

= = = =

d F [1 − τ + τ · x] ∫dx d δF (1 − τ (x) + τ (x) · x) · [1 − τ + τ · x]dx dx δx ∫ δF τ (x) · [1 − τ + τ · x]dx δx ∂F [1 − τ + τ · x] ∂τ

(3.87) (3.88) (3.89) (3.90)

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where the final equation follows from (3.31). 3.5.14

Clark’s General Chain Rule

Equation (3.85) is a special case of the most general chain rule for functional derivatives. This rule, a special case of a still more general chain rule due to D. Clark [41], is as follows [47]:10 ∑ δ ∂ |P| F ( δT ) [T [h]]. F [T [h]] = δY ∂ W ∈P δW [h] P⊟Y

(3.91)

The summation is taken over all partitions P of Y . The notation ‘P ⊟ Y ’ is shorthand for “P partitions Y into cells”; and |P| denotes the number of cells in P. Also, the following notational convention is employed: ∂ W ∈{W1 ,...,Wl }

(

δT [h] δW

)

abbr.

= ∂

(

) ( ) δT δT [h] · · · ∂ [h] . δW1 δWl

(3.92)

A summary of the theory of partitions can be found in Appendix D. Equation (3.91) is proved for functional derivatives in Section K.7 and, for the general case of topological vector spaces, in [47]. Remark 5 (Fa`a di Bruno’s formula) Clark’s general chain rule is a generalization of the following general chain rule from vector calculus. Suppose that f (x), g(x) are real-valued functions on RN . Then the directional derivatives of the composite function f (g(x)) are ∂n f (g(x)) = ∂y1 · · · ∂yn

∑ P⊟{1,...,n}

f (|P|) (g(x))

∏ I∈P

∏

∂ |I| g (x), i∈I ∂yi

(3.93)

where the ∏ summation is taken over all partitions P of the set {1, ..., n}; and where i∈I ∂yi = ∂yi1 · · · ∂yij if I = {i1 , ..., ij } ⊆ {1, ..., n} with |I| = j. 10 Clark’s general chain rule is very general, being true for the “chain derivative” of functions whose arguments are in an arbitrary topological vector space. (The chain derivative generalizes the Gˆateaux differential.) In particular, the space of “test functions” h(x) on a Hausdorff, locally compact, and completely separable space X is a topological vector space. It follows that the general chain rule applies to derivatives of functionals whose arguments are test functions.

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As a useful special case, let the functional transformation T [h] be realvalued—that is, it is a functional–and let F [h] = f (x) be an ordinary real-valued function of a real variable h = x. Then (3.91) reduces to (

∑ δ f (T [h]) = δY

∏ δT [h] δW

)

·

d|P| f (T [h]). dx|P|

(3.94)

W ∈P

P⊟Y

As a special case of this special case, let f (x) = x−1 . Assuming that F [h] > 0 for all h, we get the following inverse rule for functionals (also originally due to Clark): ∑ δ 1 = δY F [h]

(

∏ δF [h] δW

)

(−1)|P| · |P|! . F [h]|P|+1

·

(3.95)

W ∈P

P⊟Y

Example 1 (Example of Clark’s general chain rule) As a concrete example of the use of (3.91), let Y = {y1 , y2 , y3 } with |Y | = 3 and let (3.96)

T [h] = 1 − ρ + ρ · h

where 0 ≤ ρ(y) ≤ 1 is some unitless function. There are five partitions of Y : P1

=

{{y1 , y2 , y3 }},

P2 = {{y1 }, {y2 }, {y3 }}

(3.97)

P3 P5

= =

{{y3 }, {y1 , y2 }}, {{y1 }, {y2 , y3 }}.

P4 = {{y2 }, {y1 , y3 }}

(3.98) (3.99)

Thus (3.91) becomes δ F [T [h]] δY

∂ |P1 | F ( δT

=

∂ W ∈P1 +

+

δW |P3 |

[h]

) [T [h]] +

∂ |P2 | F ( δT

∂ W ∈P2

δW |P4 |

[h]

) [T [h]] (3.100)

∂ F ∂ F ( δT ) [T [h]] + ( δT ) [T [h]] ∂ W ∈P3 δW [h] ∂ W ∈P4 δW [h] ∂ |P5 | F ( δT

∂ W ∈P5

δW

[h]

) [T [h]]

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∂F = ∂

(

δ3 T δy1 δy2 δy3 [h]

) [T [h]]

(3.101)

∂3F ) ( ) ( ) [T [h]] + ( δT δT δT ∂ δy [h] ∂ δy [h] ∂ δy [h] 1 2 3 ∂2F ) ( ) [T [h]] + ( 2T δT ∂ δy [h] ∂ δyδ1 δy [h] 2 2 ∂2F ) ( ) [T [h]] + ( δT δ2 T ∂ δy [h] ∂ [h] δy1 δy3 2 ∂2F ) ( ) [T [h]]. + ( δT δ2 T ∂ δy [h] ∂ [h] δy2 δy3 1 However, δT [h] δw

=

T [h + εδw ] − T [h] ε (1 − ρ + ρ(h + εδw )) − (1 − ρ + ρh) lim ε→0 ε ερδw lim = ρδw = ρ(w) · δw ε→0 ε lim

(3.102)

ε→0

= =

(3.103) (3.104)

and so δT [h] = 0 δW

(3.105)

for all |W | > 1. Thus δ F [T [h]] δY

=

= = =

∂3F ) ( ) ( ) [T [h]] δT δT δT ∂ δy [h] ∂ [h] ∂ [h] δy δy 1 2 3 (

(3.106)

∂3F [T [h]] (3.107) ∂ (ρ(y1 )δy1 ) ∂ (ρ(y2 )δy2 ) ∂ (ρ(y3 )δy3 ) δ3F ρ(y1 ) · ρ(y2 ) · ρ(y3 ) · [T [h]] (3.108) δy1 δy2 δy3 δF ρY · [1 − ρ + ρh]. (3.109) δY

Chapter 4 Multiobject Statistics 4.1

INTRODUCTION

This chapter summarizes the basic concepts of multiobject statistics: the fundamental and ancillary multiobject statistical descriptors; and the most important multiobject processes. It is organized as follows: 1. Section 4.2: The basic fundamental and ancillary multiobject statistical descriptors—belief-mass functions, multiobject probability densities, probability generating functionals (p.g.fl.’s), cardinality distributions, probability generating functions (p.g.f.’s), probability density functions (PHDs), and multiobject factorial moment densities. 2. Section 4.3: Important multiobject processes—Poisson, independently distributed cluster (i.i.d.c.), multi-Bernoulli, and Bernoulli. 3. Section 4.4: Basic ancillary multiobject processes—censored processes and cluster processes.

4.2

BASIC MULTIOBJECT STATISTICAL DESCRIPTORS

The statistics of an RFS Ψ ⊆ Y are completely specified by three mathematically equivalent fundamental statistical descriptors, which will be described in this section.

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1. Section 4.2.1: The belief-mass function βΨ (T )—central to the derivation of multitarget Markov densities and likelihood functions from multitarget motion and measurement models (to be described in Sections 5.4 and 5.5). 2. Section 4.2.2: The multiobject probability density function fΨ (Y )—the fundamental concept for multisensor-multitarget Bayes filtering, and thus for multisensor-multitarget detection, tracking, and identification. 3. Section 4.2.3: Convolution and deconvolution formulas for multiobject probability density functions. 4. Section 4.2.4: The probability generating functional GΨ [h] or p.g.fl.— central to the derivation of approximate multisensor-multitarget Bayes filters such as the PHD filter, CPHD filter, multi-Bernoulli filter, and others (to be described in Section 5.10). 5. Section 4.2.5: Multivariate p.g.fl.’s GΨ1 ,...,Ψn [h1 , ..., hn ]—important in the derivation of RFS approximate multitarget filters and in multisensor problems. In addition to these fundamental descriptors, various ancillary statistics are also important: 1. Section 4.2.6: The cardinality distribution pΨ (n)—the probability distribution of the number |Ψ| of elements in Ψ. 2. Section 4.2.7: The probability generating function GΨ (y)—an equivalent representation of pΨ (n). 3. Section 4.2.8: The probability hypothesis density DΨ (y)—essentially, the density of objects in Ψ at y. 4. Section 4.2.9: The multiobject factorial moment density DΨ (Y ) (a generalization of the PHD to arbitrary finite sets Y ). In addition, the following three sections describe important relationships between the statistical descriptors: 1. Section 4.2.10: The equivalence of the fundamental descriptors (p.g.fl., belief-mass function, and multitarget probability density function). 2. Section 4.2.11: Radon-Nikod´ym-type relationships between the fundamental descriptors.

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3. Section 4.2.12: Linear and quadratic Campbell’s theorems. 4.2.1

Belief-Mass Functions

Let Ψ ⊆ Y be an RFS. Then its belief-mass function is defined as βΨ (T ) = Pr(Ψ ⊆ T )

(4.1)

for all measurable T .1 Suppose that Ψ1 , ..., Ψn ⊆ Y are independent RFSs and let Ψ = Ψ1 ∪ ... ∪ Ψn . Then βΨ (T ) = βΨ1 (T ) · · · βΨb (T ). (4.2) Belief-mass functions are generalizations of probability-mass functions (Section 2.2.3). Let Ψ = { Y } as in (2.37), where Y ∈ Y is a random variable. Then βΨ (T ) = Pr({Y} ⊆ T ) = Pr(Y ∈ T ) = pY (T ). (4.3) As another example, consider the “twinkling” random singleton of (2.39), and assume that Y and ∅q are independent. Then βΨ (T )

= = =

Pr({Y} ∩ ∅q ⊆ T ) Pr(∅q = ∅) + Pr(∅q ̸= ∅) · Pr(Y ∈ T ) 1 − q + q · pΨ (T ).

(4.4) (4.5) (4.6)

As a final example, consider (2.38): Ψ = {Y1 , ..., Yn } where Y1 , ..., Yn are independent: βΨ (T )

1

= =

Pr({Y1 , ..., Yn } ⊆ T ) = Pr(Y1 ∈ T, ..., Yn ∈ T ) pY1 (T ) · · · pYn (T ).

(4.7) (4.8)

The belief-mass function is also known as a belief measure. Because of the Choquet-Matheron theorem, it completely characterizes the statistics of the RFS Ψ ([179], p. 713) under the FellMatheron hit-and-miss topology. For this reason—as well as because of its utility in transforming multitarget motion and measurement models into multitarget Markov densities and likelihood functions—it is central to the finite-set statistics approach.

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4.2.2

Multiobject Probability Density Functions

A multiobject probability density function is a multiobject density function f (Y ) (see Section 3.2.4) whose set integral is unity: ∫ f (Y )δY = 1. (4.9) Refer to (3.19) for a simple example of a multiobject probability distribution. If Ψ ⊆ Y is an RFS, its multiobject probability distribution can, if it exists, be derived from the belief-mass function as follows:2 fΨ (Y ) =

δβΨ (∅). δY

(4.10)

Intuitively speaking, fΨ (Y ) is the probability (density) of the event Ψ = Y .3 As an example, consider the belief-mass function of the “twinkling random singleton,” as in (4.6). The corresponding multitarget probability distribution is given by fΨ (∅)

=

fΨ ({y})

=

fΨ (Y )

=

δβΨ (∅) = βΨ (∅) = 1 − q δ∅ δβΨ (∅) = q · fY (y) δy 0 if |Y | > 1

(4.11) (4.12) (4.13)

or, in summary, by Y fΨ (Y ) = C1,|Y | · q |Y | (1 − q)|Y | · fY Y where the power functional notation fY coefficient C1,|Y | in (2.1).

2

(4.14)

was defined in (3.5) and the binomial

In point process theory, the density functions n! · jΨ,n (y1 , ..., yn ) = fΨ ({y1 , ..., yn })

3

are known as the “Janossy densities” [55], [56] of the point process Ψ. They have also been called “joint multitarget probability densities” or “JMPDs” ([214], p. 27). If they exist (that is, are integrable finite-valued functions) then Ψ is a “simple” point process ([55], p. 138, Prop. 5.4.V)— meaning that jΨ,n (y1 , ..., yn ) = 0 whenever yi = yj for some i ̸= j. That is, a simple point process is essentially the same thing as an RFS. See Section 2.3.1. If Y is continuously infinite, this is only an intuitive interpretation since, of course, in that case Ψ = Y is a zero-probability event.

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Another example is the multitarget Dirac delta density ([179], p. 366, Eq. (11.124)). Let Y ′ = {y1′ , ..., yn′ } with |Y ′ | = n and Y = {y1 , ..., yn } with |Y | = n. Then this density is defined as ∑ ′ (y1 ) · · · δy ′ (yn ) δY ′ (Y ) = δ|Y ′ |,|Y | δyπ1 (4.15) πn π

where the summation is taken over all permutations π on the numbers 1, ..., n. A final example is the multitarget uniform distribution. Let Y = RN and S ⊆ Y a bounded closed subset with (hyper)volume |S|. Suppose that there can be no more than n0 objects in S. If Y = {y1 , ..., yn } with |Y | = n then this distribution is defined as ([94], p. 144; [179], p. 367): uS,nmax (Y ) =

|Y |! · 1YS · 1|Y |≤n0 +1 |S||Y | · (n0 + 1)

(4.16)

where 1S (y) is the indicator function of S; where 1n≤n0 +1 = 1 if n ≤ n0 + 1 and 1n≤n0 +1 = 0 otherwise; and where the power functional notation hY was defined in (3.5). 4.2.3

Convolution and Deconvolution

Let Ψ1 , ..., Ψn ⊆ Y product rule, (3.68),

be statistically independent RFSs. Then by the general

fΨ (Y ) =

∑

fΨ1 (W1 ) · · · fΨn (Wn )

(4.17)

W1 ⊎...⊎Wn =Y

where the summation is taken over all mutually disjoint (and possibly empty) subsets W1 , ..., Wn of Y , such that W1 ∪ ... ∪ Wn = Y . If n = 2, this reduces to the convolution-like formula ∑ fΨ (Y ) = fΨ1 (W ) · fΨ2 (Y − W ). (4.18) W ⊆Y

For this reason, (4.17) is called the fundamental convolution theorem for independent RFSs. As an example, let Y = {y1 , y2 } with y1 ̸= y2 . Then fΨ (Y )

=

fΨ1 (∅) · fΨ2 ({y1 , y2 }) + fΨ1 ({y1 }) · fΨ2 ({y2 }) (4.19) +fΨ1 ({y2 }) · fΨ2 ({y1 }) + fΨ1 ({y1 , y2 }) · fΨ2 (∅).

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Next, consider the problem that is inverse to (4.18), also known as the deconvolution problem: • fΨ (X) and fΨ1 (X) are known, and we are to determine fΨ2 (X). Using the quotient rule for functionals, (3.95), D. Clark has answered this question as follows: fΨ2 (X) =

∑ ∑ (−1)|P| · |P|! ∏ · fΨ (X − W ) · fΨ1 (V ). |P|+1 fΨ1 (∅) V ∈P W ⊆X

(4.20)

P⊟W

A proof of this fact can be found in Section K.2. 4.2.4

Probability Generating Functionals (p.g.fl.’s)

Let h(y) be a “test function” on Y—that is, h(y) is unitless and 0 ≤ h(y) ≤ 1.4 Let Ψ ⊆ Y be an RFS. Then its probability generating functional (p.g.fl.) is the functional defined as the expected value5 ∫ GΨ [h] = E[hΨ ] = hY · fΨ (Y )δY, (4.21) where the power functional hY was defined in (3.5). The p.g.fl. of Ψ has the following properties: 0 GΨ [1] GΨ [1T ]

≤ = =

GΨ [h] ≤ 1 1 βΨ (T ) = Pr(Ψ ⊆ T ).

Also, suppose that Ψ1 , ..., Ψn ⊆ Y are independent RFSs and let Ψ1 ∪ ... ∪ Ψn . Then the p.g.fl. factors as follows: GΨ [h] = GΨ1 [h] · · · GΨb [h]. 4 5

(4.22) (4.23) (4.24) Ψ =

(4.25)

It is sometimes additionally required that the set {y| h(y) ̸= 0} be closed and bounded. The formula for the p.g.fl. goes back to Volterra in 1927, who called it a “functional power series” ([317], p. 21, Eq. (1)). Daley and Vere-Jones attribute the introduction of the p.g.fl., in a point process sense, to the Russian physicist Bogoliubov in 1946 ([56], p.15). Moyal in 1962 attributed the p.g.fl. and other point process generating functionals to Bartlett and Kendall in the early 1950s ([207], footnote 1, p. 13). The p.g.fl. has since become a textbook-level concept in point process theory [55], [56].

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As an example, the p.g.fl. of the “twinkling random singleton,” whose multiobject probability density function was given in (4.11) through (4.13), is ∫ GΨ [h] = hY · fΨ (Y )δY (4.26) ∫ = 1 · f (∅) + h(y) · f ({y})dy + 0 (4.27) ∫ = 1 − q + q h(y) · fY (y)dy (4.28) =

1 − q + q · fY [h].

(4.29)

As another example, the p.g.fl. of the multiobject probability density function of (3.19) is Gf [h] = (1 − q + q · f1 [h]) · (1 − q + q · f2 [h]) . (4.30) For, Gf [h]

= =

=

= =

∫

hY · f (Y )δY (4.31) ∫ 1 · f (∅) + h(y) · f ({y})dy (4.32) ∫ 1 + h(y1 ) · h(y2 ) · f ({y1 , y2 })dy1 dy2 + 0 2 ∫ 2 (1 − q) + q (1 − q) h(y) · (f1 (y) + f2 (y)) dy (4.33) ∫ q2 + h(y1 ) · h(y2 ) · (f1 (y1 ) · f2 (y2 ) + f1 (y2 ) · f2 (y1 )) dy1 dy2 2 (1 − q)2 + q (1 − q) · (f1 [h] + f2 [h]) + q 2 · f1 [h] · f2 [h] (4.34) (1 − q + q · f1 [h]) · (1 − q + q · f2 [h]) . (4.35)

Remark 6 (Constructing p.g.fl.’s from b.m.f.’s) As a heuristic rule of thumb, a formula for the p.g.fl. can be constructed from a formula for the belief-mass function. Rewrite the formula for βΨ (T ) in the variable T as a formula β[1T ] in the variable 1T . Substituting h for 1T , we get a formula GΨ [h] = β[h] for the p.g.fl. Remark 7 (p.g.fl. as a probability) The belief-mass function of an RFS Ψ is defined directly as a probability: βΨ (T ) = Pr(Ψ ⊆ T ). It turns out that p.g.fl.’s

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can be regarded as generalized belief-mass functions, as follows ([179], p. 373): (4.36)

GΨ [h] = Pr(Ψ ⊆ Σα (h)).

Here, α(y) is a uniformly distributed random number in [0, 1] for every y; def. Σα (h) = {y| α(y) ≤ h(y)}; and it is assumed that, for any n ≥ 2 and any distinct y1 , ..., yn , the random variables α(y1 ), ..., α(yn ), Ψ are independent. The random set Σα (h) is the “asynchronous” random set representation of the fuzzy membership function h(x)—see (22.38). Remark 8 (Why emphasize the p.g.fl.?) The p.g.fl. is only one among many fundamental functional statistical descriptors that are employed in point process theory. Others include the characteristic functional, the Laplace functional, and the factorial moment generating functional [55], [56], [207]. So why is the p.g.fl. emphasized in finite-set statistics in preference to these others? Different functional descriptors are useful for different purposes, and it turns out that the p.g.fl. is especially useful for multitarget tracking theory and practice. This is partly because of its close relationship with the belief-mass function, which it generalizes because of (4.36). But it is useful also partly because so many other statistical descriptors, such as multiobject probability distributions and the PHD, are easily related to it via the multiobject calculus. Remark 9 (Choquet integrals) In (4.21), suppose that we replace the product hY by its fuzzy logic analog, miny∈Y h(y). Then ) ∫ ∫ ( def. h · dβΨ = min h(y) · fΨ (Y )δY (4.37) y∈Y

is known as the “Choquet integral” of the function h, with respect to the nonadditive measure (belief-mass function) βΨ (T ) (see [94], p. 179). 4.2.5

Multivariate p.g.fl.’s 1

s

1

s

Let Y1 , ..., Y be spaces, let Ψ, ..., Ψ be respective RFSs on these spaces, and 1

let f 1

s

s

i

(Y , ..., Y ) be their joint multiobject probability distribution, where Y

Ψ1 ,...,Ψ 1

i

s 1

s

denotes a finite subset of Y. Let h(y), ..., h(y) be respective test functions on 1

s

i i

Y1 , ..., Y, where y ∈ Y. Then the joint multivariate p.g.fl. of the joint process 1

s

1

s

˘ = Ψ ⊎ ... ⊎ Ψ ⊆ Ψ ⊎ ... ⊎ Ψ Ψ

(4.38)

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(where ‘⊎’ denotes disjoint union) is the expected value 1

G1

s

s

(4.39)

[h, ..., h]

Ψ1 ,...,Ψ 1 1

= =

s s

E[hΨ · · · hΨ ] ∫ 1 1 s s hY · · · hY · f 1

1 s

Ψ,...,Ψ

s

1

s

(4.40)

(Y , ..., Y )δ Y · · · δ Y .

˘ The joint multivariate p.g.fl. is equivalent to the single-variate p.g.fl. GΨ ˘ [h] ˘ ˘ of the RFS Ψ = Ψ1 ⊎ ... ⊎ Ψn . For, let h(˘ y) be a test function on the joint space i i i ˘ ˘ Y = Y ⊎ ... ⊎ Yn , with h(˘ y) = h(y) when y ˘ = y. Then 1

1

˘ GΨ ˘ [h] = G 1

s

s

(4.41)

[h, ..., h].

Ψ,...,Ψ

For, from (3.52), ˘ GΨ ˘ [h]

=

∫

˘ Y˘ · f ˘ (Y˘ )δ Y˘ h Ψ

=

∫

˘ Y ⊎...⊎Y · f 1 h

1

(4.42)

s

1 s

Ψ,...,Ψ

=

∫

1

s

1

˘Y · · · h ˘Y · f 1 h

s

Ψ,...,Ψ

=

∫

1 1

s s s

Ψ,...,Ψ

1

=

G1

s

1

s

s

1

s

1

(4.43)

s

(Y , ..., Y )δ Y · · · δ Y 1

hY · · · hY · f 1

s

(Y , ..., Y )δ Y · · · δ Y

(4.44)

s

(Y , ..., Y )δ Y · · · δ Y

(4.45)

s

[h, ..., h].

(4.46)

Ψ,...,Ψ

Two examples of multivariate p.g.fl.’s will be considered: joint multitarget, single-sensor multivariate p.g.fl.’s (Section 4.2.5.1) and joint multitargetmultisensor multivariate p.g.fl.’s (Section 4.2.5.2). 4.2.5.1

Example: Joint Target/Sensor p.g.fl.’s

The following bivariate p.g.fl. plays a central role in the principled approximation of RFS filters. Let Ξk+1|k ⊆ X be the random predicted multitarget state set at time tk+1 , and let Σk+1 ⊆ Z be the random multitarget measurement set at time tk+1 . Assume that current measurements do not depend on the previous measurement

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history: fΣk+1 |Ξk+1|k (Z|X) = fΣk+1 |Ξk+1|k ,Σ1 ,...,Σk (Z|X, Z1 , ..., Zk ).

(4.47)

Let fk+1 (Z|X) abbr. = fΣk+1 |Ξk+1|k (Z|X)

(4.48)

be the multitarget likelihood function at time tk+1 and let fk+1|k (X|Z (k) ) abbr. = fΞk+1|k |Σ1 ,...,Σk (X|Z1 , ..., Zk )

(4.49)

be the distribution of the predicted target process Ξk+1|k , conditioned on the measurement-stream Z (k) . Then the joint distribution of Σk+1 and Ξk+1|k is fΣk+1 ,Ξk+1|k (Z, X)

=

fk+1 (Z|X) · fk+1|k (X|Z (k) )

(4.50)

=

(k)

(4.51)

fk+1 (Z, X|Z

).

Thus the joint single-sensor, multitarget p.g.fl. of Ξk+1|k and Σk+1 , conditioned on Z (k) , is the bivariate p.g.fl. def.

F [g, h]

= = = =

GΣk+1 ,Ξk+1|k |Σ1 ,...,Σk [g, h|Z (k) ] ∫ g Z · hX · fk+1 (Z, X|Z (k) )δZδX ∫ g Z · hX · fk+1 (Z|X) · fk|k (X|Z (k) )δZδX ∫ hX · Gk+1 [g|X] · fk|k (X|Z (k) )δZδX

where Gk+1 [g|X] =

∫

g Z · fk+1 (Z|X)δZ

(4.52) (4.53) (4.54) (4.55)

(4.56)

is the p.g.fl. of fk+1 (Z|X). Note that F [g, h] is normal—that is, F [1, 1] = 1— and is, therefore, a p.g.fl. 4.2.5.2

Example: Joint Target/Multisensor p.g.fl.’s

Generalize the previous example by assuming that, instead of a single sensor, we 1

have s sensors with respective measurement spaces

s

Z, ..., Z. At time tk+1 ,

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1

s

suppose that we have respective random measurement sets Σk+1 , ..., Σk+1 . If the sensors are independent (measurements are conditionally independent of target states), then 1

1

s

s

1

s

(4.57)

fk+1 (Z, ..., Z|X) = f k+1 (Z|X) · · · f k+1 (Z|X). 1

s

Thus the joint multitarget-multisensor p.g.fl. of Ξk+1|k , Σk+1 , ..., Σk+1 is 1

=

∫

=

·fk+1|k (X|Z (k) )δ Z · · · δ ZδX ∫ 1 s 1 s hX · Gk+1 [g|X] · · · Gk+1 [g|X]

s

F [g, ..., g, h]

1

s

1

s

1

s

g Z · · · g Z · hX · f k+1 (Z|X) · · · f k+1 (Z|X) (4.58) 1

s

(4.59)

·fk+1|k (X|Z (k) )δX where j j

Gk+1 [g|X] =

∫

j

j

j

j

j

g Z · f k+1 (Z|X)δ Z.

(4.60)

Remark 10 (An erroneous factorization) The claim has been made, in Eq. (20) of 1 s [282], that, if the sensors are independent, then F [g, ..., g, h] factors as follows: 1

s

1

s

(4.61)

F [g, ..., g, h] = F [g, h] · · · F [g, h].

This equation is true only if s = 1—that is, only when there is a single sensor.6 Furthermore, it is untrue even under the assumptions made in [282]—namely, that the multiobject state process has a special “traffic process” form. Specifically, a “traffic process” Ξ ⊆ X with PHD DΞ (x) is defined as follows: (1) it has the 1

s

1

j

s

form Ξ = Ξ∪...∪ Ξ; (2) the Ξ, ..., Ξ are independent; (3) each Ξ is Poisson with j

ℓ

PHD D j (x) = β(x)·DΞ (x) ([282], Eq. (17)); and (4) the β(x) are nonnegative Ξ ∑s ℓ functions such that ℓ=1 β(x) = 1 identically for all x. The erroneousness of (4.61) can be demonstrated via a counterexample—see Section K.3. 6

As a consequence, the claimed main result of [282]—that is, the intensity-function measurementupdate in Eq. (26)—is untrue. In particular, the claimed computational linearity of Eq. (26) in the number s of sensors is spurious, because Eq. (26) is not, as claimed, a theoretically valid measurement-update equation.

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Cardinality Distributions

The cardinality distribution of an RFS Ψ ⊆ Y is pΨ (n)

= = =

Pr(|Ψ| = n) ∫ fΨ (Y )δY |Y |=n ∫ 1 fΨ ({x1 , ..., dxn })dx1 · · · dxn . n!

(4.62) (4.63) (4.64)

The number pΨ (n) is the probability that Ψ contains n elements. As an example, consider the multiobject probability density function of (4.14). Then its cardinality distribution is, for n = 0, 1, pΨ (n) = C1,n · q n (1 − q)1−n

(4.65)

where Cn′ ,n is the binomial coefficient as defined in (2.1). 4.2.7

Probability Generating Functions (p.g.f.’s)

Setting h = y where 0 ≤ y ≤ 1 is a scalar, the p.g.fl. reduces to the probability generating function (p.g.f.) of Ψ: ∫ ∑ GΨ (y) = [GΨ [h]]h=y = y |Y | · fΨ (Y )δY = pΨ (n) · y n . (4.66) n≥0

The p.g.f. is so-named because, for all n ≥ 0, the cardinality distribution can be generated from it, 1 (n) pΨ (n) = GΨ (0), (4.67) n! (n) where GΨ (y) denotes the nth derivative of GΨ (y). The expected value µΨ , the second factorial moment µΨ,2 , and the variance 2 σΨ of GΨ (y) (equivalently, of pΨ (n)) are given by: ∑ (1) µΨ = GΨ (1) = n · pΨ (n) (4.68) n≥1

µΨ,2

=

(2) GΨ (1)

=

∑

(4.69)

n(n − 1) · pΨ (n)

n≥2 2 σΨ

=

µΨ,2 − µ2Ψ + µΨ = −µ2Ψ +

∑ n≥1

n2 · pΨ (n).

(4.70)

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Suppose that Ψ1 , ..., Ψn ⊆ Y are independent RFSs and let Ψ = Ψ1 ∪ ... ∪ Ψn . Then the p.g.f. factors as GΨ (y) = GΨ1 (y) · · · GΨn (y)

(4.71)

and the mean and variance of Ψ are given by µΨ 2 σΨ 4.2.8

= =

µΨ1 + ... + µΨn 2 2 σΨ + ... + σΨ . 1 n

(4.72) (4.73)

Probability Hypothesis Densities (PHDs)

The probability hypothesis density (PHD) of an RFS Ψ is an ordinary density function DΨ (y) on single objects y ∈ Y (see [179], Section 16.2). Intuitively speaking: • The number DΨ (y) is the density of the objects at y. • DΨ (y)dy is the number of objects contained in the infinitesimal region dy centered at y. • DΨ (y) is the probability (density) of the zero-probability event y ∈ Ψ, in the same sense that an ordinary probability density function fY (y) of a random variable Y ∈ Y is the probability (density) of the zero-probability event Y = y. (If Y is a discrete space then DΨ (y) = Pr(y ∈ Ψ).) See [179], pp. 576-580. Formally, the PHD is defined as a set integral ([179], Eq. (16.26)): ) ∫ ∫ (∑ DΨ (y) = fΨ ({y} ∪ W )δW = δw (y) · fΨ (Y )δY.

(4.74)

w∈Y

The PHD can also be expressed in terms of functional derivatives and set derivatives ([179], Eqs. (16.35,16.36)): DΨ (y) =

δGΨ δ log GΨ δβΨ δ log βΨ [1] = [1] = (Y) = (Y). δy δy δy δy

(4.75)

The two logarithmic formulas are often to be preferred, since they often result in algebraically simpler formulas. The PHD has the following important properties:

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• For any closed T ⊆ Y, the expected number E[|Ψ∩T |] of objects contained in T is given in terms of the PHD by ∫ ∫ |Ψ ∩ T | · fΨ (Y )δY = DΨ (y)dy. (4.76) T

• In particular, if T = Y then the total expected number of objects E[|Ψ|] in the scene is the integral of the PHD: ∫ NΨ = DΨ (y)dy. (4.77) • Suppose that Ψ1 , ..., Ψn are independent RFSs and let Ψ = Ψ1 ∪ ... ∪ Ψn . Then DΨ (y) = DΨ1 (y) + ... + DΨn (y). (4.78) As a simple example of a PHD, consider the multitarget probability density f (Y ) of (3.19). Its PHD is given by ∫ 1 1 D(y) = f ({y}) + f ({y, w})dw + 0 (4.79) 0! 1! = q (1 − q) · (f1 (y) + f2 (y)) (4.80) ∫ +q 2 (f1 (y) · f2 (w) + f1 (w) · f2 (y)) dw = =

q (1 − q) · (f1 (y) + f2 (y)) + q 2 · (f1 (y) + f2 (y)) q · f1 (y) + q · f2 (y).

(4.81) (4.82)

Remark 11 (PHDs: Terminology and interpretations) In point process theory, the PHD is more commonly known as a “first-moment density,” “intensity density,” or “intensity function” [55], [56]. To avoid possible confusion, the lattermost usage has been avoided because of the very large number of alternative meanings of “intensity” in engineering and physics. In classical thermodynamics, the PHD is known as a “phase-space density” (see, for example, [235], p. 31). As such, the PHD turns out to play a central role in the traffic-flow theory (TFT) approach to multitarget tracking in urban environments [187]. The name “probability hypothesis density” is unique to multitarget tracking for historical reasons [165]. The concept and name were first proposed, at an intuitive level, by M. Stein and C. Winter [274], [275]. It was subsequently shown by Mahler to be the same thing as a first-moment density in the point process sense ([94], pp. 168-169).

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95

Factorial Moment Density

Let Ψ ⊆ Y be an RFS. Then the multitarget factorial moment density of the multiobject distribution fΞ (Y ) is ([168],[165, p.1162, Eq. (60)], [179], Section 16.2.5): ∫ δGΨ δβΨ DΨ (Y ) = fΨ (Y ∪ W )δW = [1] = (Y). (4.83) δY δY It generalizes the concept of a PHD, since DΨ (x) = DΨ ({x}).

(4.84)

The multiobject distribution fΨ (Y ) can be recovered from its factorial moment density DΨ (Y ) via the following inversion formula (see [165, p.1162, Eq. (60)], Eq. (68); and [179], Eq. (16.88)): fΨ (Y ) =

∫

(−1)|W | · DΨ (Y ∪ W )δW.

(4.85)

The factorial moment density also obeys the identity ([168], p. 142, [55], p. 222): GΨ [1 + h] = 4.2.10

∫

hY · DΨ (Y )δY.

(4.86)

Equivalence of the Fundamental Descriptors

Let Ψ ⊆ Y be an RFS. Then the following are all equivalent representations of the probability law of Ψ: the probability density fΨ (Y ), the belief-mass function βΨ (T ), and the p.g.fl. GΨ [h]. This is because they can all be recovered from each other: ∫ βΨ (T ) = fΨ (Y )δY = GΨ [1T ] (4.87) T

fΨ (Y )

=

GΨ [h]

=

δβΨ δGΨ (∅) = [0] δY δY ∫ ∫ δβΨ hY · fΨ (Y )δY = hY · [0]δY. δY

(4.88) (4.89)

Thus fΨ (Y ), βΨ (T ), and GΨ [h] all completely characterize the statistics of Ψ and can be used interchangeably without losing any information about Ψ.

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4.2.11

Radon-Nikod´ym Formulas

Let Ψ ⊆ Y be an RFS and let fΨ (Y ), βΨ (T ), and GΨ [h] be, respectively, its probability distribution, belief-mass function, and p.g.fl. The general RadonNikod´ym theorem for p.g.fl.’s is ([179], Eq. (11.251)): ∫ δGΨ [h] = hY · fΨ (W ∪ Y )δY (4.90) δW where the power functional hY was defined in (3.5). A special case for belief-mass functions is obtained by setting h = 1T : ∫ δβΨ (T ) = fΨ (W ∪ Y )δY. (4.91) δW T When W = ∅ this reduces to ([179], Eq. (11.248)): ∫ βΨ (T ) = fΨ (Y )δY.

(4.92)

T

This is the multiobject analog of the single-target equation ∫ pY (T ) = fY (y)dy.

(4.93)

T

4.2.12

Campbell’s Theorems

These theorems show us how to simplify formulas involving sums of test functions. Let us be given a function h(y). Then: • Linear form of Campbell’s theorem: The expected value E[A] of the random number A = hΨ (4.94) where hY abbr. =

∑

h(y)

(4.95)

h(y) · DΨ (y)dy

(4.96)

y∈Y

is:

∫

hY · fΨ (Y )δY =

∫

where DΨ (y) is the PHD of Ψ as defined in (4.75). Equation (4.96) is proved in Section K.6.

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As an example, (4.76) is a direct consequence of Campbell’s theorem. For, set h(y) = 1T (y) and let Y ⊆ Y be finite. Then |Y ∩ T | =

∑

(4.97)

1T (y)

y∈Y

and so by Campbell’s theorem,

E[|Ψ ∩ T |]



∫

=

=

∫

=

∫



∑ y∈Y



1T (y) · fΨ (Y )δY

(4.98)

(4.99)

1T (y) · DΨ (y)dy

(4.100)

DΨ (y)dy. T

• Quadratic form of Campbell’s theorem (scalar and vector versions): Given a second test function h′ (y), the expected value E[A] of the random number 

A = hΨ · h′Ψ =  is ∫

hΨ · h′Ψ · fΨ (Y )δY

=

∑ y∈Ψ



h(y) 

∑ y∈Ψ



h′ (y)

(4.101)

∫

h(y) · h′ (y) · DΨ (y)dy (4.102) ∫ + h(y1 ) · h′ (y2 ) · DΨ ({y1 , y2 })dy1 dy2

where DΨ (Y ) is the factorial moment density of Ψ as defined in (4.83). Equation (4.102) is proved in Section K.6. Equation (4.102) immediately generalizes to the case when h(x) is vector-valued and thus A is matrixvalued:    ∑ ∑ A = hΨ · (h′Ψ )T =  h(y)  h′ (y)T  (4.103) y∈Ψ

y∈Ψ

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4.3

IMPORTANT MULTIOBJECT PROCESSES

The purpose of this section is to introduce four families of RFSs that will have prominent roles throughout the book: 1. Section 4.3.1: Poisson RFSs—central to the theory of PHD filters. 2. Section 4.3.2: i.i.d.c. RFSs—central to the theory of CPHD filters. 3. Section 4.3.3: Bernoulli RFSs—central to the theory of Bernoulli filters. 4. Section 4.3.4: Multi-Bernoulli RFSs—central to the theory of multi-Bernoulli filters. 4.3.1

Poisson RFSs

Intuitively speaking, an RFS Ψ is Poisson if its instantiations are constructed as follows. We are given a PHD DΨ (y) and therefore also the expected number of targets ∫ NΨ =

DΨ (y)dy.

(4.104)

NΨn n!

(4.105)

From the Poisson distribution pΨ (n) = e−NΨ ·

draw an integer ν ∼ pΨ (·). Then from the spatial distribution of objects, sΨ (y) =

DΨ (y) , NΨ

(4.106)

draw ν elements y1 , ..., yν ∼ sΨ (·). Given this, Ψ = {y1 , ..., yν } is a particular instantiation of Ψ. Stated somewhat differently, a Poisson RFS is one in which the objects of Ψ are spatially distributed according to the distribution sΨ (y), and in which the number |Ψ| of objects in Ψ is Poisson-distributed. More formally, an RFS Ψ ⊆ Y is Poisson if its p.g.fl. has the form GΨ [h] = eDΨ [h−1] where the linear-functional notation DΨ [h − 1] was defined in (3.4). Other statistics associated with a Poisson RFS Ψ are:

(4.107)

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99

• Multiobject probability distribution: Y fΨ (Y ) = e−NΨ · DΨ

(4.108)

Y where the power-functional notation DΨ was defined in (3.5).

• p.g.f.: GΨ (y) = eNΨ ·(y−1) .

(4.109)

2 σΨ = NΨ .

(4.110)

• Variance: Remark 12 (“Independent increments” property) Let T1 , T2 ⊆ Y be closed subsets such that T1 ∩ T2 = ∅. Let Ψ be Poisson with GΨ [h] = eDΨ [h−1] and define two new RFSs Ψ 1 = Ψ ∩ T1 ,

Ψ 2 = Ψ ∩ T2 .

(4.111)

Ordinarily, it should not be possible for Ψ1 and Ψ2 to be independent, since they are both defined in terms of Ψ and thus are correlated. Because Ψ is a Poisson RFS, however, Ψ1 and Ψ2 are actually independent. This follows from the fact that Ψ1 , Ψ2 are Poisson with GΨ1 [h] = e(1T1 DΨ )[h−1] and GΨ2 [h] = e(1T2 DΨ )[h−1] —see Section K.5. 4.3.2

Identical, Independently Distributed Cluster (i.i.d.c.) RFSs

Intuitively speaking, an RFS Ψ is i.i.d.c. if its instantiations are constructed as follows. We are given a probability distribution pΨ (n) on the number of targets, and a probability density sΨ (y) on the targets themselves. From pΨ (n), draw an integer ν ∼ pΨ (·). Then from sΨ (y), draw ν elements y1 , ..., yν ∼ sΨ (·). Then Ψ = {y1 , ..., yν } is a particular instantiation of Ψ. Stated somewhat differently, an i.i.d.c. RFS is one in which the objects of Ψ are spatially distributed according to the distribution sΨ (y), and in which the probability distribution of the number of objects in Ψ is pΨ (n). An i.i.d.c. RFS is thus a direct generalization of the concept of a Poisson RFS. If the cardinality distribution pΨ (n) is Poisson then an i.i.d.c. RFS Ψ is a Poisson RFS. Formally, an RFS Ψ ⊆ Y is i.i.d.c. if its p.g.fl. has the form GΨ [h] = GΨ (sΨ [h])

(4.112)

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where GΨ (y) =

∑

pΨ (n) · y n

(4.113)

n≥0

is the p.g.f. of Ψ and where the linear-functional notation sΨ [h] was defined in (3.4). Other statistics associated with an i.i.d.c RFS Ψ are: • Multiobject probability distribution: fΨ (Y ) = |Y |! · pΨ (|Y |) · sYΨ

(4.114)

where the power functional notation sYΨ was defined in (3.5). • Expected number of objects: (1)

NΨ = GΨ (1) =

∑

n · pΨ (n).

(4.115)

n≥0

• PHD: DΨ (y) = NΨ · sΨ (y). • Variance:

(2)

2 σΨ = GΨ (1) − NΨ2 + NΨ .

(4.116) (4.117)

Note that the only interaction between the p.g.f. GΨ (y) and the PHD DΨ (y) is given by (4.115). That is, the integral of the PHD and the expected value of the cardinality distribution must be equal. 4.3.3

Bernoulli RFSs

An RFS Ψ is a Bernoulli RFS if |Ψ| ≤ 1, in which case we define Pr(|Ψ| = 1) = qΨ .

(4.118)

It is characterized by two items: a spatial distribution (a probability density) sΨ (y) and a probability of existence qΨ . The statistical descriptors of a Bernoulli RFS are: • p.g.fl.: GΨ [h] = 1 − qΨ + qΨ · sΨ [h].

(4.119)

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101

• Multiobject probability distribution:

fΞ (Y ) =

 

1 − qΨ qΨ · sΨ (y)  0

if if if

Y =∅ Y = {y} . |Y | ≥ 2

(4.120)

• p.g.f.: GΨ (y) = 1 − qΨ + qΨ · y.

(4.121)

DΨ (y) = qΨ · sΨ (y).

(4.122)

n pΨ (n) = C1,n · qΨ · (1 − qΨ )1−n

(4.123)

• PHD: • Cardinality distribution:

where the binomial coefficient Cn,i was defined in (2.1). • Expected number of objects: NΨ = qΨ .

(4.124)

2 σΨ = qΨ · (1 − qΨ ).

(4.125)

• Variance:

4.3.4

Multi-Bernoulli RFSs

An RFS is multi-Bernoulli if it is the union (superposition) of a finite number nΨ of independent Bernoulli RFSs. Its p.g.fl. therefore has the form ( ) nΨ nΨ 1 1 GΨ [h] = 1 − qΨ + qΨ · s1Ψ [h] · · · (1 − qΨ + qΨ · snΨΨ [h]) .

(4.126)

A multi-Bernoulli process can be more intuitively understood as follows: nΨ 1 • There are nΨ independent random objects YΨ , ..., YΨ ∈ Y with respecnΨ 1 tive probability distributions sΨ (y), ..., sΨ (y) and respective probabilities nΨ 1 of existence qΨ , ..., qΨ .

Let Y = {y1 , ..., yn } with |Y | = n. Then the other statistics associated with a multi-Bernoulli RFS Ψ are:

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• Multiobject probability distribution: (n ) Ψ ∏ i fΨ (Y ) = (1 − qΨ )

(4.127)

i=1

·

∑ 1≤i1 ̸=...̸=in ≤nΨ

i1 qΨ · siΨ1 (y1 ) q in · sin (yn ) · · · Ψ Ψ in . i1 1 − qΨ 1 − qΨ

• p.g.f.: ( ) nΨ nΨ 1 1 GΨ (y) = 1 − qΨ + qΨ · y · · · (1 − qΨ + qΨ · y) .

(4.128)

• PHD: nΨ 1 DΨ (y) = qΨ · s1Ψ (y) + ... + qΨ · snΨΨ (y).

(4.129)

• Cardinality distribution: If n > nΨ then pΨ (n) = 0 and, if otherwise, (n ) ( 1 ) Ψ nΨ ∏ qΨ qΨ i pΨ (n) = (1 − qΨ ) · σnΨ ,n (4.130) 1 , ..., 1 − q nΨ 1 − qΨ Ψ i=1 where σn,i (x1 , ..., xn ) is the elementary homogeneous symmetric function of degree i in n variables. • Expected number of objects: nΨ 1 NΨ = qΨ + ... + qΨ .

(4.131)

nΨ nΨ 2 1 1 (1 − qΨ ). σΨ = qΨ (1 − qΨ ) + ... + qΨ

(4.132)

• Variance: Equation (4.127) follows from [179], Eq. (11.133).7 Note that the variance of a multi-Bernoulli RFS cannot exceed the expected number of objects: 2 σΨ ≤ NΨ . (4.133) One consequence of this is that multi-Bernoulli filters cannot provide high-accuracy approximations of the multitarget Bayes filter, if target number is being poorly estimated. Equation (3.19) provides a specific example of (4.127), assuming that nΨ = 2 1 2 and qΨ = qΨ = q. 7

Erratum: Eq. (11.134) in [179] does not follow from Eq. (11.133) and is not correct.

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4.4

103

BASIC DERIVED RFSs

Certain important RFSs can be derived from those already considered. Two that will play important roles later in this book are: 1. Section 4.4.1: Censored RFSs—important for the theory of target prioritization as described in Section 25.14. 2. Section 4.4.2: Cluster RFSs—important for the theory of extended targets and group targets, as considered in Chapter 21. 4.4.1

Censored RFSs

Let Ψ ⊆ Y be an RFS and let T0 ⊆ Y be a closed subset. Then Ψ ∩ T0 is also an RFS, one in which Ψ has been “censored” by excluding from it all of the elements of T0c . The statistical descriptors of Ψ ∩ T0 can be shown to be ([94], pp. 164-165; [179], Eq. 14.302): • Multiobject probability distribution: fΨ∩T0 (Y ) = 1YT0 ·

δβΨ c (T0 ) δY

(4.134)

where the power functional notation 1YT0 was defined in (3.5). • p.g.fl.: GΨ∩T0 [h] = GΨ [1 − 1T0 + h · 1T0 ].

(4.135)

• Belief-mass function: βΨ∩T0 (T ) = βΨ∩T0 (T ∪ T0c ).

(4.136)

GΨ∩T0 (y) = GΨ [1 − 1T0 + y · 1T0 ].

(4.137)

• p.g.f.: • Cardinality distribution: ∫ δ n βΨ 1 n pΨ∩T0 (n) = (T c )dy1 · · · dyn . ?? ? δy1 · · · δyn 0 n! ? T0 × ... × T0

(4.138)

• PHD: DΨ∩T0 (x) = 1T0 (x) · DΨ (x).

(4.139)

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4.4.2

Cluster RFSs

Let us be given two spaces Y (a “mother space”) and D (a “daughter space”). A cluster RFS is an RFS of D that consists of two parts: • A mother RFS (also known as “germ process”) Ψ ⊆ Y. • A family ∆y ⊆ D of daughter RFSs (also known as “grain process”), parametrized by y ∈ Y. It is usually assumed that, for any n ≥ 2 and any distinct y1 , ..., yn , the ∆y1 , ..., ∆yn are independent (“independent daughters”). The cluster process determined by the mother and daughter processes is defined to be the RFS ∆ of D defined by ∆=

∪

∆y .

(4.140)

y∈Ψ

That is, an instantiation of ∆ is constructed by 1. Drawing a sample Y = {y1 , ..., yn } ∼ fΨ (·) from the mother process; 2. Drawing samples Dyi = {dyi ,1 , ..., dyi ,n(yi ) } ∼ f∆yi (·) from the daughter processes; and 3. Constructing the sample D = Dy1 ∪ ... ∪ Dyn of the cluster process. Because of the independence assumption, it can be shown that the joint probability distribution of Ψ and ∆ is GΨ,∆ [g, h] = GΨ [h · G∆∗ [g]]

(4.141)

where GΨ [h] is the p.g.fl. of the mother RFS Ψ; where abbr.

T [g](y) = G∆y [g] =

∫

g D · f∆y (D)δD

(4.142)

are the p.g.fl.’s of the daughter RFSs; and where GΨ [h · G∆∗ [g]] is shorthand for GΨ [h · T [g]]. In turn, the p.g.fl. of the cluster process ∆ itself is G∆ [g] = GΨ,∆ [g, 1] = GΨ [G∆∗ [g]].

(4.143)

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Equation (4.141) is true because ∫ GΨ,∆ [g, h] = g D · hY · fΨ,∆ (Y, D)δDδY ∫ = g D · hY · f∆|Ψ (D|Y ) · fΨ (Y )δDδY ∫ = hY · G∆|Ψ [g|Y ] · fΨ (Y )δY

(4.144) (4.145) (4.146)

If Ψ = Y = {y1 , ..., yn } with |Y | = n, then G∆|Ψ [g|Y ] is the p.g.fl. of ∆ = ∆y1 ∪ ... ∪ ∆yn . Since the daughters are independent, G∆|Ψ [g|Y ] factors into the product of the p.g.fl.’s of the daughters:

G∆|Ψ [g|Y ] =

n ∏

G∆yi [g] =

i=1

∏

T [g](y).

(4.147)

y∈Y

Thus

GΨ,∆ [g, h]

=

∫



hY · 

∏ y∈Y

=

∫

=

GΨ [h · T [g]].



T [g](y) · fΨ (Y )δY

(hT [g])Y · fΨ (Y )δDδY

(4.148)

(4.149) (4.150)

Example 2 (Measurement RFSs) The random measurement set Σ generated by a random target process Ξ is one example of a cluster RFS, with the state space X being the mother space and the measurement space Z being the daughter space. Let Σx be the set of measurements generated by a target with state x. Then Ξ is the mother RFS, the Σx are the daughter RFSs, and the total cluster RFS is (see (7.7)): ∪ Σ= Σx . (4.151) x∈Ξ

See Remark 17 in Section 8.2 for a more concrete example. Example 3 (Spawning models) The random target set Ξk+1|k of those targets at time tk+1 that are spawned by a random target set Ξk|k at time tk is another example of a cluster RFS. In this case the state space X is both the mother space

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and the daughter space. Let Ξx be the set of targets that were spawned by a target with state x. Then Ξk|k is the mother RFS, the Ξx are the daughter RFSs, and the cluster RFS is ∪ Ξk+1|k = Ξx . (4.152) x∈Ξk|k

Example 4 (Group targets) A group target (see Section 5.5) is a set of targets that belong to a single coordinated tactical group. Let X be the state space for the conventional targets, and let ˚ X be the state space for the group targets. Let Ξ˚ ⊆ X be the RFS of targets associated with the group target with state ˚ x, x and let ˚ Ξ ⊆ X be the RFS of group targets. Then the RFS of all conventional targets is ∪ Ξ= Ξ˚ (4.153) x. ˚ x∈˚ Ξ

Chapter 5 Multiobject Modeling and Filtering 5.1

INTRODUCTION

The finite-set statistics approach provides an explicit methodology for deriving the optimal solutions for multisource-multitarget problems. It consists of three main steps: 1. Step 1: Construction of multisensor-multitarget motion and measurement models in terms of RFSs. 2. Step 2: Construction, using multiobject calculus, of “true” multitarget Markov densities and “true” multisensor-multitarget likelihood functions from these RFS models. 3. Step 3: Utilization of the true multitarget Markov density and the true multitarget likelihood function in a multitarget Bayes filter. The purpose of this chapter is to describe these three steps in greater detail. It is organized as follows: 1. Section 5.2: The multisensor-multitarget Bayes filter. 2. Section 5.3: Multitarget Bayes optimality and Bayes-optimal multitarget state estimators. 3. Section 5.4: RFS multitarget motion models. 4. Section 5.5: RFS multitarget measurement models. 5. Section 5.6: Multitarget Markov density functions.

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6. Section 5.7: Multisensor-multitarget likelihood functions. 7. Section 5.8: The multisensor-multitarget Bayes filter in p.g.fl. form. 8. Section 5.9: The Bayes filter for “mixed-state” systems—that is, systems with a state variable of the form (˚ x, X) where ˚ x∈˚ X is the state of a single object and X ⊆ X is a finite set of conventional target states. 9. Section 5.10: A summary of principled approximate multisource-multitarget filters—PHD, CPHD, multi-Bernoulli, and Bernoulli.

5.2

THE MULTISENSOR-MULTITARGET BAYES FILTER

The multisensor-multitarget recursive Bayes filter is the theoretical foundation for multisensor-multitarget detection, tracking and identification ([179], Chapter 14). Let Z (k) : Z1 , ..., Zk be a time sequence (sample path) of measurement sets collected by all of the sensors. Then this filter propagates a multitarget posterior distribution fk|k (X|Z (k) ) through time:1 ... →

fk|k (X|Z (k) )

fk+1|k (X|Z (k) )

→

→

fk+1|k+1 (X|Z (k+1) )

It is defined by the time-update and measurement-update equations ∫ fk+1|k (X|Z (k) ) = fk+1|k (X|X ′ ) · fk|k (X ′ |Z (k) )δX ′ fk+1|k+1 (X|Z (k+1) )

=

→ ...

(5.1)

fk+1 (Zk+1 |X) · fk+1|k (X|Z (k) ) fk+1 (Zk+1 |Z (k) )

(5.2)

fk+1 (Zk+1 |X) · fk+1|k (X|Z (k) )δX

(5.3)

where fk+1 (Zk+1 |Z 1

(k)

)=

∫

Note: In two other commonly used systems of notation, the multitarget probability density fk+1|k (X|Z (k) ) can be written as fk+1|k (X|Z (k) ) = f (Xk+1 |Z1:k ) or as fk+1|k (X|Z (k) ) = fΞk+1|k |Σ1 ,....,Σk (X|Z1 , ..., Zk ) where Ξk+1|k is the predicted multitarget RFS at time tk+1 and Σj is the measurement RFS at time tj .

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is the Bayes normalization factor and where the integrals are set integrals, as defined in (3.11). The multitarget Bayes filter requires two a priori distributions: 1. The multitarget Markov transition density MX (X ′ ) = fk+1|k (X|X ′ ),

(5.4)

which is the probability (density) that targets with state set X will be present at time tk+1 if the targets at time tk had state set X ′ . 2. The multitarget likelihood function LZ (X) = fk+1 (Z|X),

(5.5)

which is the probability (density) that the measurement set Z will be collected at time tk+1 if the targets at time tk+1 have state set X. The question then becomes: How do we construct “true” formulas for these two items (in a sense to be defined shortly)? In analogy with Section 2.2, we first construct: 1. A RFS multitarget motion model Ξk+1|k = Ξk+1|k (X ′ ), which is the RFS of targets at time tk+1 , given that the targets at time tk have state set X ′ . This process is described in Section 5.4. 2. An RFS multitarget measurement model Σk+1 = Σk+1 (X), which is the RFS of measurements at time tk+1 , given that the targets at time tk+1 have state set X. This process is described in Section 5.5. Then we: • Convert the RFS motion model into a “true” multitarget Markov density. • Convert the RFS measurement model into a “true” multitarget likelihood function. By “true” is meant the following: • fk+1|k (X|X ′ ) contains exactly the same information as the original RFS multitarget motion model—no more and no less. That is, – No information in the model has been lost. If information has been lost, then this means that fk+1|k (X|X ′ ) does not faithfully preserve the information contained in the model.

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– No information extraneous to the motion model has been inadvertently added. For if information has been added, then an unrecognized statistical bias has been inserted into fk+1|k (X|X ′ ). • fk+1 (Z|X) contains exactly the same information as the original RFS multisensor-multitarget measurement model. That is, – No information has been lost. If information has been lost, then fk+1 (Z|X) does not faithfully preserve the model. – No extraneous information has been introduced. If information has been introduced, fk+1 (Z|X) will introduce an unrecognized statistical bias into the Bayesian analysis. The construction of true multitarget Markov densities and true multitarget likelihood functions requires the multitarget integral and differential calculus, which was introduced in Chapter 3. The two procedures are described in Sections 5.6 and 5.7, respectively.

5.3

MULTITARGET BAYES OPTIMALITY

Information of interest—the number of targets, their positions, velocities, types, and so on—can be extracted from the posterior distribution fk+1|k+1 (X|Z (k+1) ) using a Bayes-optimal multitarget state estimator. ˆ A multitarget state estimator is a function X(Z), the values of which are state sets and the argument of which is a measurement set Z. Let C(X, Y ) ≥ 0 be a multitarget cost function defined on state sets X, Y . This means that C(X, Y ) = C(Y, X) and that C(X, Y ) = 0 implies X = Y . The posterior ˆ given the measurement set Z, is the cost averaged with respect to the cost of X, posterior distribution: ∫ ¯ X|Z) ˆ ˆ C( = C(X, X(Z)) · fk+1|k+1 (X|Z (k) , Z)δX. (5.6) The Bayes risk is the average posterior cost (with respect to all possible measurement sets): ∫ ˆ ¯ X|Z)] ˆ ¯ X|Z) ˆ R(X) = EZ [C( = C( · fk+1 (Z|Z (k) )δZ (5.7) ∫ ˆ = C(X, X(Z)) · fk+1 (Z|X) · fk+1|k (X|Z (k) )δXδZ. (5.8)

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ˆ is Bayes-optimal (with respect to the cost The multitarget state estimator X function C(X, Y )) if it minimizes the Bayes risk ([94], pp. 189-190, and [179], p. 63). This is the only theoretically rigorous meaning of the term “Bayes-optimal” when applied to multitarget filtering. An example of a Bayes-optimal multitarget state estimator is the joint multitarget (JoM) estimator ([179], Section 14.5.3, pp. 498-505): |X| ˆ k+1|k+1 = arg sup c X · fk+1|k+1 (X|Z (k+1) ) X∈X∞ |X|!

(5.9)

where c > 0 is a constant with the same units of measurement as x. Another example is the marginal multitarget (MaM) estimator ([179], Section 14.5.2, pp. 497-498) ˆ k+1|k+1 = X

arg sup

fk+1|k+1 ({x1 , ..., xnˆ k+1|k+1 }|Z (k+1) )

(5.10)

x1 ,...,xn ˆ k+1|k+1 ∈X

where n ˆ k+1|k+1 = arg sup pk+1|k+1 (n|Z (k+1) )

(5.11)

n≥0

and where pk+1|k+1 (n|Z (k+1) ) is, as defined in (4.62), the cardinality distribution of fk+1|k+1 (X|Z (k+1) ). Remark 13 (Transient events) In a few applications, measurements are available only at a single instant of time—at tk , say. This occurs, for example, in static data-clustering (see Section 21.6). The targets are considered to be motionless and the goal is to localize rather than track them. If the multitarget prior distribution f0 (X) is unknown, two approaches can be employed. The first is to assume that f0 (X) is a multitarget uniform distribution—that is, a distribution that is uniform in regard to both target number and target state (see [179], pp. 366-367). Then the JoM or MaM estimators can be applied to the corresponding posterior distribution. Alternatively, one can abandon the Bayesian approach in favor of a maximumlikelihood approach. If Z0 is the measurement set that has been collected, the multitarget state can be estimated using the multitarget version of the maximum likelihood estimator (MMLE): ˆ 0 = arg sup f0 (Z0 |X) X

(5.12)

X

where f0 (Z|X) is the multitarget likelihood function for the sensor (see [179], pp. 500-501).

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RFS MULTITARGET MOTION MODELS

Just as single-target motion can be modeled using a motion model Xk+1|k = φk (x, Wk ),

(5.13)

so the motion of multitarget systems can be modeled using a multitarget motion model of the general form previous targets

Ξk+1|k

new targets

? ?? ? ? ?? ? = Tk+1|k (X ′ ) ∪ Bk+1|k .

(5.14)

Let X ′ = {x′1 , ..., x′n } with |X ′ | = n. Then it is typically assumed that Tk+1|k (X ′ ) = Tk+1|k (x′1 ) ∪ ... ∪ Tk+1|k (x′n )

(5.15)

where Tk+1|k (x′ ) is the RFS of targets at time tk+1 that originated in some fashion from a target at time tk with state x′ . In turn, Tk+1|k (x′ ) can have the form per sp Tk+1|k (x′ ) = Tk+1|k (x′ ) ∪ Tk+1|k (x′ ) (5.16) sp where Tk+1|k (x′ ) is the RFS of new targets spawned by x′ ; and where per per ′ Tk+1|k (x ) models the persistence of x′ itself—that is, either Tk+1|k (x′ ) = ∅ per (the target x′ disappeared from the scene), or |Tk+1|k (x′ )| = 1 (the target x′ persisted). There are two cases to consider:

1. Uncoordinated multitarget motion: The motions of all targets are statistically independent—that is, unrelated to each other moment-to-moment. A simple example will be presented shortly in Section 5.6. 2. Coordinated multitarget motion ([179], pp. 478-482): The motions of the targets are related in some manner, and thus are not statistically independent. The following special case of uncoordinated motion is of particular interest: • Standard Multitarget Motion Model without Spawning: sp per – Tk+1|k (x′ ) = ∅ for all x′ , in which case Tk+1|k (x′ ) = Tk+1|k (x′ ).

– Tk+1|k (x′1 ), ..., Tk+1|k (x′n ), Bk+1|k are statistically independent.

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– Bk+1|k is Poisson. • Standard Multitarget Motion Model with Spawning: sp – Tk+1|k (x′ ) ̸= ∅ for all x′ . per per sp sp – Tk+1|k (x′1 ), ..., Tk+1|k (x′n ), Tk+1|k (x′1 ), ..., Tk+1|k (x′n ), Bk+1|k statistically independent.

are

– Bk+1|k is Poisson. It should also be pointed out that the target appearance RFS Bk+1|k can be used to implement optimal search, the purpose of which is to detect currently undetected targets. In conjunction with sensor management and platform management, Bk+1|k can be chosen so as to implement whatever search strategy has been chosen.

5.5

RFS MULTITARGET MEASUREMENT MODELS

Just as single-sensor, single-target data can be modeled using a measurement model of the form Zk+1 = ηk+1 (x, Vk ), (5.17) so multitarget multisensor data can be modeled using a multisensor-multitarget measurement model. Assume that the measurements from s sensors are collected at approximately the same time, tk+1 . First, the random RFS Σk+1 of measurements has the form 1

s

(5.18)

Σk+1 = Σk+1 ⊎ ... ⊎ Σk+1 j

j

where Σk+1 ⊆ Z is the random measurement set generated by the jth sensor, 1

s

and where ‘⊎’ indicates disjoint union. Typically, Σk+1 , ..., Σk+1 are assumed to be conditionally independent of the multitarget state. j

Second, each of the Σk+1 has the form target-generated measurements j

Σk+1 =

?

j

??

?

Υk+1 (X)

clutter

?

j

??

?

∪ C k+1 (X)

(5.19)

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where Υk+1 (X) is the random set of target-generated measurements and Ck+1 (X) is the random set of clutter-generated measurements. The latter functionally depends on X because, in some applications, clutter statistics can depend on the states of the targets. The basic issue, then, is the form of Υk+1 (X)—where hereafter the sensor index j will be suppressed for the sake of notational clarity. Most familiar sensors rely on wave phenomena such as acoustic waves and electromagnetic waves. When these waves impinge on targets, sensor signatures are generated and collected by the sensor. Such signatures can be modeled using a superpositional measurement model that generalizes the single-target (2.22): ∑ Zk+1 = ηk+1 (x) + Vk+1 . (5.20) x∈X

Here, X = {x1 , ..., xn }, n ≥ 0, is a set of target states; and Zk+1 is a random signature (for example, an image) or a random real- or complex-valued vector. That is, the generated measurement is a superposition of signals generated by all of the targets present (if any). Superpositional models will be discussed in more detail in Chapter 19. Equation (5.20) is typically so computationally involved that it must first be simplified using some sort of preprocessing methodology. Typical examples are: • In radar, computing the real part Re(Zk+1 ) of Zk+1 and then applying some sort of peak detector (threshold) to it. • Applying a “blob detector” to a camera image, and selecting the centroids of the blobs. • Applying a wavelet-coefficient detector to a high range-resolution radar (HRRR) signature. In each of these cases, each signature will typically generate a finite set of “detection measurements” or “detections.” A detection measurement model results if we assume that any measurement is generated by at most a single target. In this case, the measurement model will have the general form measurements

meas’s (target x1 )

meas’s (target xn )

? ?? ? ? ?? ? ?clutter ? ?? ? ?? ? Σk+1 = Υk+1 (x1 ) ∪... ∪ Υk+1 (x1 ) ∪ Ck+1

(5.21)

where Υk+1 (x) is the RFS of measurements generated by a target with state x; and where Ck+1 is the RFS of remaining measurements—that is, false detections and/or clutter.

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Given (5.21), there are four possibilities (see [179], p. 433, Figure 12.2): 1. Point (also known as “small”) targets: Targets are far enough away from the sensor that each generates at most a single measurement—that is, |Υk+1 (x)| is no larger than 1. At the same time, they are separated enough (relative to sensor resolution) that they are resolvable as separate targets. Point targets are the focus of Part II. 2. Extended targets ([179], pp. 427–432): Each measurement originates with a single physical target, but this target can generate multiple measurements. That is, the number |Υk+1 (x)| of measurements generated by a single target can be arbitrarily large or small. This is because, typically, the target is close enough to the sensor that multiple measurements are generated by discernible scatterers distributed on the target’s surface. (a) The state variables of an extended target can include centroid, centroidal velocity, target type, and target-shape parameters. (b) Extended targets are addressed in Chapter 21. 3. Group targets. Once again |Υk+1 (x)| can be arbitrary. But in this case, measurements are generated by autonomous or semi-autonomous point targets that, collectively, constitute a tactically integrated “meta-target.” Examples include platoons or regiments, aircraft sorties, and aircraft carrier groups. (a) The state variables of a group target can include centroid, centroidal velocity, formation type, number of targets in the group, and groupshape parameters. (b) Group targets differ from extended targets primarily in that the former can interpenetrate whereas the latter, having physical extent, cannot overlap. (c) Group targets are addressed in Chapter 21. 4. Unresolved targets ([179], pp. 432-444): Targets are so far away that they appear to be colocated at a single point. Mathematically speaking, they can be modeled as pairs ˚ x = (n, x) where n is the number of targets colocated at the same state x. Thus the value of |Υ(n, x)| can be arbitrary. Unresolved targets are one of the subjects of Chapter 21. Two special cases of these general models are of particular interest:

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• Standard Multitarget Measurement Model (described in more detail in Section 7.2): – Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 are statistically independent. – |Υk+1 (x)| ≤ 1

for all

x (the small-target case).

– Ck+1 is Poisson. • Generalized Standard Multitarget Measurement Model: – Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 are statistically independent. – |Υk+1 (x)| can be arbitrary (the extended-target or group target case), for all x. – Ck+1 can be arbitrary. It is necessary to discuss another situation that can give rise to the generalized standard model. In general, it is possible for clutter to be dependent on the state of the target. For example, clutter may be more dense in the vicinity of a target than elsewhere. Or, multiple returns may be created because of multipath conditions. In this case, the clutter-generated targets can have the form

Ck+1 (X)

state-dependent clutter

independent clutter

? ?? ? 1 Ck+1 (X)

? ?? ? 0 Ck+1

= =

(

∪

1 Ck+1 (x)

∪ )

0 ∪ Ck+1

(5.22) (5.23)

x∈X 1 where Ck+1 (x) is the random set of clutter measurements associated with 0 the target with state x. Also, Ck+1 is the random set of clutter measurements that have no dependence on target states. Typically, if |X| = n then 1 1 0 Ck+1 (x1 ), ..., Ck+1 (xn ), Ck+1 are assumed to be independent. But most typically, 0 it is assumed that Ck+1 (X) = Ck+1 —that is, clutter has no dependence on the target states. A simple example will be presented shortly in Section 5.7. State-dependent clutter is considered in greater detail in Section 8.7.

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5.6

117

MULTITARGET MARKOV DENSITIES

Just as the Markov transition density fk+1|k (x|x′ ) can be derived from the probability mass function pk+1|k (S|x′ ) = Pr(Xk+1|k ∈ S|Xk|k = x′ )

(5.24)

of the single-target motion model, so the true multitarget Markov transition density fk+1|k (X|X ′ ) can be derived from the belief-mass function βk+1|k (S|X ′ ) = Pr(Ξk+1|k ⊆ S|Ξk|k = X ′ )

(5.25)

of the RFS multitarget motion model. This is accomplished using the set derivative: fk+1|k (X|X ′ ) =

δβk+1|k (∅|X ′ ). δX

(5.26)

As an example, assume that targets do not spawn other targets. That is, a per target either persists or it disappears, in which case |Tk+1|k (x′ )| = |Tk+1|k (x′ )| ≤ 1. Define the probability of target survival to be pS (x′ ) abbr. = pS,k+1|k (x′ ) def. = Pr(Tk+1|k (x′ ) ̸= ∅).

(5.27)

Then the belief-mass function of Tk+1|k (x′ ) is

=

βk+1|k (S|x′ ) Pr(Tk+1|k (x′ ) ⊆ S)

= =

Pr(Tk+1|k (x′ ) = ∅) + Pr(Tk+1|k (x′ ) ̸= ∅, Tk+1|k (x′ ) ⊆ S) (5.29) 1 − pS (x′ ) + pS (x′ ) · Pr(Tk+1|k (x′ ) ⊆ S|Tk+1|k (x′ ) ̸= ∅). (5.30)

(5.28)

Since Tk+1|k (x′ ) is a singleton set if it is nonempty, the final factor is a probabilitymass function pk+1|k (S|x′ ) with density function fk+1|k (x|x′ ), and so βk+1|k (S|x′ ) = 1 − pS (x′ ) + pS (x′ ) · pk+1|k (S|x′ ).

(5.31)

There are different formulas for fk+1|k (X|X ′ ), depending on the number of elements in X ′ . Only the cases X ′ = ∅ and X ′ = {x′ } will be considered here—for the general case, see (7.66).

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• X ′ = ∅: fk+1|k (X|∅) =

{

1 0

if if

X′ = ∅ X ′ ̸= ∅

(5.32)

with p.g.fl. (5.33)

Gk+1|k [h|∅] = 1. ′

′

• X = {x }:  

1 − pS (x′ ) pS (x ) · fk+1|k (x|x′ ) fk+1|k (X|{x }) =  0 ′

′

if if if

X=∅ X = {x}} |X| ≥ 2

(5.34)

with p.g.fl. Gk+1|k [h|x′ ] = 1 − pS (x′ ) + pS (x′ ) · Mh (x′ ) where ′

def.

Mh (x ) =

5.7

∫

h(x) · fk+1|k (x|x′ )dx.

(5.35)

(5.36)

MULTISENSOR-MULTITARGET LIKELIHOOD FUNCTIONS

Just as the single-sensor, single-target likelihood function fk+1 (z|x) derived from the probability mass function pk+1 (T |x) = Pr(Zk+1 ∈ T |Xk+1|k = x)

can be

(5.37)

of the measurement model, so the single-sensor, multitarget likelihood function fk (Z|Xk ) can be derived from the belief-mass function βk+1 (T |X) = Pr(Σk+1 ⊆ T |Ξk+1|k = X)

(5.38)

of the multisensor-multitarget measurement model. This is accomplished using the set derivative: δβk+1 LZ (X) abbr. = fk+1 (Z|X) = (∅|X). (5.39) δZ Multisensor case: In the multisensor case, a measurement set will have the form 1 s Zk+1 = Z k+1 ⊎ ... ⊎ Z k+1 (5.40)

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j

where Z k+1 is the measurement set collected by the jth sensor. Assuming that measurements are conditionally independent of state, the corresponding multisensor likelihood function will be 1

s

1

s

fk+1 (Z|X) = f k+1 (Z k+1 |X) · · · f k+1 (Z k+1 |X) j

(5.41)

j

where f k+1 (Z k+1 |X) is the likelihood function for the jth sensor. As an example, assume that a target generates either a single measurement or no measurement at all. Define the probability of target detection to be pD (x) abbr. = pD,k+ (x) def. = Pr(Υk+1 (x) ̸= ∅).

(5.42)

Then the belief-mass function of Υk+1 (x) is βk+1 (T |x) = = =

Pr(Υk+1 (x) ⊆ T ) Pr(Υk+1 (x) = ∅) + Pr(Υk+1 (x) ̸= ∅, Υk+1 (x) ⊆ T ) 1 − pD (x) + pD (x) · Pr(Υk+1 (x) ⊆ T |Υk+1 (x) ̸= ∅)

(5.43) (5.44) (5.45)

=

1 − pD (x) + pD (x) · pk+1 (T |x)

(5.46)

where pk+1 (T |x) is a probability-mass function with density function Lz (x) = fk+1 (z|x). There are different formulas for fk+1 (Z|X), depending on the number of elements in X. Only the cases X = ∅ and X = {x} will be considered here— for the general case, see (7.21). • X = ∅: fk+1 (Z|∅) =

{

1 0

if if

Z=∅ Z ̸= ∅

(5.47)

with p.g.fl. (5.48)

Gk+1 [g|∅] = 1. • X = {x}:

fk+1 (Z|{x}) =

 

1 − pD (x) pD (x) · fk+1 (z|x)  0

if if if

Z=∅ Z = {z}} |Z| ≥ 2

(5.49)

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with p.g.fl. Gk+1 [g|x] = 1 − pD (x) + pD (x) · Lg (x) where Lg (x) =

5.8

∫

g(z) · fk+1 (z|x)dz.

(5.50)

(5.51)

THE MULTITARGET BAYES FILTER IN p.g.fl. FORM

The finite-set statistics approach is based on the derivation of approximate multisensormultitarget filters from the p.g.fl. form of the multisensor-multitarget Bayes filter. The first step is to replace the multitarget Bayes filter ... →

fk|k (X|Z (k) )

→

fk+1|k (X|Z (k) )

→

fk+1|k+1 (X|Z (k+1) )

→ ...

with a filter on the p.g.fl.’s of its multitarget distributions ... →

Gk|k [h|Z (k) ]

→

Gk+1|k [h|Z (k) ]

→

Gk+1|k+1 [h|Z (k+1) ]

→ ...

The p.g.fl. form of the filter neither loses information nor inadvertently introduces extraneous information, as compared with the multitarget Bayes filter. This is because of the relationship (see (4.88)) that relates p.g.fl.’s with their corresponding multitarget probability distributions: fk|k (X|Z

(k)

[ ] δGk|k (k) def. δGk|k (k) )= [0|Z ] = [h|Z ] . δX δX h=0

(5.52)

The second step is to express Gk+1|k [h|Z (k) ] in terms of Gk|k [h|Z (k) ] and Gk+1|k+1 [h|Z (k+1) ] in terms of Gk+1|k [h|Z (k) ], as explained in the next two subsections. 5.8.1

The p.g.fl. Time Update Equation

This is Gk+1|k [h|Z

(k)

]=

∫

Gk+1|k [h|X ′ ] · fk|k (X ′ |Z (k) )δX ′

where ′

Gk+1|k [h|X ] =

∫

hX · fk+1|k (X|X ′ )δX

(5.53)

(5.54)

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is the p.g.fl. of the multitarget Markov transition density fk+1|k (X|X ′ ). Equation (5.53) immediately follows from set-integration of both sides of (5.1): ∫ Gk+1|k [h|Z (k) ] = hX · fk+1|k (X|Z (k) )δX (5.55) ) ∫ (∫ = fk+1|k (X|X ′ )δX · fk|k (X ′ |Z (k) )δX ′ (5.56) ∫ = Gk+1|k [h|X ′ ] · fk|k (X ′ |Z (k) )δX ′ . (5.57) 5.8.2

The p.g.fl. Measurement Update Equation

The p.g.fl. form of Bayes’ rule is

Gk+1|k+1 [h|Z (k+1) ] =

δFk+1 δZk+1 [0, h] δFk+1 δZk+1 [0, 1]

[

δFk+1 δZk+1 [g, h]

def.

= [

δFk+1 δZk+1 [g, h]

]

] g=0

,

(5.58)

g=0,h=1

where the bivariate p.g.fl. Fk+1 [g, h] of the joint target-measurement RFS Σk+1 ⊎ Ξk+1|k ⊆ Z ⊎ X is defined as2 Fk+1 [g, h] =

∫

hX · Gk+1 [g|X] · fk+1|k (X|Z (k) )δX

where Gk+1 [g|X] =

∫

g Z · fk+1 (Z|X)δZ

is the p.g.fl. of the multisensor-multitarget likelihood function fk+1 (Z|X). 2

See Section 4.2.5 for a discussion of multivariate p.g.fl.’s).

(5.59)

(5.60)

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Equation (5.2) follows from [

δFk+1 [g, h] δZk+1

]

=

[∫

g=0

=

∫

hX ·

δGk+1 [g|X] · fk+1|k (X|Z (k) )δX δZk+1

]

hX · fk+1 (Zk+1 |X) · fk+1|k (X|Z (k) )δX

(5.61) g=0

(5.62)

=

fk+1 (Zk+1 |Z (k) ) ∫ · hX · fk+1|k+1 (X|Z (k+1) )δX

(5.63)

=

fk+1 (Zk+1 |Z (k) ) · Gk+1|k+1 [h|Z (k+1) ].

(5.64)

For, we then get δFk+1|k+1 [0, h] δZ δFk+1|k+1 [0, 1] δZ

5.9

=

fk+1 (Zk+1 |Z (k) ) · Gk+1|k+1 [h|Z (k+1) ] fk+1 (Zk+1 |Z (k) ) · Gk+1|k+1 [1|Z (k+1) ]

(5.65)

=

Gk+1|k+1 [h|Z (k+1) ].

(5.66)

THE FACTORED MULTITARGET BAYES FILTER

In some applications, the state of the system will have the mixed form (˚ x, X) where ˚ x ∈˚ X is the state of a single object and X is a finite subset of states x ∈ X. Examples of such applications include: • Simultaneous localization and mapping (SLAM) [210], [208], [1] in which case: – X is the set of reference landmarks or reference features (usually assumed to be static). – ˚ x is the state-vector of the robot. • Detection and tracking of single group targets (Section 21.9.3), in which case: – ˚ x is the state of the group target. – X is the set of (conventional) targets of which it is comprised.

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• Joint multitarget tracking and sensor registration (Chapter 12), in which case: – X is the usual multitarget state set. – ˚ x is the vector of sensor biases for all sensors. • Sensor management (Part V), in which case – X is the multitarget state. – ˚ x is the joint state of all of the sensors. The optimal Bayes filter for this problem has the form ... → fk|k (˚ x, X|Z (k) ) → fk+1|k (˚ x, X|Z (k) ) → fk+1|k+1 (˚ x, X|Z (k+1) ) → ... where fk+1|k (˚ x, X|Z

(k)

)

=

∫

fk+1|k (˚ x, X|˚ x′ , X ′ )

(5.67)

·fk|k (˚ x′ , X ′ |Z (k) )d˚ x′ δX ′ fk+1|k+1 (˚ x, X|Z (k+1) )

=

fk+1 (Zk+1 |Z (k) )

=

fk+1 (Zk+1 |˚ x, X) · fk+1|k (˚ x, X|Z (k) ) (5.68) (k) fk+1 (Zk+1 |Z ) ∫ fk+1 (Zk+1 |˚ x, X) (5.69) ·fk+1|k (˚ x, X|Z (k) )d˚ xδX

and where fk+1|k (˚ x, X|˚ x′ , X ′ ) and fk+1 (Z|˚ x, X) are, respectively, the Markov transition density and likelihood function for the hybrid-state system. This mixed-state Bayes filter can, under certain assumptions, be restated in an often more useful “factored” form.3 By Bayes’ rule, fk|k (˚ x, X|Z (k) ) = fk|k (˚ x|Z (k) ) · fk|k (X|Z (k) ,˚ x)

(5.70)

where fk|k (˚ x|Z (k) ) is a probability distribution on ˚ x and fk|k (X|˚ x, Z (k) ) is a multitarget probability distribution on X. The Markov density can similarly be 3

The concept of dimensional reduction using density factorization appears to be due to Murphy and Russell in 1996, who employed it for Rao-Blackwellization of particle filters [213]. In robotics, it is the theoretical basis for both the FastSLAM algorithm [204] and the RFS approach to SLAM [210], [208].

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factored as fk+1|k (˚ x, X|X ′ ,˚ x′ ) = fk+1|k (˚ x|˚ x′ , X ′ ) · fk+1|k (X|˚ x,˚ x′ , X ′ ).

(5.71)

Assume that the future single-object state does not depend on the original multiobject state, and that the future multiobject state does not depend on the original single-object state: fk+1|k (˚ x|˚ x′ , X ′ ) fk+1|k (X|˚ x,˚ x′ , X ′ )

= =

fk+1|k (˚ x|˚ x′ ) fk+1|k (X|X ′ ,˚ x′ ).

(5.72) (5.73)

Then it is shown in Section K.4 that the hybrid-state Bayes filter can be written in factored form as two coupled filters: ... → fk|k (˚ x|Z (k) ) ... → fk|k (X|Z (k) ,˚ x)

→ fk+1|k (˚ x|Z (k) ) → ↑↓ → fk+1|k (X|Z (k) ,˚ x) →

fk+1|k+1 (˚ x|Z (k+1) ) → ... ↑↓ fk+1|k+1 (X|Z (k+1) ,˚ x) → ...

These filters are defined by the following equations: • Mixed-state time-update: fk+1|k (˚ x|Z (k) ) fk+1|k (X|Z

(k)

,˚ x)

=

∫

fk+1|k (˚ x|˚ x′ ) · fk|k (˚ x′ |Z (k) )d˚ x′ (5.74)

=

∫

f˜k+1|k (X|Z (k) ,˚ x′ )

(5.75)

·fk|k+1 (˚ x′ |˚ x, Z (k) )d˚ x′ where the top equation is a conventional single-target time-update; where, for fixed ˚ x′ , ∫ f˜k+1|k (X|Z (k) ,˚ x′ ) = fk+1|k (X|X ′ ,˚ x′ ) · fk|k (X ′ |Z (k) ,˚ x′ )δX ′ (5.76) is a conventional multitarget time-update; and where fk|k+1 (˚ x′ |˚ x, Z (k) ) =

fk+1|k (˚ x|˚ x′ ) · fk|k (˚ x′ |Z (k) ) fk+1|k (˚ x|Z (k) )

is a reverse Markov density (“retrodictive density”).

(5.77)

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• Mixed-state measurement-update: (

fk+1|k+1 (˚ x|Z (k+1) )

=

fk+1|k+1 (X|Z (k+1) ,˚ x)

=

) fk+1 (Zk+1 |˚ x, Z (k) ) ·fk+1|k (˚ x|Z (k) ) fk+1 (Zk+1 |Z (k) ) ( ) fk+1 (Zk+1 |˚ x, X) ·fk+1|k (X|Z (k) ,˚ x) (k) fk+1 (Zk+1 |Z ,˚ x)

(5.78)

(5.79)

where the top equation is a conventional single-target measurement-update; where the bottom equation is, for fixed ˚ x, a conventional multitarget measurement-update; and where fk+1 (Zk+1 |Z

(k)

,˚ x)

fk+1 (Zk+1 |Z (k) )

=

∫

fk+1 (Zk+1 |˚ x, X)

(5.80)

=

·fk+1|k (X|Z (k) ,˚ x)δX ∫ fk+1 (Zk+1 |Z (k) ,˚ x)

(5.81)

·fk+1|k (˚ x|Z (k) )d˚ x.

5.10

APPROXIMATE MULTITARGET FILTERS

The purpose of this section is to summarize the finite-set statistics strategy for deriving approximate multitarget filters. It is organized as follows: 1. Section 5.10.1: The p.g.fl. time-update equation, assuming conditionally independent Time evolution of targets. 2. Section 5.10.2: The p.g.fl. measurement-update equation, assuming conditionally independent generation of measurements. 3. Section 5.10.3: The finite-set statistics methodology for devising approximate multitarget filters. 4. Section 5.10.4: PHD filters in the general sense of the term. 5. Section 5.10.5: CPHD filters in the general sense of the term. 6. Section 5.10.6: Multi-Bernoulli filters in the general sense of the term.

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7. Section 5.10.7: Bernoulli filters in the general sense of the term. 5.10.1

The p.g.fl. Time Update for Independent Targets

The p.g.fl. form of the multitarget Bayes filter was described in Section 5.8. The purpose of this section is to derive, under fairly general assumptions, a concrete formula for the predicted p.g.fl. Gk+1|k [h|Z (k) ] in terms of the previous p.g.fl. Gk|k [h|Z (k) ]—see (5.83). RFS multitarget motion models were discussed in Section 5.4. Let X ′ = ′ {x1 , ..., x′n } with |X| = n and assume that the RFS motion model has the form Ξk+1|k = Tk+1|k (x′1 ) ∪ ... ∪ Tk+1|k (x′n ) ∪ Bk+1|k

(5.82)

where: • Tk+1|k (x′ ) is the RFS of targets at time tk+1 , originating with a target with state x′ at time tk ; • Bk+1|k is the random set of newly appearing targets. • Tk+1|k (x′1 ), ..., Tk+1|k (x′n ), Bk+1|k are statistically independent. Given this, the predicted p.g.fl. Gk+1|k [h] is related to the original p.g.fl. Gk|k [h] by Gk+1|k [h] = GB (5.83) k+1|k [h] · Gk|k [Qk+1|k [h]], where GB k+1|k [h] is the p.g.fl. of Bk+1|k ; where h ?→ Qk+1|k [h] is the functional transformation defined by Qk+1|k [h](x′ ) = Gk+1|k [h|x′ ]; and where Gk+1|k [h|x′ ] is the p.g.fl. of Tk+1|k (x′ ). To see why (5.83) is true, note that from (5.53), Gk+1|k [h] =

∫

Gk+1|k [h|X ′ ] · fk|k (X ′ |Z (k) )δX ′ .

(5.84)

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Because of independence, Gk+1|k [h|X ′ ]

GB k+1|k [h]

=

∏

Gk+1|k [h|x′ ]

(5.85)

Qk+1|k [h](x′ )

(5.86)

x′ ∈X ′

GB k+1|k [h]

=

∏

x′ ∈X ′ X GB k+1|k [h] · Qk+1|k [h]

=

′

(5.87)

where the power-functional notation hX was defined in (3.5). Thus Gk+1|k [h]

∫

′

Qk+1|k [h]X · fk|k (X ′ |Z (k) )δX ′

=

GB k+1|k [h]

=

GB k+1|k [h] · Gk|k [Qk+1|k [h]].

(5.88) (5.89)

As a simple example, assume that there are no target appearances of any kind. In this case Bk+1|k = ∅ and |Tk+1|k (x′ )| ≤ 1. (This is what was assumed for the example presented at the end of Section 5.6.) From (5.35) we know that the p.g.fl. of Tk+1|k (x′ ) is Qk+1|k [h](x′ ) = Gk+1|k [h|x′ ] = 1 − pS (x′ ) + pS (x′ ) · Mh (x′ )

(5.90)

or, alternatively, (5.91)

Qk+1|k [h] = 1 − pS + pS · Mh where Mh (x′ ) =

∫

h(x) · fk+1|k (x|x′ )dx.

(5.92)

Thus Gk+1|k [h|X ′ ] = (1 − pS + pS Mh )

X′

(5.93)

and so the p.g.fl. of the predicted-target RFS Ξk+1|k is: Gk+1|k [h]

=

GB k+1|k [h] · Gk|k [1 − pS + pS Mh ]

(5.94)

=

Gk|k [1 − pS + pS Mh ].

(5.95)

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5.10.2

The p.g.fl. Measurement Update for Independent Measurements

The p.g.fl. form of the multitarget Bayes filter was described in Section 5.8. The purpose of this section is to derive, under fairly general assumptions, a concrete formula for the measurement-updated p.g.fl. Gk+1|k+1 [h|Z (k+1) ] in terms of the predicted p.g.fl. Gk+1|k [h|Z (k) ]—see (5.99). RFS multitarget measurement models were discussed in Section 5.5. Let X = {x1 , ..., xn } with |X| = n and assume that the single-sensor, multitarget measurement model has the form Σk+1 = Υk+1 (x1 ) ∪ ... ∪ Υk+1 (xn ) ∪ Ck+1

(5.96)

where: • Υk+1 (x) is the RFS of measurements (including both target-generated measurements and state-dependent clutter measurements) associated with a target that has state x at time tk+1 . • Ck+1 is the random set of clutter measurements generated by the background at time tk+1 . • Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 are statistically independent. Given this, the measurement-updated p.g.fl. Gk+1|k+1 [h] can be expressed in terms of the predicted p.g.fl. Gk+1|k [h]. Specifically, the bivariate p.g.fl. of (5.59) has the form Fk+1 [g, h] = Gκk+1 [g] · Gk+1|k [h · Tk+1 [g]] (5.97) where Gκk+1 [g] denotes the p.g.fl. of Ck+1 , and where the functional transformation g ?→ Rk+1 [g] is defined as (5.98)

Rk+1 [g](x) = Gk+1 [g|x],

where Gk+1 [g|x] is the p.g.fl. of Υk+1 (x). Consequently, from (5.58) the measurement-updated p.g.fl. has the form [ ( κ )] δ G [g] · G [h · R [g]] k+1 k+1|k k+1 δZk+1 g=0 . (5.99) Gk+1|k+1 [h] = [ ( κ )] δ G [g] · G [h · R [g]] k+1 k+1|k k+1 δZk+1 g=0,h=1

As a simple example, suppose that there is no clutter of any kind, so that Ck+1 = ∅ and |Υk+1 (x)| ≤ 1. (This is what was assumed for the example at the

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end of Section 5.7.) From (5.50) we know that the p.g.fl. of Υk+1 (x) is (5.100)

Rk+1 [g](x) = Gk+1|k [g|x] = 1 − pD (x) + pD (x) · Lg (x) or, alternatively,

(5.101)

Rk+1 [g](x) = 1 − pD + pD Lg where Lg (x) =

∫

(5.102)

g(z) · fk+1 (z|x)dz.

Thus the p.g.fl. of Σk+1 is X

(5.103)

Gk+1 [g|X] = (1 − pD + pD Lg ) and so Fk+1 [g, h]

5.10.3

= =

Gκk+1 [g] · Gk+1|k [h(1 − pD + pD Lg )] Gk+1|k [h(1 − pD + pD Lg )].

(5.104) (5.105)

A Principled Approximation Methodology

Given suitable independence assumptions, we know from (5.83) and (5.99) that the time-update and measurement-update equations for the p.g.fl. Bayes filter are

=

Gk+1|k [h] = GB k+1|k [h] · Gk|k [Qk+1|k [h]] Gk+1|k+1 [h] [ ( κ )] δ G [g] · G [h · R [g]] k+1 k+1|k k+1 δZk+1 g=0 [ ( κ )] δ G [g] · G [h · R [g]] k+1 k+1|k k+1 δZk+1

(5.106) (5.107)

.

g=0,h=1

Suppose that we are given a multitarget motion model and a single-sensor, multitarget measurement model. Then we can derive various approximate multitarget filters by: 1. Choosing carefully specified simplifying approximate forms for Gk+1|k [h] and Gk+1|k+1 [h]. 2. Applying the product rule for functional derivatives, (3.70), followed by 3. Application of Clark’s general chain rule, (3.91). The most common simplifying assumptions employ the Poisson, i.i.d.c., and multi-Bernoulli processes as discussed in Sections 4.3.1, 4.3.2, and 4.3.4.

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5.10.4

Poisson Approximation: PHD Filters

This section introduces the following concepts: • PHD filters in the general sense. • The generalized classical PHD filter. • The classical PHD filter. • Nonclassical PHD filters. 5.10.4.1

PHD Filters in the General Sense

The following six-step procedure is used to derive PHD filters in the general sense of the term: 1. Assume that the evolving multitarget RFS is approximately Poisson, as in (4.107). That is, for every k ≥ 0, Gk|k [h]

=

eDk|k [h−1]

(5.108)

Gk+1|k [h]

=

eDk+1|k [h−1]

(5.109)

fk|k (X|Z (k) )

=

X e−Nk|k · Dk|k

(5.110)

fk+1|k (X|Z (k) )

=

X e−Nk+1|k · Dk+1|k .

(5.111)

or, equivalently, that

2. Given that Gk|k [h] is Poisson, use the formulas for the multitarget motion model to determine the formula for Gk+1|k [h]. 3. Use (4.75) to determine the PHD of Gk+1|k [h]: Dk+1|k (x) =

δGk+1|k [1]. δx

(5.112)

4. Given that Gk+1|k [h] is Poisson with the PHD as determined in (5.112), use the multitarget measurement model to determine the formula for Gk+1|k+1 [h]. 5. Use (4.75) to determine the PHD of Gk+1|k+1 [h]: Dk+1|k+1 (x) =

δGk+1|k+1 [1]. δx

(5.113)

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6. Given the previous steps, the Time evolution ... →

Dk|k (x)

→

Dk+1|k (x)

→

Dk+1|k+1 (x)

→ ...

defines a PHD filter in the general sense, corresponding to the given target and sensor models. In general, closed-form formulas for PHD filters can be derived only if simplifying assumptions are made in regard to the multitarget motion and measurement models. 5.10.4.2

The Generalized Classical PHD Filter

Specifically, suppose that: • The motion model is the “standard” one with spawning, as described at the end of Section 5.4. • The measurement model is the generalized standard one described at the end of Section 5.5. • Gk|k [h] is not assumed to be Poisson. Then it is possible to derive closed-form formulas for a generalized classical PHD filter for these models. This filter is described in Section 8.2. The measurement-update equation for the general PHD filter involves a combinatorial sum and is not computationally tractable in general. 5.10.4.3

The Classical PHD Filter

Assume in addition that: • The measurement model is the “standard” model, as described at the end of Section 5.5. Then the resulting PHD filter is the “classical” PHD filter, which will be discussed further in Section 8.4. It should be emphasized that, as with the general PHD filter, • The time-update formula for the classical PHD filter is exact—that is, it is not necessary to assume that Gk|k [h] is Poisson.

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5.10.4.4

Nonclassical PHD Filters

The following variants of the PHD filter—ones that address nonstandard multitarget motion or measurement models—will be addressed later in the book: • Multisensor versions of the general PHD filter (Chapter 10). • Generalizations of the classical PHD filter for addressing rapidly maneuvering targets (Section 11.4). • Generalizations of the classical PHD filter for addressing nontraditional (for example, human-mediated) measurements (Section 22.10). • Variants of the PHD filter for addressing extended, cluster, group, or unresolved targets (Chapter 21). • Variants of the PHD filter for addressing superpositional sensors (Chapter 19). 5.10.5

i.i.d.c. Approximation: CPHD Filters

This section introduces the following concepts: • CPHD filters in the general sense. • The classical CPHD filter. • Nonclassical CPHD filters. 5.10.5.1

CPHD Filters in the General Sense

The following six-step procedure is used to derive CPHD filters in the general sense of the term: 1. Assume that the evolving multitarget RFS is approximately i.i.d.c., as in (4.112). That is, for every k ≥ 0, Gk|k [h]

=

Gk|k (sk|k [h])

(5.114)

Gk+1|k [h]

=

Gk+1|k (sk+1|k [h])

(5.115)

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or, equivalently, fk|k (X|Z (k) ) fk+1|k (X|Z

(k)

)

= =

|X|! · pk|k (|X|) · sX k|k |X|! · pk+1|k (|X|) ·

sX k+1|k .

(5.116) (5.117)

2. Given that Gk|k [h] is i.i.d.c., use the formulas for the multitarget motion model and (5.106) to determine the formula for Gk+1|k [h]. 3. Use (4.62) and (4.75) to determine the cardinality distribution and PHD of Gk+1|k [h]: pk+1|k (n)

=

Dk+1|k (x)

=

[ ] 1 dn Gk+1|k [x] n! dxn x=0 δGk+1|k [1]. δx

(5.118) (5.119)

4. Given that Gk+1|k [h] is i.i.d.c. with the cardinality and spatial distributions as in (5.118) and (5.119), use the formulas for the multitarget measurement model and (5.108) to determine the formula for Gk+1|k+1 [h]. 5. Use (4.62) and (4.75) to determine the cardinality distribution and PHD of Gk+1|k+1 [h]: pk+1|k+1 (n)

=

Dk+1|k+1 (x)

=

[ ] 1 dn G [x] k+1|k+1 n! dxn x=0 δGk+1|k+1 [1]. δx

(5.120) (5.121)

6. Given this, the Time evolution ... →

Dk|k (x)

... →

pk|k (n)

→ ↓ →

Dk+1|k (x) pk+1|k (n)

→ ↑↓ →

Dk+1|k+1 (x)

→ ...

pk+1|k+1 (n)

→ ...

is a CPHD filter in the general sense, corresponding to the given target and sensor models. Once again, closed-form formulas for CPHD filters can be derived only if simplifying assumptions are made for the multitarget motion and measurement models.

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5.10.5.2

The Classical CPHD Filter

Assume in addition that: • The motion model is the “standard” one without spawning, as described at the end of Section 5.4. • The measurement model is the “standard” model described at the end of Section 5.5. Then the resulting CPHD filter is the “classical” CPHD filter, which is discussed further in Section 8.5. 5.10.5.3

Nonclassical CPHD Filters

In addition, the following variants of the CPHD filter—addressing nonstandard multitarget motion or measurement models—will be addressed in this book: • Multisensor versions of the general and classical CPHD filters (Chapter 10). • Generalizations of the classical CPHD filter for addressing rapidly maneuvering targets (Section 11.5). • Generalizations of the classical CPHD filter for addressing nontraditional measurements (Section 22.10). • Variants of the CPHD filter for unknown clutter and unknown detection profiles (Chapters 18 and 17, respectively). • Variants of the CPHD filter for superpositional sensors (Chapter 19). 5.10.6

Multi-Bernoulli Approximation: Multi-Bernoulli Filters

This section introduces the following concepts: • Multi-Bernoulli filters in the general sense. • The MeMBer filter. • The CBMeMBer filter.

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5.10.6.1

135

Multi-Bernoulli Filters in the General Sense

The following six-step procedure is used to derive multi-Bernoulli filters in the general sense of the term: 1. Assume that the evolving multitarget RFS is approximately multi-Bernoulli, as in (4.126). That is, for every k ≥ 0, νk|k

Gk|k [h]

=

∏(

i i 1 − qk|k + qk|k · sik|k [h]

i=1 νk+1|k

Gk+1|k [h]

=

∏ (

)

(5.122)

) i i 1 − qk+1|k + qk+1|k · sik+1|k [h] . (5.123)

i=1

2. Given that Gk|k [h] is multi-Bernoulli, use the formulas for the multitarget motion model and (5.106) to determine the formula for Gk+1|k [h]. 3. Use some procedure to, at least approximately, determine from Gk+1|k [h] i the multi-Bernoulli parameters νk+1|k and qk+1|k , sik+1|k (x) for i = 1, ..., νk+1|k . 4. Given that Gk+1|k [h] is multi-Bernoulli with these parameters, use the formulas for the multitarget measurement model and (5.108) to determine the formula for Gk+1|k+1 [h]. 5. Use some procedure to, at least approximately, determine from Gk+1|k+1 [h] i the multi-Bernoulli parameters νk+1|k+1 and qk+1|k+1 , sik+1|k+1 (x) for i = 1, ..., νk+1|k+1 . 6. Given this, the Time evolution ... → νk|k v

k|k i ... → {qk|k }i=1

v

k|k ... → {sik|k (x)}i=1

→ νk+1|k → ↑↓ vk+1|k i → {qk+1|k }i=1 → ↑↓ vk+1|k → {sik+1|k (x)}i=1 →

νk+1|k+1 → ... ↑↓ vk+1|k+1 i {qk+1|k+1 }i=1 → ... ↑↓ vk+1|k+1 {sik+1|k+1 (x)}i=1 → ...

is a multi-Bernoulli filter in the general sense, corresponding to the given target and sensor models.

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5.10.6.2

The Multitarget Multi-Bernoulli (MeMBer) Filter

This filter was proposed by Mahler in Chapter 17 of [179]. The derivation there of the MeMBer filter measurement-update equations included an ill-considered approximation—namely the first-order Taylor’s linearization in Eq. (17.176) on p. 681 of [179]. This approximation resulted in a significant upward bias in the estimate of target number. 5.10.6.3

The Cardinality Balanced MeMBer (CBMeMBer) Filter

This bias was noticed by Vo, Vo, and Cantoni [310]. They subsequently devised a corrected MeMBer filter, the cardinality-balanced multi-Bernoulli (CBMeMBer) filter [310]. This filter is described in Section 13.4. 5.10.6.4

Other Multi-Bernoulli Filters

The following multi-Bernoulli filter variant will be addressed later in the book: • A multi-Bernoulli filter for “raw” image data (Chapter 20). 5.10.7

Bernoulli Approximation: Bernoulli Filters

Assume that the evolving target RFS is approximately Bernoulli, as in (4.119)—that is, for every k ≥ 0, Gk|k [h] Gk+1|k [h]

= =

(5.124) (5.125)

1 − qk|k + qk|k · sk|k [h] 1 − qk+1|k + qk+1|k · sk+1|k [h].

Then the Time evolution ... →

qk|k

... →

sk|k (x)

→ ↑↓ →

qk+1|k sk+1|k (x)

→ ↑↓ →

qk+1|k+1

→ ...

sk+1|k+1 (x)

→ ...

is the Bernoulli filter in the general sense corresponding to the given target and sensor models. For the standard multitarget motion and measurement models, the “classical” Bernoulli filter was independently proposed by B.-T. Vo [298] and by Mahler

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([179], pp. 514-528). 4 The time-update and measurement-update steps for this filter can be found in Section 13.2. A tutorial introduction to the Bernoulli filter can be found in Ristic et al. [262]. See also Ristic’s book, Particle Filters for Random Set Models [250].

4

In [179], the Bernoulli filter was called the “joint target-detection and tracking (JoTT)” filter. This book adopts Vo’s more technically descriptive and correct terminology, “Bernoulli filter.”

Chapter 6 Multiobject Metrology 6.1

INTRODUCTION

Metrology refers to the process of determining the degree of similarity or dissimilarity of entities of interest. It is central to information fusion, whether we are to compare competing algorithms with each other or, within an algorithm, to determine the influence of internal parameters on the algorithm’s performance. In the single-sensor, single-target realm, two general metrological paradigms dominate: 1. Measurement of the distance between points: Suppose that a single-target tracking algorithm generates a time sequence of state estimates that are to be compared to ground truth: Tracker: Ground truth:

x1|1 , ..., xk|k g1 , ..., gk .

(6.1) (6.2)

The tracker’s instant-by-instant performance can be compared using a distance metric such as Euclidean distance: d(xk|k , gk ) = ∥xk|k − gk ∥. Its trackwise performance can be measured using a suitable generalization of this, such as the root-mean-square (RMS) miss distance: ? ? k ?1 ∑ k k d({xi|i }i=1 , {gk }i=1 ) = ? d(xi|i , gi )2 . k i=1

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(6.3)

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2. Measurement of the distance between probability distributions: Suppose that the tracker generates track distributions, such as: fk|k (x) = NPk|k (x − xk|k ).

(6.4)

Suppose also that ground truth can be represented as a distribution gk (x) = NCk (x − gk )

(6.5)

where Ck is determined using (for example) the Cramer-Rao bound. Then instant-by-instant performance can be compared using, as two possible illustrations, the Hellinger distance ∫ (√ )2 √ d(fk|k , gk ) = fk|k (x) − gk (x) dx (6.6) or the Kullback-Leibler cross-entropy ( ) ∫ fk|k (x) KL(fk|k ; gk ) = fk|k (x) · log dx. gk (x)

(6.7)

This chapter outlines analogous approaches for multisensor-multitarget problems: • Comparison of the distance between multitarget state sets X = {x1 , ..., xn }. • Comparison of the distance between multitarget distributions fk|k (X). The chapter is organized as follows: 1. Section 6.2: Multiobject miss distance—Hausdorff distance, Wasserstein distance, optimal subpattern assignment (OSPA) distance, and generalizations of OSPA. 2. Section 6.3: Multiobject information-theoretic functionals: the Csisz´ar family of information-theoretic functionals and the Cauchy-Schwartz functional; and specific formulas for them for use with PHD and CPHD filters.

6.2

MULTIOBJECT MISS DISTANCE

In single-sensor, single-target statistics, target states x and sensor measurements z are points. If we want to determine the performance of a single-sensor, single-target

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filter, the most common approach is to first choose a distance metric d(x, x′ ) on states. Most commonly, x, x′ are Euclidean vectors and d(x, x′ ) is a Mahalanobis distance √ d(x, x′ ) = (x − x′ )T C −1 (x − x′ ). (6.8) The instantaneous performance of the filter can then be determined by calculating the distance d(xk|k , gk ) between the filter’s current state estimate xk|k and the current ground truth gk . In multitarget problems, however, multitarget states X and multisensormultitarget measurements Z are finite sets of points. This section addresses the following question: • How can the concept of distance be usefully extended to multitarget states X, X ′ —or, more generally, to the finite subsets of any space that is equipped with an underlying distance metric? Three possible answers to this question are described in what follows: Hausdorff distance, Wasserstein distance, and the optimal subpattern assignment (OSPA) metric. The section is organized as follows: 1. Section 6.2.1: A short history of the development of the concept of “multiobject miss distance.” 2. Section 6.2.2: An introduction to the optimal subpattern assignment (OSPA) multiobject miss distance. 3. Section 6.2.3: The generalization of OSPA to algorithms that produce estimates of state uncertainty as well as states. 4. Section 6.2.4: The generalization of OSPA to labeled tracks. 5. Section 6.2.5: The generalization of OSPA to temporally-connected tracks. 6.2.1

Multiobject Miss Distance: A History

Finite-set statistics is based on hyperspaces Y∞ , whose elements are the finite subsets of some underlying space Y. For metrological purposes, it is desirable to have a way of computing the distance d(Y, Y ′ ) between two finite subsets Y, Y ′ ⊆ Y. Here the term “distance” has a specific meaning—a metric in the mathematical sense. That is, it must have the following properties:

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• Nonnegativity: d(Y, Y ′ ) ≥ 0. • Symmetry: d(Y, Y ; ) = d(Y ′ , Y ). • Definiteness: d(Y, Y ′ ) = 0 if and only if Y = Y ′ . • Triangle inequality: d(Y, Y ′′ ) ≤ d(Y, Y ′ ) + d(Y ′ , Y ′′ ) for any Y, Y ′ , Y ′′ . Remark 14 (The triangle inequality and practice) Because of the triangle inequality’s seeming abstractness, some practitioners have been inclined to dismiss it as a fussy mathematical nicety. For this reason, it is necessary to clarify its practical meaning and importance.1 Suppose that two multitarget trackers A and B are being fed the same data. Let their respective outputs be XA and XB . These are to be compared using a distance-type Measure of Performance (MoP), denoted d(X, Y ). Suppose that A’s estimate is “close” to ground truth G—that is, d(XA , G) is small. Suppose further that B’s estimate is “close” to A’s— that is, d(XB , XA ) is small. If d(·, ·) is to be meaningful from a practical point of view, then B’s estimate must also be “close” to ground truth—that is, d(XB , G) must also be small. Any distance-type MoP d(·, ·) that does not satisfy this property would be “metrically incoherent”—it would not measure “closeness” in an intuitively reasonable manner. The triangle inequality ensures that d(·, ·) is metrically coherent, because it forces d(XB , G) to be small: d(XB , G) ≤ d(XB , XA ) + d(XA , G). 6.2.1.1

(6.9)

Hausdorff Distance

The Hausdorff distance is the most familiar distance metric for subsets, finite or otherwise. It is defined by dH (Y, Y ′ ) = ∞ if either Y = ∅ or Y ′ = ∅ and, if otherwise, { } H ′ ′ ′ d (Y, Y ) = max max min d (y, y ) , max min d (y, y ) (6.10) ′ ′ ′ ′ y∈Y y ∈Y

y ∈Y y∈Y

where d(y, y′ ) is some underlying metric on Y. The Hausdorff distance is statistically consistent with finite-set statistics, since its metric topology is the Fell-Matheron topology ([201], pp. 3,12).2 However, it is not entirely adequate as a metric for performance estimation in multitarget 1 2

This discussion is adapted from Ristic, Vo, and Clark [261], Section 1. For a general discussion of metrics that generate the Fell-Matheron topology, see [225].

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tracking. This is because it tends to be insensitive to cardinality and to the effect of statistical outliers ([267], Section II-A). In particular, it is possible for d(Y, Y ′ ) to be small even if |Y | and |Y ′ | are very different. 6.2.1.2

Wasserstein Distance

Let Y = {y1 , ..., yn } and Y ′ = {y1′ , ..., yn′ } with |Y | = n, |Y ′ | = n′ , and n = n′ . Then the following definition of miss distance, originally proposed by Drummond [62], is intuitively natural: ? ? n ?1 ∑ ′ ′ ∥2 d(Y, Y ) = min ? ∥yi − yπi π n i=1

(6.11)

where the minimum is taken over all permutations π on the numbers 1, ..., n. That is, d(Y, Y ′ ) is the smallest root-mean-square (RMS) error between the elements of Y and the elements of Y ′ . How might this definition be extended to the situation when n ̸= n′ —but in such a manner that one ends up with a true metric on Y? In 2002, Mahler proposed the family of Wasserstein distances as an answer. Let d(y, y′ ) be a metric on y, y′ ∈ Y. Then the Wasserstein distance of power p is defined by (see [109], [110], [179], Section 14.6.3, pp. 510-512):3 ? ? n n′ ?∑ ∑ p W ′ def. dp (Y, Y ) = inf ? Ci,i′ · d(yi , yi′ ′ )p

(6.12)

C

i=1 i′ =1

where the infimum is taken over all n × n′ “transportation matrices” C. A matrix C is a transportation matrix if, for all i = 1, ..., n and i′ = 1, ..., n′ , Ci,i′ ≥ 0 and n n′ ∑ ∑ 1 1 ′ Ci,i = ′ , Ci,i′ = . (6.13) n n ′ i=1 i =1

When n = n′ , p = 2, and d(y, y′ ) = ∥y − y′ ∥, (6.12) reduces to Drummond’s (6.11). 3

The metric topology of the Wasserstein metric is not the restriction of the Matheron topology to finite subsets.

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6.2.2

The Optimal Sub-Pattern Assignment (OSPA) Metric

Schuhmacher, Vo, and Vo discovered that, from the point of view of practical performance evaluation, Wasserstein distance4 suffers from a number of subtle non-intuitive behaviors (see [267], Section II-B). In its place, they proposed a new Wasserstein-like metric that not only avoids these difficulties, but is also more mathematically intuitive and more easily computed [267]. Let us be given the following: • A baseline metric d(x, x′ ) defined on single-target states x, x′ . • A real number c > 0—the “association cutoff radius”—which has the same units of measurement as x. • A unitless real number p ≥ 1. Let dc (x, x′ ) = min{c, d(x, x′ )}

(6.14)

′

be the “cutoff metric” associated with d(x, x ). Let X = {x1 , ..., xn } be the estimated track set and G = {g1 , ..., gm } the ground-truth track set, with |X| = n and |G| = m. First assume that 0 < n ≤ m. Then the OSPA distance is defined as:5

OSPA dp,c (X, G) =

(

n

1 ∑ cp dc (xi , gπi )p + · (m − n) min π m m i=1

)1/p

(6.15)

where the minimum is taken over all permutations π on 1, ..., m. If 0 = n ≤ m, OSPA then by convention set dp,c (∅, G) = c. If n > m, then define dp,c (X, G) = dp,c (G, X). Also, define dOSPA p,c (X, G) = 0 if n = m = 0. It follows that OSPA 0 ≤ dp,c (X, G) ≤ c. (6.16) The number c determines the importance assigned to target-number accuracy, as compared to the importance assigned to localization accuracy. The number p determines the sensitivity of the metric to statistical “outliers.” The larger the value 4 5

In [267], Schuhmacher et al. refer to the Wasserstein metric as the “optimal mass transfer” (OMAT) metric. As with the Wasserstein metric, the metric topology of the OSPA metric is not the restriction of the Fell-Matheron topology to finite subsets. Rather, it is the “vague topology”—the topology most commonly used in the counting-measure formulation of point process theory. See [267], p. 3451.

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of p, the more that a multitarget tracking algorithm is “punished” for arriving at poor target state estimates. Equation (6.15) is easily computed using standard optimal-assignment algorithms such as Munkres, JVC, and so on [267]. The definition of the OSPA metric has two parts. The first part n

min π

1 ∑ dc (xi , gπi )p m i=1

(6.17)

measures truth-to-track localization accuracy, in essentially the same manner as Drummond’s formula (6.11). However, dc (x, g) does not distinguish between state-vectors x, g that are too far apart to be candidates for association. The second part cp · (m − n) (6.18) m measures the degree of accuracy in estimating target number. If c is small then localization accuracy is more strongly emphasized than cardinality accuracy. 6.2.2.1

Constructive Interpretation of OSPA

OSPA Intuitively speaking, if 0 < n ≤ m then dp,c (X, G) is the distance between the elements of G that are most tightly associated with the elements of X—but with the proviso that an element of G must be qualified to associate with an element of X—that is, it must be within distance c of it. This interpretation can be understood more completely using the following three-step procedure for constructing dOSPA p,c (X, G). Suppose that 0 < n ≤ m. Then:

1. Find that subset GX = {gπˆ 1 , ..., gπˆ n } ⊆ G

(6.19)

of G with n elements which is closest to X, using the following generalization n 1 ∑ dp (X, G) = min d(xi , gπi )p (6.20) π m i=1 of Drummond’s truth-to-track association formula (6.11); and where π ˆ is the corresponding optimal assignment xi ↔ gπˆ i for i = 1, ..., n.

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2. For each g ∈ G, let δg =

{

c min{c, d(xi , gπˆ i )}

if if

g∈ / GX g = gπˆ i

(6.21)

be the “association cutoff distance” between g and its optimal-association match in X. That is, if g has no match then δg = c. If g has a match but is insufficiently near it then δg = c. If g has a match and is sufficiently near it, then δg is the distance to its match. 3. Compute the pth-order average of the cutoff distances δg : √ p

1 ∑ p δg |G|

(6.22)

g∈G

=

?   ? ? ∑ p ∑ p 1 ? p  ? δg + δg  |G| g∈GX

6.2.2.2

g∈G / X

=

? ( ) ? n ∑ ?1 p ? min dc (xi , gπi )p + cp · (m − n) π m i=1

(6.23)

=

dOSPA p,c (X, G).

(6.24)

The “Components” of OSPA

The OSPA metric can be decomposed into two “components.” These are not metrics, since they do not satisfy the triangle inequality. Nevertheless, they provide valuable additional information about the degree to which localization versus cardinality contribute to an OSPA score. The first, the localization-error component loc ep,c (X, G), measures the contribution of localization accuracy alone. When n ≤ m it is defined by

loc ep,c (X, G) =

(

n

1 ∑ min dc (xi , gπi )p π m i=1

)1/p

(6.25)

loc and, if n > m, by ep,c (G, X) = eloc p,c (X, G). The second, the cardinality-error crd component ep,c (X, G), measures the contribution of cardinality accuracy alone.

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When n ≤ m it is defined by crd

ep,c (X, G) = c ·

(

m−n m

)1/p

(6.26)

crd crd and, if n > m, by ep,c (G, X) = ep,c (X, G).

6.2.3

Extension of OSPA to Covariance (COSPA)

Most modern multitarget tracking algorithms produce track outputs of the form X = {(x1 , P1 ), ..., (xn , Pn )}, where Pi is the error-covariance matrix corresponding to the state estimate xi . To accommodate such algorithms, it is necessary to generalize the OSPA metric in a suitable manner. The purpose of this section is to describe techniques for doing so. The basic approach is to extend the base metric d(x, x′ ) to a metric d((x, P ), (x′ , P ′ )) defined on pairs (x, P ), where x is a state-vector and P is a covariance matrix. This extension must have the following consistency property: d((x, P ), (x′ , P ′ )) → d(x, x′ )

(6.27)

as P → 0 and P ′ → 0. The simplest such extension is d((x, P ), (x′ , P ′ )) = d(x, x′ ) + d(P, P ′ )

(6.28)

where d(P, P ′ ) is any metric defined on positive-definite matrices P, P ′ . The most obvious choice for d(P, P ′ ) is the Frobenius metric dF (P, P ′ ) =

√

tr(P − P ′ )2

(6.29)

arising from the Frobenius matrix scalar product ⟨P, P ′ ⟩F = tr(P T P ′ ). There is a potential difficulty with (6.28), however. One would expect that the closeness of x and x′ should be influenced by the closeness of P and P ′ , and vice versa. For example, in practice (x, P ) arises from a track distribution fk|k (x|Z k ) which inherently imposes a statistical coupling between x and P : fk|k (x|Z k ) = NPk|k (x − xk|k ). In (6.28), however, x and P are completely independent of each other.

(6.30)

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A second metric on covariance matrices has found application in image processing (see [38], Eq. (11)). It is the metric on the Riemannian manifold of covariance matrices defined by [87], [203]: ? ? d ?∑ −1 d(C1 , C2 ) = ∥ log(C1 C2 )∥F = ? (log λi )2

(6.31)

i=1

√ where log C denotes the matrix logarithm, ∥C∥F = ⟨C, C⟩F is the Frobenius norm, and λi are the eigenvalues of C1−1 C2 . Besides being a distance metric, d(C1 , C2 ) is invariant under affine transformation and under matrix inversion. That is, d(BC1 B T , BC2 B T ) = d(C1 , C2 ) (6.32) for any regular matrix B, and d(C1−1 , C2−1 ) = d(C1 , C2 ).

(6.33)

A third approach is to compare the track distribution fk|k (x|Z k ) with another distribution f (x|Gk ) that represents the current ground truth Gk = {g1 , ..., gγ }. This is the approach adopted by Nagappa, Clark, and Mahler [221]. They define d((x, P ), (x′ , P ′ )) to be the Hellinger distance (see (6.67)) between two track distributions. For linear-Gaussian distributions NP (y − x) and NP ′ (y − x′ ), the Hellinger distance can be computed in closed form: ′

′

d((x, P ), (x , P ))

=

=

1−

∫ √ √

NP (y − x) · NP ′ (y − x′ )dy

(6.34)

√

det P P ′ (6.35) det 12 (P + P ′ ) ( ) 1 · exp − (x − x′ )T (P + P ′ )−1 (x − x′ ) . 4

1−

To compare an estimated track (x, P ) with a ground truth track (g, C), one must first choose C. Nagappa et al. proposed that C should be specified in terms of the Cramer-Rao lower bound (CRLB), as determined using an estimation process. See [221] for more details. There is a potential problem with (6.34) and (6.35): they do not necessarily satisfy the consistency property of (6.27). This failure occurs, for example, when

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both covariances P, P ′ are very small. Thus, more generally, let E be a fixed covariance matrix and define ∫ √ ′ ′ dE ((x, P ), (x , P )) = 1 − NP +E (y − x) · NP ′ +E (y − x′ )dy (6.36) √√ det(P + E)(P ′ + E) = 1− (6.37) det 12 (P + P ′ + 2E) ( ) 1 ′ T ′ −1 ′ · exp − (x − x ) (P + P + 2E) (x − x ) . 4 Then dE (x, x′ )

=

lim dE ((x, P ), (x′ , P ′ )) ( ) 1 ′ T −1 ′ 1 − exp − (x − x ) E (x − x ) 8

(6.38)

P,P ′ →0

=

(6.39)

is essentially just a Mahalanobis distance on x, x′ . 6.2.4

OSPA for Labeled Tracks (LOSPA)

The OSPA metric measures only the instantaneous distance, at a particular time tk , between the current track set Xk|k and the current ground-truth target set Gk . In general, we must determine how well a multitarget tracker estimates ground truth over an extended period of time. This is the problem addressed by the temporal generalization of OSPA due to Ristic, Vo, and Clark [261], to be described shortly in Section 6.2.5. For convenience it will be referred to here as the “TOSPA metric.” Before we can understand the TOSPA metric, however, we must first consider an intermediary metric: OSPA for labeled tracks. This is the purpose of this section. The track of a single target is not just a time sequence x1|1 , ..., xK|K or, alternatively, (x1|1 , P 1|1 ), ..., (xk|k , P K|K ). (For the sake of notational clarity, in what follows we will consider only the former situation.) It is, rather, a time sequence of the form ([179], Section 14.5.6, pp. 505-508) (x1|1 , ℓ), ..., (xK|K , ℓ)

(6.40)

where the integer label ℓ ∈ {1, ...., L} uniquely identifies each (xk|k , ℓ) as belonging to a single, temporally connected track-trajectory. Thus the output of a

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multitarget tracker will be a time sequence X (K) : X1|1 , ..., XK|K

(6.41)

of track sets, where now each Xi|i has the form X = {(x1 , ℓ1 ), ..., (xn , ℓn )}

(6.42)

with labels ℓi ∈ {1, ..., L}. It follows that, for each k, the instantaneous track set Xk|k at time tk can be partitioned as 1 L Xk|k = Xk|k ⊎ ... ⊎ Xk|k

(6.43)

ℓ ℓ where Xk|k is the set of all (x, ℓ) such that (x, ℓ) ∈ Xk|k . Either Xk|k contains ℓ a single element (the unique track with label ℓ at time tk ) or Xk|k = ∅ (that is, the track with label ℓ has been dropped). Stated differently, the sequence ℓ ℓ X1|1 , ..., XK|K

(6.44)

is the ℓth track sequence as determined by the multitarget tracker. Now assume that ground truth has been provided. Then we have a time sequence G(K) : G1|1 , ..., GK|K (6.45) of ground truth track sets, each of which consists of elements of the form (g, γ) where g is a ground truth state and γ ∈ {1, ..., L} is a ground-truth label. Each ground truth track set can be partitioned as Gk|k = G1k|k ⊎ ... ⊎ GL k|k ,

(6.46)

Gγ1|1 , ..., GγK|K

(6.47)

and is the track sequence for the ground-truth track with label γ. Next, suppose for the moment that the tracker correctly and exactly estimated all ground truth track sequences. Even given this fact, the tracker’s track labeling convention will not be the same as the ground truth labeling convention. That is, ℓ there will be a permutation σ on 1, ..., L such that Xk|k = Gσℓ for all k|k k = 1, ..., K and all ℓ = 1, ..., L.

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More generally, suppose that the tracker’s estimates of ground truth are not ℓ exact. Then there will be a σ such that Xk|k and Gσℓ k|k are as “close to each other as possible” (in an OSPA sense) for all k, ℓ. The question then becomes: When it comes to labeled track sequences, what does “close to each other” mean in a quantifiable sense? Ristic et al. begin by replacing the original base metric d(x, x′ ) on target states with a new base metric on labeled states (x, ℓ) as follows: √ d˜p,α ((x, ℓ), (x′ , ℓ′ )) = p d(x, x′ )p + αp · (1 − δℓ,ℓ′ ) (6.48) where δℓ,ℓ′ is the Kronecker delta. The number α > 0 controls the relative weighting of the labeling error 1 − δℓ,ℓ′ . Now suppose that X G

= =

{(x1 , ℓ1 ), ..., (xn , ℓn )} {(g1 , γ1 ), ..., (gm , γm )}

(6.49) (6.50)

with |X| = n and |G| = m. Here, ℓj ∈ {1, ...., N } are the estimated labels of the estimated tracks; and γj ∈ {1, ...., N } are the known labels of the ground truth tracks. Denote the result of stripping labels from the tracks as X↓ G↓

= =

{x1 , ..., xn } {g1 , ..., gm }.

(6.51) (6.52)

Substitute d˜p,α (·, ·) in (6.48) into (6.15) whenever d(·, ·) occurs. What results is an explicit formula for the OSPA metric with labels, denoted as “LOSPA”: ( )1/p n p ∑ α ↓ ↓ p dLOSPA dOSPA (1 − δℓi ,γπˆ i ) (6.53) p,c,α (X, G) = p,c (X , G ) + m i=1 where the permutation π ˆ on 1, ..., n is defined by π ˆ = arg min π

n ∑

dc (xi , gπi )p .

(6.54)

i=1

Thus, one first finds the optimal association π ˆ of tracks in X with ground truth tracks∑ in G, irrespective of labels. If all labels were estimated correctly, then the LOSPA sum ) will be zero and dp,c,α (X, G) = dOSPA p,c,α (X, G). If none are ˆi i (1 − δℓi ,γπ correct, the sum will be n and the value of dLOSPA p,c,α (X, G) will be increased by an amount determined by the size of α.

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6.2.5

Temporal OSPA (TOSPA)

The LOSPA metric measures the distance between labeled track sets at a particular instant of time. To address the general track-fusion problem, we must be able to measure the distance between time sequences of labeled track sets. In particular, we must address the fact that the assignment of labels to evolving track sequences is essentially arbitrary. This is the purpose of the temporal OSPA (TOSPA) metric, introduced by Ristic, Vo, and Clark [261]. Given the labeled ground truth track set G = {(g1 , γ1 ), ..., (gm , γm )} and a permutation σ on 1, ..., L, define σ

(6.55)

G = {(g1 , σγ1 ), ..., (gm , σγm )}.

Then given the ground truth multitrack sequence G(K) and the multitrack sequence X (K) produced by a multitarget tracker, the TOSPA metric is defined by

dTOSPA p,c,α (X

(K)

,G

(K)

)=

(

min σ

K ∑

σ

dLOSPA p,c,α (Xk|k , Gk|k )

p

)1/p

.

(6.56)

k=1

Intuitively speaking, the TOSPA metric searches for the best match, in a temporally global sense, between the ground truth tracks and the tracks created by the multitarget tracker. It measures the following aspects of tracker performance: • Localization accuracy. • Accuracy in determining target number (which encompasses accuracy in regard to dropped tracks or false tracks). • Accuracy in track labeling throughout the timespan of an entire scenario. Suppose that the algorithm being measured provides error-covariance matrices as well as states. Then the approach described in Section 6.2.3 immediately allows the TOSPA metric to be suitably generalized. The minimization operation in (6.56) is an optimal track-to-track association procedure. It requires minimization of the total distances between all ground truth and all estimated tracks, in both space and time. It is therefore computationally intractable in general, and some approximation must be devised. The approach of Ristic et al. [261] is to reassign labels to the estimated track sequences, with the purpose of correctly aligning them with the labels already assigned to the ground truth track sequences.

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Let Gγ1|1 , ..., GγK|K

(6.57)

be a ground truth track sequence with label γ—which means that, for any k, either Gγk|k = ∅ or Gγk|k = {(gk|k , γ)}. Likewise, let ℓ ℓ X1|1 , ..., XK|K

(6.58)

ℓ be an estimated track sequence with label ℓ—so that, for any k, either Xk|k =∅ ℓ or Xk|k = {(xk|k , ℓ)}. Let

eℓk =

{

1 0

if if

ℓ Xk|k ̸= ∅ , otherwise

e˜γk =

{

1 0

if if

Gγk|k ̸= ∅ . otherwise

(6.59)

Then define the pairwise cost of assigning a ground truth track γ with an estimated track ℓ to be: { ∑K ℓ γ ∑K ℓ γ ˜ ·∥xk −gk ∥ k=1 ek e ∑Kk if ˜k > 0 γ ℓe k=1 ek e exp e ˜ ( ) c(ℓ, γ) = . (6.60) k=1 k k ∞ if otherwise The c(l, γ) are used to form the assignment matrix in the two-dimensional assignment algorithm. Once truth-to-track assignments have been made using this heuristic, an estimated track sequence has either (1) been given the label of the ground truth track sequence that was associated with it; or (2) assigned a completely new label if it could not be associated with any ground truth track sequence. Once this has been accomplished, labeled track sets Xk|k for the estimated tracks are created for each time tk —but now equipped with the newly-assigned labels. The LOSPA or labeled-COSPA metrics are then applied to these labeled track sets; and their value(s) are displayed, instant-by-instant, for the entire scenario. Equation (6.60) causes longer duration estimated tracks to be assigned to ground truth tracks.

6.3

MULTIOBJECT INFORMATION FUNCTIONALS

Single-target distance metrics address the problem of measuring the similarity or dissimilarity of two states x, x′ . A more general concept is that of measuring the similarity or dissimilarity of two probability distributions f1 (x), f0 (x) defined on

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states. Perhaps the most familiar approach is information-theoretic—for example, the Kullback-Leibler discrimination functional: ( ) ∫ f1 (x) KL(f1 ; f0 ) = f1 (x) · log dx. (6.61) f0 (x) This section addresses the following question: • How can information-theoretic functionals be extended to multitarget probability distributions f1 (X), f0 (X) defined on multitarget states X—or, more generally, to probability distributions defined on the finite subsets of any space? The approach described in this section is the infinite family of multiobject Csisz´ar information-theoretic functionals, introduced by Zajic and Mahler in 1999 ([331], pp. 96-97). They are useful not only for performance evaluation, but also for the approach to sensor and platform management to be described later in Part V. The section is organized as follows: 1. Section 6.3.1: The family of Csisz´ar information functionals. 2. Section 6.3.2: Csisz´ar functionals for Poisson processes. 3. Section 6.3.3: Csisz´ar functionals for i.i.d.c. processes. 6.3.1

Csisz´ar Information Functionals

Let c(x) be a convex kernel—that is, a unitless, nonnegative convex function of a nonnegative variable x ≥ 0, such that c(1) = 0 and such that c(x) is strictly convex at x = 1 (that is, c(2) (1) > 0). Let f1 (Y ) and f0 (Y ) be two multiobject probability distributions on some space Y. Then the multiobject Csisz´ar information-discrimination functional associated with c(x) is [331]: Ic (f1 ; f0 ) =

∫

c

(

f1 (Y ) f0 (Y )

)

· f0 (Y )δY.

(6.62)

It has the property that Ic (f1 ; f0 ) ≥ 0, with equality occurring only if f1 (Y ) = f0 (Y ) almost everywhere [52], [53].6 6

Caution: If f1 or f0 is not a multiobject probability density function, then neither of these properties remains valid. See [52], [53].

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The correspondence c ?→ Ic (f1 ; f0 ) is not one-to-one, since (6.63)

Ic2 (f1 ; f0 ) = Ic1 (f1 ; f0 ). whenever c2 (x) = c1 (x) + K · (x − 1) 7

for any constant K. Examples of multiobject Csisz´ar information functionals are as follows: • Kullback-Leibler—c(x) = 1 − x + x log x: Ic (f1 ; f0 ) =

∫

f1 (Y ) · log

(

f1 (Y ) f0 (Y )

)

(6.64)

δY.

• Chi-squared—c(x) = (x − 1)2 : Ic (f1 ; f0 ) = −1 +

∫

f1 (Y )2 δY. f0 (Y )

(6.65)

• L1 metric—c(x) = |x − 1| : Ic (f1 ; f0 ) =

∫

|f1 (Y ) − f0 (Y )|δY.

(6.66)

∫ √

(6.67)

√ 2 • Hellinger—c(x) = ( x − 1) : Ic (f1 ; f0 ) = 2 − 2 • Information deviation—c(x) = (3,4)): Ic (f1 ; f0 ) = 7

(1)

1 α (1 − α)

(

f1 (Y ) · f0 (Y )δY.

αx+1−α−xα α(1−α)

1−

∫

([323], Eq. (6), and [117], Eqs.

f1 (Y )α · f0 (Y )1−α δY

)

.

(6.68)

If one sets K = −c1 (1), then c2 (x) will have a unique minimum at x = 1. In this case, c2 (x) = 0 implies x = 1.

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If the Kullback-Leibler functional is expanded in f1 around f0 in a Taylor’s series expansion, then the chi-square discrimination is seen to be the second-order Taylor’s approximation of the Kullback-Leibler discrimination. Also, chi-square discrimination and Hellinger discrimination provide bounds for the Kullback-Leibler discrimination ([92], pp. 12,13): ( ) I(√x−1)2 (f1 ; f0 )2 ≤ I1−x+x log x (f1 ; f0 ) ≤ log 1 + I(x−1)2 (f1 ; f0 ) . (6.69) The information deviation functional is defined for 0 ≤ α ≤ 1. It converges to Kullback-Leibler discrimination when α → 0 or α → 1. It is closely related to Chernoff information, which in multiobject form is defined as ( ) ∫ ω 1−ω C(f1 ; f0 ) = sup − log f1 (Y ) · f0 (Y ) δY . (6.70) 0≤ω≤1

It is also closely related to the R´enyi α-divergence, which in multiobject form is defined as ∫ 1 Rα (f1 ; f0 ) = log f1 (Y )α · f0 (Y )1−α δY (6.71) α−1 for α > 0. Specifically, if c(x) is the convex function corresponding to information deviation then Rα (f1 ; f0 ) =

1 · log [1 − α (1 − α) · Ic (f1 ; f0 )] . α−1

(6.72)

A final information functional does not seem to be a Csisz´ar divergence, but should be discussed because of increasing interest in it for information fusion applications [64]. This is the Cauchy-Schwartz divergence functional. Its popularity is due to the fact that (1) it behaves much like Kullback-Leibler discrimination; and (2) can be evaluated essentially in closed form if f1 (y) and f0 (y) are Gaussian mixtures [132]. The multiobject version of the Cauchy-Schwartz divergence is: ∫ |Y | c · f1 (Y ) · f0 (Y )δY √∫ CS(f1 ; f0 ) = − log √∫ (6.73) c|Y | · f1 (Y )2 δY · c|Y | · f0 (Y )2 δY where c is a positive real number that has the same units of measurement as the elements y of Y.8 Equation (6.73) has the following geometric meaning: CS(f1 ; f0 ) = − log cos θf1 ,f0 8

(6.74)

This constant is not required in the single-object version of the Cauchy-Schwartz discrimination. It is necessary for the multiobject version because, without it, the indicated set integrals would not be mathematically well defined.

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where θf1 ,f0 is the angle between f1 and f0 , when they are regarded as vectors in the inner product space L2 (Y∞ ) of square-integrable multiobject density functions f (Y ) on Y ∈ Y∞ , with respect to the inner product ⟨f1 , f2 ⟩ =

6.3.2

∫

c|Y | · f1 (Y ) · f2 (Y )δY.

(6.75)

Csisz´ar Functionals for Poisson Processes

The purpose of this section is to provide explicit formulas for the information functionals described in the previous section, given that f1 , f0 are both Poisson (as defined in Section 4.3.1). That is, f1 (Y ) = e−N1 · D1Y ,

f0 (Y ) = e−N0 · D0Y

(6.76)

where D1 (y), D0 (y) are the respective PHDs of f1 (Y ), f0 (Y ); N1 , N0 are the respective integrals of D1 (y), D0 (y); and the power-functional notation D Y was defined in (3.5). Also, define s1 (y) = N1−1 · D1 (y),

s0 (y) = N0−1 · D0 (y).

(6.77)

The mathematical derivations of the following formulas can be found in Section K.9. • Kullback-Leibler divergence ( c(x) = 1 − x + x log x ): Ic (f1 ; f0 )

= =

N0 − N1 + Ic (D1 ; D0 ) ( ( ) ) N1 N1 N0 · c + · Ic (s1 ; s0 ) . N0 N0

(6.78) (6.79)

• Chi-squared divergence ( c(x) = (x − 1)2 ): log (1 + Ic (f1 ; f0 ))

= =

∫ D1 (x)2 N0 − 2N1 + dx D0 (x) ( ( ) ) N12 N0 · c + Ic (s1 ; s0 ) . N0 N1

(6.80) (6.81)

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• Information deviation ( c(x) = α−1 (1 − α)

−1

· (αx + 1 − α − xα ) ):

log (1 − α (1 − α) · Ic (f1 ; f0 )) −αN1 − (1 − α)N0 ∫ + D1 (y)α · D0 (y)1−α dy ( ) ( ) 1 N0 · c N N0 −α(1 − α) · . +N1α N01−α · Ic (s1 ; s0 )

=

=

(6.82)

(6.83)

• R´enyi α-divergence:9 Rα (f1 ; f0 )

=

=

α · N1 + N0 α−1∫ 1 + D1 (y)α · D0 (y)1−α dy α−1 ( ) N1 αN0 · c + αN1α N01−α · Ic (s1 ; s0 ) N0 −

(6.84)

(6.85)

where c(x) is the convex kernel corresponding to information deviation. For completeness, the following result is included: • Cauchy-Schwartz divergence: c CS(f1 ; f0 ) = 2 6.3.3

∫

(D1 (y) − D0 (y))2 dy.

(6.86)

Csisz´ar Functionals for i.i.d.c. Processes

The purpose of this section is provide explicit formulas for various information functionals, assuming that f1 , f0 are both i.i.d.c. (as defined in Section 4.3.2): f1 (Y ) = |Y |! · p1 (|Y |) · sY1 ,

f0 (Y ) = |Y |! · p0 (|Y |) · sY0 .

(6.87)

The derivations of the following formulas are given in Section K.10. • Kullback-Leibler divergence—c(x) = 1 − x + x log x: Ic (f1 ; f0 ) = Ic (p1 ; p0 ) + N1 · Ic (s1 ; s0 ) 9

Equation (6.84) is originally due to Ristic, Vo, and Clark—see [260], Eq. (18).

(6.88)

Multiobject Metrology

where Ic (p1 ; p0 ) =

∑

p1 (n) · log

159

(

p1 (n) p0 (n)

)

.

(6.89)

n≥0

• Chi-squared divergence—c(x) = (x − 1)2 : Ic (f1 ; f0 ) = −1 + I˜c (p1 ; p0 ) · Gp˜(I˜c (s1 ; s0 ))

(6.90)

I˜c (f1 ; f0 ) = I˜c (p1 ; p0 ) · Gp˜(I˜c (s1 ; s0 ))

(6.91)

or equivalently

where Gp˜(y) is the p.g.f. of the probability distribution p˜(n) defined by x(n) =

1 p1 (n)2 · I˜c (p1 ; p0 ) p0 (n)

(6.92)

and where I˜c (f1 ; f0 )

=

I˜c (s1 ; s0 )

=

I˜c (p1 ; p0 )

=

∫

f1 (Y )2 δY f0 (Y ) ∫ s1 (y)2 dy s0 (y) ∑ p1 (n)2 . p0 (n)

(6.93) (6.94) (6.95)

n≥0

• Information deviation—c(x) = α−1 (1 − α) Ic (f1 ; f0 ) =

−1

· (αx + 1 − α − xα ):

[ ] 1 · 1 − I˜c (p1 ; p0 ) · Gp˜(I˜c (s1 ; s0 ) α (1 − α)

(6.96)

or, equivalently, I˜c (f1 ; f0 ) = I˜c (p1 ; p0 ) · Gp˜(I˜c (s1 ; s0 )

(6.97)

where Gp˜(y) is the p.g.f. of the probability distribution p˜(n) defined by p˜(n) =

p1 (n)α · p0 (n)1−α I˜c (p1 ; p0 )

(6.98)

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and where I˜c (f1 ; f0 )

=

I˜c (p1 ; p0 )

=

∫

f1 (Y )α · f0 (Y )1−α δY ∑ p1 (n)α · p0 (n)1−α

(6.99) (6.100)

n≥0

I˜c (s1 ; s0 )

=

∫

s1 (y)α · s0 (y)1−α dy.

(6.101)

• R´enyi α-divergence:10 Rα (f1 ; f0 ) =

1 1 · log I˜c (p1 ; p0 ) + · log Gp˜(I˜c (s1 ; s0 )) (6.102) α−1 α−1

where c(x), I˜c (p1 ; p0 ), I˜c (s1 ; s0 ), and Gp˜(y) are defined as in (6.96).

10 Equation (6.102) is originally due to Ristic, Vo, and Clark—see [260], Eq. (14).

Part II

RFS Filters: Standard Measurement Model

161

Chapter 7 Introduction to Part II The chapters in Part II describe multitarget algorithms that presume the “standard” multitarget motion and measurement models (introduced, respectively, in Sections 5.4 and 5.5. These algorithms are: • The “classical” PHD and CPHD filters and their properties and behavior—for example, “spooky action at a distance.” • Multisensor classical PHD and CPHD filters. • The cardinality-balanced multi-Bernoulli (CBMeMBer) filter. • Jump-Markov PHD and CPHD filters for rapidly maneuvering targets. • PHD smoothers. • Extension of the PHD filter to joint multitarget tracking and estimation of unknown spatial biases in the sensors. • The Vo-Vo exact closed-form solution of the general multitarget Bayes filter. The purpose of this introductory chapter is to set the stage for Part II by describing the standard measurement model in greater detail, and by elucidating the relationship between it and the conventional measurement-to-track association (MTA) approach to multitarget tracking (which was introduced in Section 1.1.3). The remainder of the chapter is organized as follows: 1. Section 7.1: A summary of the major lessons learned in this chapter. 2. Section 7.2: The standard multitarget measurement model.

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3. Section 7.3: An approximation of the multitarget likelihood function for the standard multitarget measurement model. 4. Section 7.4: The standard multitarget motion model. 5. Section 7.5: The standard multitarget motion model with target spawning. 6. Section 7.6: The organization of Part II.

7.1

SUMMARY OF MAJOR LESSONS LEARNED

The following are the major concepts, results, and formulas that the reader will learn in this chapter: • The formula for the p.g.fl. of the standard multitarget measurement model (see (7.19)): Gk+1 [g|X] = eκk+1 [g−1] · (1 − pD + pD Lg )X .

(7.1)

• The formula for the corresponding multitarget likelihood function (see (7.21)): fk+1 (Z|X)

=

κk+1 (Z) · (1 − pD )X ∑ ∏ pD (xi ) · fk+1 (zθ(i) |xi ) · . (1 − pD (xi )) · κk+1 (zθ(i) ) θ

(7.2)

i:θ(i)>0

• An approximate formula for this likelihood function, assuming that targets are not too close together (see (7.50)): 

fk+1 (Z|X) ∼ = κk+1 (Z) · 1 − pD +

∑ z∈Zk+1

X

pD L z  . κk+1 (z)

(7.3)

• A formula expressing the relationship between measurement-to-track association (MTA) and the multitarget likelihood function for the standard multitarget measurement model (see (7.48)): quasi-uniform prior likelihood of the MTA θ ∫ ? RFS likelihood ?? ? ? ?? ? ?? ? ∑ ? fk+1 (Zk+1 |X) · f0 (X) δX = ℓZk+1 |Xk+1|k (θ) . θ

(7.4)

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165

• The formula for the p.g.fl. of the standard multitarget motion model, without spawning (see (7.64)): ′

Gk+1 [h|X ′ ] = ebk+1 [h−1] · (1 − pS + pS Mh )X .

(7.5)

• The formula for the multitarget Markov density function for the standard multitarget motion model, without spawning (see (7.66)): fk+1|k (X|X ′ )

′

=

bk+1|k (X) · (1 − pS )X (7.6) ′ ∑ ∏ pS (xi ) · fk+1|k (xθ(i) |xi ) · . (1 − pS (x′i )) · bk+1|k (xθ(i) ) θ

7.2

i:θ(i)>0

STANDARD MULTITARGET MEASUREMENT MODEL

The standard multitarget measurement model ([179], pp. 408-422) was introduced in Section 5.5. At time tk+1 , the measurement RFS is (see (5.21)) Σk+1 = Υk+1 (x1 ) ∪ ... ∪ Υk+1 (xn ) ∪ Ck+1 ,

(7.7)

where: • x1 , ..., xn are the distinct states of n targets present at time tk+1 . • Ck+1 is the Poisson clutter RFS. • The Bernoulli RFS Υk+1 (xi ) is the set of measurements generated by the ith target, with |Υk+1 (xi )| ≤ 1. • Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 are statistically independent. It follows that Σk+1 is the union of a multi-Bernoulli RFS (the targetgenerated measurements) and a Poisson RFS (clutter). Since |Υk+1 (xi )| ≤ 1, each target either generates a measurement (a “target detection” or just “detection”) or it does not (it is “not detected” or it is a “missed detection”). This section describes the standard measurement model in more detail than in Section 5.5. It consists of the following subsections: 1. Section 7.2.1: The component submodels of the standard multitarget measurement model.

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2. Section 7.2.2: The p.g.fl. and multitarget likelihood function of the standard multitarget measurement model. 3. Section 7.2.3: Some special cases of the standard multitarget measurement model. 4. Section 7.2.4: A review of the theory of measurement-to-track association (MTA). 5. Section 7.2.5: The relationship between the standard RFS measurement model and MTA. 7.2.1

Standard Multitarget Measurement Submodels

From the RFS measurement model we get the following model functions: • Probability of detection. The probability of collecting a (single) measurement from a target with state x: pD (x) abbr. = pD,k+1 (x) = Pr(Υk+1 (x) ̸= ∅).

(7.8)

• Single-target likelihood function. This is the probability density function Lz (x) abbr. = fk+1 (z|x) =

δpk+1 (∅|x) δz

(7.9)

of the probability measure pk+1 (T |x) = Pr(Υk+1 (x) ⊆ T |Υk+1 (x) ̸= ∅).

(7.10)

It is the probability (density) that a target with state x at time tk+1 will generate measurement z. • Clutter intensity function. The PHD (first-moment density) of the clutter RFS: κk+1 (z) =

δβCk+1 (∅) δz

where

βCk+1 (T ) = Pr(Ck+1 ⊆ T ). (7.11)

• Clutter rate. The expected number of clutter measurements: ∫ λk+1 = κk+1 (z)dz.

(7.12)

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167

• Clutter spatial distribution. The spatial distribution of the clutter measurements: κk+1 (z) ck+1 (z) = . (7.13) λk+1 • Clutter p.g.f. and cardinality distribution. These are Gκk+1 (z)

=

pκk+1 (m)

=

Gκk+1 [z] = eλk+1 ·(z−1) , m 1 dm Gκk+1 −λk+1 λk+1 (0) = e · m! dz m m!

(7.14) (7.15)

where Gκk+1 [g] = eκk+1 [g−1]

(7.16)

is the p.g.fl. of the Poisson clutter RFS, where κk+1 [g − 1] =

∫

(g(z) − 1) · κk+1 (z)dz

and where it must be the case that ] [ κ dGk+1 λk+1 = (z) . dz z=1 7.2.2

(7.17)

(7.18)

Standard Multitarget Measurement Model: p.g.fl. and Likelihood

Given this, the fundamental statistical descriptors for the standard model are as follows. Let X = {x1 , ..., xn } with |X| = n and Zk+1 = {z1 , ..., zm } with |Z| = m. Then: • p.g.fl. of the standard measurement model ([179], Eq. (12.151)): Gk+1 [g|X] = eκk+1 [g−1] · (1 − pD + pD Lg )X

(7.19)

where the power functional notation hX was defined in (3.5); and where Lg (x) =

∫

g(z) · fk+1 (z|x)dz.

(7.20)

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• Multitarget likelihood function for the standard measurement model ([179], Eq. (12.139)): fk+1 (Z|X)

=

κk+1 (Z) · (1 − pD )X ∑ ∏ pD (xi ) · fk+1 (zθ(i) |xi ) · (1 − pD (xi )) · κk+1 (zθ(i) ) θ

(7.21)

i:θ(i)>0

where −λk+1 κk+1 (Z) = e−λk+1 · κZ k+1 = e

∏

κk+1 (z);

(7.22)

z∈Z

and where the summation is taken over all measurement-to-track associations (MTAs) or association hypotheses θ. MTAs are functions θ : {1, ..., n} → {0, 1, ..., m} such that θ(i) = θ(i′ ) > 0 implies i = i′ . For a given θ, • θ(i) = 0 indicates that the target xi is hypothesized to have been undetected. • By convention, the product in (7.21) equals 1 for the unique association such that θ(i) = 0 identically (that is, when none of the targets are detected). • θ(i) > 0 indicates that the target xi is hypothesized to have generated measurement zθ(i) . 7.2.3

Standard Multitarget Measurement Model: Special Cases

Three special cases of the multitarget likelihood function are of interest. • Multitarget likelihood function—no clutter, λk+1 = 0: Then fk+1 (Z|X) = 0 if m > n and, otherwise ([179], Eq. (12.136)): fk+1 (Z|X)

=

(1 − pD )X ∑ · 1≤i1 ̸=...̸=im ≤n

=

(7.23) m ∏ pD (xij ) · fk+1 (zj |xij ) 1 − pD (xij ) j=1

(1 − pD )X ∑ ∏ pD (τ (z)) · fk+1 (z|τ (z)) · 1 − pD (τ (z)) τ :Z→X z∈Z

(7.24)

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169

where the second summation is taken over all injections (that is, one-to-one functions) τ : Z → X. • Multitarget likelihood function—no missed detections, pD (x) = 1: fk+1 (Z|X) = κk+1 (Z)

∑

∏

θ

i:θ(i)>0

fk+1 (zθ(i) |xi ) . κk+1 (zθ(i) )

(7.25)

• Multitarget likelihood function—no clutter and no missed detections, λk+1 = 0, pD = 1. In this case ([179], Eq. (12.1108)): fk+1 (Z|X) = δn,m

∑

fk+1 (z|xπ1 ) · · · fk+1 (z|xπn )

(7.26)

π

where the summation is taken over all permutations π on the numbers 1, ..., n. 7.2.4

Measurement-to-Track Association (MTA)

The multitarget likelihood function for the standard multitarget measurement model has close connections with the theory underlying conventional multitarget tracking algorithms, such as the multiple-hypothesis tracker (MHT). The purpose of this and the following section is to describe this relationship in greater detail. As in Section 1.1.3, the following discussion is conceptual. It is not intended to be a description of the internal logic of any particular conventional multitarget tracking approach. For further information about such approaches, see the book by Blackman and Popoli [24]. It is first necessary to establish some notation. If θ is a MTA, then: • Zθ def. = {zi | θ(i) > 0} is the set of target detections (target-generated measurements), with mθ def. = |Zθ |. • Z − Zθ is the set of false detections and/or clutter measurements, with |Z − Zθ | = m − mθ .1 • Xθ def. = {xi | θ(i) > 0} is the set of detected tracks, with |Xθ | = mθ . • X − Xθ = {xi | θ(i) = 0} is the set of undetected tracks, with |X − Xθ | = n − mθ . 1

Note that Z − Zθ could also contain measurements from previously unknown targets. This possibility is ignored for the sake of conceptual clarity.

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Suppose that Xk+1|k = {x1 , ..., xn } with |Xk+1|k | = n are the predicted tracks at time tk+1 , with respective track distributions fk+1|k (x|1), ..., fk+1|k (x|n). Also suppose that Zk+1 = {z1 , ..., zm } with |Z| = m is the set of new measurements collected at time tk+1 . In conventional multitarget tracking theory, the goal is to determine which measurements originated with which predicted tracks or, alternatively, which measurements are clutter-generated. In what follows, let z ∈ Zk+1 . Then the total likelihood that z is associated with the ith predicted track, given that the track was detected, is ∫ ℓk+1 (z|i) = pD (x) · fk+1 (z|x) · f (x|i)dx. (7.27) This measures the degree to which the measurement distribution matches the track distribution, given the degree to which the track can be detected. Also, ∫ ℓk+1 (∅|i) = (1 − pD (x)) · f (x|i)dx (7.28) is the total likelihood that the ith track will not be detected at all. The following discussion summarizes the basic elements of the theory of MTA (also known as data association). • No clutter, no missed detections: In this case m = n and a MTA θ is just a permutation on the numbers 1, ..., n, and pD (x) = 1. The quantity ℓZk+1 |Xk+1|k (θ)

= =

ℓk+1 (zθ(1) |1) · · · ℓk+1 (zθ(n) |n) ∏ ℓk+1 (zθ(i) |i)

(7.29) (7.30)

i:θ(i)>0

is the likelihood—the global association likelihood—that the following association is true: zθ(1) associates with x1 , zθ(2) associates with x2 , and so on. The larger the number of good associations, the larger the value of the global association likelihood. • With clutter but no missed detections: In this case, m ≥ n and clutter

detections

?? ? ?? ? ?∏ ? Zk+1 −Zθ · ℓk+1 (zθ(i) |i) ℓZk+1 |Xk+1|k (θ) = e−λk+1 κk+1 i:θ(i)>0

(7.31)

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is the associated global association likelihood, where the power functional notation κZ was defined in (3.5). It is the likelihood that the association zθ(i) ⇔ xi is true for all i, and thus that the remaining measurements in Zk+1 are clutter-generated. • With clutter and missed detections: In this case the global association likelihood as three contributing factors: missed detections

clutter

detections

? ?∏ ?? ? ? ?? ? ? ∏ ?? −λk+1 Zk+1 −Zθ ℓZk+1 |Xk+1|k (θ) = e κk+1 · ℓk+1 (∅|i) · ℓk+1 (zθ(i) |i) . i:θ(i)=0

i:θ(i)>0

(7.32) • With clutter and constant probability of detection: Suppose that pD (x) = pD is constant. Then (7.27) and (7.28) become ℓk+1 (z|i) ℓk+1 (∅|i) where ℓ˜k+1 (z|i) =

∫

pD · ℓ˜k+1 (z|i) 1 − pD

(7.33) (7.34)

fk+1 (z|x) · f (x|i)dx.

(7.35)

= =

Thus (7.32) reduces to ∏

n−mθ θ ℓZk+1 |Xk+1|k (θ) = κk+1 (θ)·pm D (1−pD )

ℓ˜k+1 (zθ(i) |i) (7.36)

i:θ(i)>0

where Z

k+1 κk+1 (θ) = e−λk+1 · κk+1

−Zθ

.

(7.37)

• Global association probabilities: There is no a priori reason to prefer one association over another. Thus the prior distribution p0 (θ) on associations can be assumed to be uniform. The global association probability—the posterior probability that θ is the correct association—is the posterior distribution ℓZk+1 |Xk+1|k (θ) pZk+1 |Xk+1|k (θ) = ∑ . (7.38) ′ θ ′ ℓZk+1 |Xk+1|k (θ ) • Linear-Gaussian case with constant probability of detection. In this case, assume that fk+1 (z|x) = NR (z − Hx),

fk+1|k (x|i) = NPi (x − xi ).

(7.39)

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Then (7.27) and (7.28) reduce to ℓk+1 (z|i)

=

pD · NR+HPi H T (z − Hxi )

(7.40)

ℓk+1 (∅|i)

=

1 − pD .

(7.41)

In this case the global association probability becomes pZk+1 |Xk+1|k (θ)

n−mθ θ κk+1 (θ) · pm D (1 − pD )

∝

·QZk+1 |Xk+1|k (θ) · e

(7.42)

− 12 dZk+1 |Xk+1|k (θ)2

where κk+1 (θ) was defined in (7.37) and where

=

dZk+1 |Xk+1|k (θ)2 (7.43) ∑ T T −1 (zθ(i) − Hxi ) (R + HPi H ) (zθ(i) − Hxi ) i:θ(i)>0

and where 1 QZk+1 |Xk+1|k (θ) = ∏

i:θ(i)>0

√

.

(7.44)

det 2π(R + HPi H T )

The quantity dZk+1 |Xk+1|k (θ) is the global association distance. The association θ that minimizes dZk+1 |Xk+1|k (θ) is the best association in a global nearest-neighbor sense. To see why (7.42) is true, note that under linear-Gaussian assumptions, (7.36) becomes (7.45)

ℓZk+1 |Xk+1|k (θ) θ pm D (1

n−mθ

=

κk+1 (θ) · − pD ) ∏ · NR+HPi H T (zθ(i) − Hxi )

=

n−mθ θ κk+1 (θ) · pm (7.46) D (1 − pD )   ∏ 1  √ · det 2π(R + HPi H T ) i:θ(i)>0   ∑ 1 · exp − (zθ(i) − Hxi )T (R + HPi H T )−1 (zθ(i) − Hxi ) 2

i:θ(i)>0

i:θ(i)>0

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from which (7.42) follows. 7.2.5

Relationship Between the MTA and RFS Approaches

Assume a priori that the n predicted tracks are equally likely to be actual targets. Then the multitarget prior distribution that describes the predicted tracks is the “quasi-uniform” multitarget prior distribution. It is defined as f0 (X) = 0 if |X| ̸= n and, if otherwise with X = {x1 , ..., xn } and |X| = n, ∑ f0 (X) = fk+1|k (x1 |π1) · · · fk+1|k (xn |πn) (7.47) π

where the summation is taken over all permutations π on the numbers 1, ..., n. If fk+1 (Z|X) is defined as in (7.21), then the following equation, which is proved in Section K.11, establishes the basic relationship between RFS theory and the conventional MTA approach: RFS theory

?∫

??

?

fk+1 (Zk+1 |X) · f0 (X)δX =

MTA theory

?∑

?? ? ℓZk+1 |Xk+1|k (θ) .

(7.48)

θ

That is, the probability (density) that the measurement set Zk+1 will be collected from the predicted tracks Xk+1|k is the same as the total likelihood of association between Zk+1 and Xk+1|k , taken over all possible associations. Remark 15 (MTA with track existence probabilities) Note that (7.47) is not the only possible prior distribution for the predicted tracks. For example, suppose that each track has an associated probability of existence qi . In this case, f0 (X) should be chosen to be a multi-Bernoulli distribution as defined in (7.48) . In this case, (7.48) would give rise to a significantly more complicated formula for ℓZk+1 |Xk+1|k (θ). Even so, it should be recognized that any MTA approach that presumes a fixed form for f0 (X) is inherently a heuristic approximation. The only theoretically rigorous choice for f0 (X) is f0 (X) = fk+1|k (X|Z (k) ).

7.3

AN APPROXIMATE STANDARD LIKELIHOOD FUNCTION

The multitarget likelihood function in (7.21) is intended for use in the measurementupdate step of the multitarget Bayes filter, (5.2). Even in particle implementations— see [179], Chapter 15—this step will be, in general, computationally challenging.

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For this reason Reuter and Dietmayer proposed a computational approximation of (7.21) for use with multitarget particle filters when targets are not too close together [246], [247]. This approximation, which presumes that there is no clutter, is: ( )X ∑ ∼ fk+1 (Z|X) = 1 − pD + pD L z (7.49) z∈Z X

where the power-functional notation h was defined in (3.5). The expression in the parentheses on the right is not mathematically well defined, however. This is because the term 1 − pD (x) is unitless, whereas the units of measurement of the summation are inverse to those of z. Nevertheless, the underlying idea can both be rendered valid and generalized. Assume that there is clutter and that it is Poisson. If targets are not too close together, an approximate multitarget likelihood for the standard measurement model is: 

fk+1 (Z|X) ∼ = κk+1 (Z) · 1 − pD +

∑ z∈Zk+1

X pD L z  κk+1 (z)

(7.50)

where the Poisson-clutter factor κk+1 (Z) is as in (7.21): |Z|

κk+1 (Z) = e−λk+1 · λk+1

∏

ck+1 (z).

(7.51)

z∈Z

This approximate equality is established in Section K.12. Note that if κk+1 (z) is constant on some bounded region (compact subset) of X, then (7.49) becomes a special case of (7.50). Remark 16 Note the right side of (7.50) ∫ does not define an actual multitarget likelihood function. This is because fk+1 (Z|X)δZ is not necessarily equal to 1. Rather, (7.50) defines an approximation of a multitarget likelihood function.

7.4

STANDARD MULTITARGET MOTION MODEL

The standard multitarget motion model ([179], pp. 466-474) was introduced in Section 5.4. It is a direct mathematical analog of the standard multitarget measurement

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model. Recall from (5.4) that it has the form Ξk+1|k = Tk+1|k (x′1 ) ∪ ... ∪ Tk+1|k (x′n′ ) ∪ Bk+1|k

(7.52)

where • x′1 , ..., x′n′ are the target states at the earlier time tk . • Bk+1|k is the Poisson target appearance (“target birth”) RFS. • The Bernoulli RFS Tk+1|k (x′i ) is the set of targets generated at time tk+1 by the ith target at time tk , with |Tk+1|k (x′i )| ≤ 1. • Tk+1|k (x′1 ), ..., Tk+1|k (x′n′ ), Bk+1|k are independent. Thus Ξk+1|k is the union of a multi-Bernoulli RFS (the existing targets) and a Poisson RFS (the newly appearing or “birth” targets). Since |Tk+1 (x′i )| ≤ 1, any existing target either survives into the next time-step, or it disappears (“target death”). From the RFS motion model we get the following model functions: • Probability of target survival. This is the probability that a target with state x′ at time tk will survive into time tk+1 : pS (x′ ) abbr. = pS,k+1|k (x′ ) = Pr(Tk+1|k (x′ ) ̸= ∅).

(7.53)

• Single-target Markov transition density. This is the probability density function δpk+1|k fk+1|k (x|x′ ) = (∅|x′ ) (7.54) δz of the probability measure pk+1|k (S|x′ ) = Pr(Tk+1|k (x′ ) ⊆ S|Tk+1|k (x′ ) ̸= ∅). It is the probability (density) that a target with state x′ at time transition to a target with state x at time tk+1 .

(7.55) tk will

• Target-appearance intensity function. The PHD (first-moment density) of the target-birth RFS: bk+1 (x) =

δβBk+1|k (∅) δx

where βBk+1|k (S) = Pr(Bk+1|k ⊆ S). (7.56)

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• Target-birth rate. The expected number of newly appearing targets: ∫ B Nk+1|k = bk+1|k (x)dx.

(7.57)

• Target-birth spatial distribution. The spatial distribution of the newly appearing targets: bk+1|k (x) sB . (7.58) k+1|k (x) = B Nk+1|k • Target-appearance p.g.f. and cardinality distribution. These are, respectively, GB k+1|k (x)

=

Nk+1|k ·(x−1) GB k+1|k [x] = e

(7.59)

pB k+1|k (n)

=

n B B (Nk+1|k )n B 1 d Gk+1|k (0) = e−Nk+1|k · n! dx n!

(7.60)

where bk+1|k [h−1] GB k+1|k [h] = e

(7.61)

is the p.g.fl. of the Poisson target appearance RFS with PHD bk+1|k (x); where ∫ bk+1|k [h − 1] = (h(x − 1) · bk+1|k (x)dx (7.62) and where it must be the case that [ B Nk+1|k

dGB k+1|k

=

(x)

]

(7.63)

.

dx x=1

Given this, the fundamental statistical descriptors for the standard motion model are as follows. Let X ′ = {x′1 , ..., x′n′ } with |X ′ | = n′ be the targets at time tk and and X = {x1 , ..., xn } with |X| = n be the targets at time tk+1 . Then: • p.g.fl. of the standard multitarget motion model ([179], Eq. (13.61)): Gk+1 [h|X ′ ] = ebk+1 [h−1] · (1 − pS + pS Mh )X where the notation hX was defined in (3.5) and where ∫ ′ Mh (x ) = h(x) · fk+1|k (x|x′ )dx.

′

(7.64)

(7.65)

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• Multitarget Markov transition density for the standard multitarget motion model ([179], Eqs. (113.42,13.43)): fk+1|k (X|X ′ )

′

bk+1|k (X) · (1 − pS )X (7.66) ′ ∑ ∏ pS (xi ) · fk+1|k (xθ(i) |xi ) · (1 − pS (x′i )) · bk+1|k (xθ(i) )

=

θ

i:θ(i)>0

where B

bk+1|k (X) = e−Nk+1|k

∏

bk+1|k (x)

(7.67)

x∈X

∫ B is the Poisson target appearance process; where Nk+1|k = bk+1|k (x)dx; and where the summation is taken over all functions θ : {1, ..., n′ } → {0, 1, ..., n} such that θ(i) = θ(i′ ) > 0 implies i = i′ . For a given θ, θ(i) = 0 indicates that the target with state xi disappeared; whereas if θ(i) > 0 then it is hypothesized to have transitioned to a target with state xθ(i) . By convention, the product in (7.66) equals 1 for the unique association such that θ(i) = 0 identically (that is, all of the targets disappear). Three special cases should be pointed out: B • Multitarget Markov density—no target births, Nk+1|k = 0 ([179], Eqs. ′ (13.38,13.39)): Then fk+1|k (X|X ) = 0 if n > n′ and, otherwise,

fk+1|k (X|X ′ ) =

(7.68)

X′

(1 − pS ) ∑ ·

n ∏ pS (x′ij ) · fk+1|k (xj |x′ij )

1≤i1 ̸=...̸=im ≤n′ j=1

=

(1 − pS )X

′

1 − pS (x′ij )

∑ ∏ pD (τ (x)) · fk+1|k (x|τ (x)) 1 − pS (τ (x)) τ

(7.69)

x∈X

where the second summation is taken over all one-to-one functions τ : X → X ′ . That is, for a given τ and X = {x1 , ..., xn } with |X| = n, the targets with states {τ (x1 ), ..., τ (xn )} ⊆ X ′ at time tk+1 are hypothesized to have transitioned to the targets x1 , ..., xn at time tk+1 ; and the other targets in X ′ are hypothesized to have disappeared.

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• Multitarget Markov density—no target deaths, pS (x′ ) = 0: fk+1|k (X|X ′ ) = bk+1 (X) ·

∑

∏

θ

i:θ(i)>0

fk+1|k (xθ(i) |x′i ) . bk+1|k (xθ(i) )

(7.70)

B • Multitarget Markov density—no target births or deaths, Nk+1 = 0, pS = 1: In this case ([179], Eq. (12.35)): ∑ fk+1|k (X|X ′ ) = δn,n′ fk+1|k (x|x′π1 ) · · · fk+1|k (x|x′πn ) (7.71) π

where the summation is taken over all permutations π on the numbers 1, ..., n.

7.5

STANDARD MOTION MODEL WITH TARGET SPAWNING

A variant of the standard motion model, (7.52), involves relaxing the assumption that Tk+1|k (x′ ) is Bernoulli, allowing |Tk+1|k (x′ )| to have values other than 0 and 1. In this case, the target x′i at time tk is said to have spawned the targets in Tk+1|k (x′i ) at time tk+1 . (A target can, of course, spawn itself.) The formulas for the p.g.fl. and the Markov transition density for the standard motion model with spawning can be found in [179], pp. 472-474.

7.6

ORGANIZATION OF PART II

Part II is organized as follows: 1. Chapter 8: The “classical” PHD and CPHD filters. This includes the general PHD filter and a zero-false-alarms (ZFA) CPHD filter. 2. Chapter 9: Practical implementation of the classical PHD and CPHD filters. 3. Chapter 10: PHD and CPHD filters for multiple sensors. 4. Chapter 11: Jump-Markov versions of the PHD and CPHD filters—a method for increasing the performance of the PHD and CPHD filters, given the presence of rapidly maneuvering, “noncooperative” targets. 5. Chapter 12: Extension of the PHD filter to estimation of unknown spatial biases in the sensors.

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6. Chapter 13: Multi-Bernoulli filters, including the Bernoulli and CBMeMBer filters. 7. Chapter 14: RFS Bayes multitarget smoothers. 8. Chapter 15: The Vo-Vo exact closed-form solution of the multitarget Bayes filter.

Chapter 8 Classical PHD and CPHD Filters 8.1

INTRODUCTION

The fundamental reasoning that leads to the “classical” PHD and CPHD filters was described in Sections 5.10.4 and 5.10.5. The purpose of this chapter is to describe these filters, as well as certain special cases and generalizations, in greater detail. 8.1.1

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter: • A general PHD filter, which allows general models for clutter and general models for target-generated measurements. Its measurement-update equation has the form Dk+1|k+1 (x) = 1 − p˜D (x) + Dk+1|k (x)

∑ P⊟Zk+1

ωP

∑

LW (x) κW + τW

(8.1)

W ∈P

where the summation is taken over all partitions P of the current measurement set Zk+1 . This filter is a consequence of Clark’s general chain rule (3.91), and can be generalized to the multisensor case (see Section 10.3). • Equation (8.1) is the underlying (but previously unrecognized) theoretical basis for previously reported PHD filters—for example, the extended-target PHD filter ([174], [226], Chapter 21) and the unresolved-target PHD filter of [175], to be described in Chapter 21.

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• While (8.1) is combinatorial in nature, researchers are devising serviceable approximations (as will be explained in more detail in Section 21.4.3.3). • An immediate special case of (8.1) is a generalization of the classical PHD filter that is valid for arbitrary clutter processes and has the same computational complexity as the classical PHD filter. Its measurement-update equation is ∑ Dk+1|k+1 (x) = 1 − pD (x) + Dk+1|k (x)

z∈Zk+1

pD (x) · Lz (x) κk+1 ({z}) κk+1 (∅) + τk+1 (x)

(8.2)

where κk+1 (Z), the multiobject density function for the clutter RFS, is evaluated at Z = ∅ and Z = {z}. • An immediate and cautionary consequence of (8.2) is (see Remark 21 of Section 8.3.3): – Merely because a clutter-like term (such as κk+1 ({z})/κk+1 (∅)) occurs in a PHD filter-like formula (such as (8.2)), this does not necessarily mean that this term is actually the intensity function (PHD) of the clutter RFS. • The measurement-update equation for the classical PHD filter is a special case of (8.2): ∑ Dk+1|k (x) pD (x) · Lz (x) = 1 − pD (x) + . Dk+1|k (x) κk+1 (z) + τk+1 (x)

(8.3)

z∈Zk+1

• The time-update and measurement-update equations for the “classical” CPHD filter (Section 8.5). • If targets are assumed to be not too close together, the classical CPHD filter can be replaced by an approximate CPHD filter that has the same computational complexity as the classical PHD filter (Section 8.5.7). • A useful special case of the classical CPHD filter, the zero false alarms (ZFA) CPHD filter, is valid when there is no clutter. It has the same

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computational complexity as the classical PHD filter. It has the measurementupdate equations (Section 8.6): (m+1)

Nk+1|k · Dk+1|k+1 (x) Dk+1|k (x)

Gk+1|k (ϕk ) =

(8.4)

(1 − pD (x)) ·

(m)

Gk+1|k (ϕk ) +

∑ pD (x) · Lz (x) τˆk+1 (z) z∈Zk+1

pk+1|k+1 (n) 8.1.2

∝

Cn,m · ϕn−m . · pk+1|k (n). k

(8.5)

Organization of the Chapter

The chapter is organized as follows: 1. Section 8.2: The general PHD filter for arbitrary clutter models and arbitrary target-generated measurements. 2. Section 8.3: The generalization of the classical PHD filter to arbitrary clutter processes. 3. Section 8.4: The classical PHD filter. 4. Section 8.5: The classical CPHD filter. 5. Section 8.6: The zero-false-alarms (ZFA) CPHD filter. 6. Section 8.7: A generalization of the PHD filter to the case when the Poisson clutter depends on the states of the targets.

8.2

A GENERAL PHD FILTER

The classical PHD filter is based on the standard multitarget measurement model described in Section 7.2. This model—targets generate at most single measurements and the clutter RFS is Poisson—ensures that the classical PHD filter is computationally tractable. However, there is a much more general PHD filter, originally reported in [47], that is based on the generalized standard multitarget measurement model described at the end of Section 5.5. It allows both the clutter and target measurementgeneration processes to be arbitrary.

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This filter has the following limitation: the PHD measurement-update equation involves a combinatorial sum over all partitions of the current measurement set. Despite this fact, serviceable approximations are currently being devised for measurement-update equations of this type. The purpose of this section is to summarize this general PHD filter. A multisensor version of the filter is described in Section 10.3. The classical PHD filter is based on a multitarget measurement model of the form Σk+1 = Υk+1 (x1 ) ∪ ... ∪ Υk+1 (xn ) ∪ Ck+1 (8.6) where the clutter RFS Ck+1 is Poisson, where Υk+1 (x) is a Bernoulli RFS for all x, and where Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 are statistically independent. In this section, we will allow both Ck+1 and Υk+1 (x) to be arbitrary. The general PHD filter has the same motion model as the classical PHD filter, namely Ξk+1|k = Tk+1|k (x′1 ) ∪ ... ∪ Tk+1|k (x′n ) ∪ Bk+1|k (8.7) where x′1 , ..., x′n are the target states at time tk ; where the target appearance RFS Bk+1|k is Poisson; where Tk+1|k (x′ ) is Bernoulli, and where Tk+1|k (x′1 ), ..., Tk+1|k (x′n ), Bk+1|k are independent. Remark 17 (Cluster processes) The RFS of target-generated measurements provides a specific instance of a cluster process, as briefly discussed in Example 2 of Section 4.4.2. In the notation of that section, Ψ = Ξk+1|k is the parent process, ∆x = Υx abbr. = Υk+1 (x) is the daughter process, and the target-generated measurement RFS ∪ ∆ = Σk+1 = Υk+1 (x) (8.8) x∈Ξk+1|k

is the total cluster process. The joint measurement-target p.g.fl. is, from (4.141), F [g, h] = GΞk+1|k ,Σk+1 [g, h] = GΞk+1|k [h · GΥ∗ [g]] where T [g](x) abbr. = GΥx [g] =

∫

g Z · fΥx (Z)δZ

(8.9)

(8.10)

are the p.g.fl.’s of the daughter RFSs, and where GΞk+1|k [h · GΥ∗ [g]] is shorthand for GΞk+1|k [h · T [g]]. Similar remarks apply to the RFS of surviving targets.

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The section is organized as follows: 1. Section 8.2.1: Motion modeling for the general PHD filter (as well as for the classical PHD filter). 2. Section 8.2.2: The time-update equation for the general PHD filter (as well as for the classical PHD filter). 3. Section 8.2.3: Measurement modeling for the general PHD filter. 4. Section 8.2.4: The measurement-update equation for the general PHD filter. 8.2.1

General PHD Filter: Motion Modeling

The following motion models apply not just to this filter, but also to the classical PHD filter (Section 8.4.1) and its generalization to arbitrary clutter processes (Section 8.3.1). 1. Single-target Markov transition density: fk+1|k (x|x′ ). This is the probability (density) that a target with state x′ at time tk will have state x at time tk+1 . 2. Target-survival probability: pS (x′ ) abbr. = pS,k+1 (x′ ). This is the probability ′ that a target with state x at time tk will not disappear at time tk+1 . 3. Target-appearance PHD: bk+1|k (x). This is the PHD (intensity function) of Birth the multitarget distribution fk+1|k (X)—that is, of the probability (density) that a set X of new targets will appear in the scene at time tk+1 . The quantity ∫ B Nk+1|k =

bk+1|k (x)dx

(8.11)

is the target-birth rate (the expected number of newly appearing targets); and sB k+1|k (x) =

bk+1|k (x) B Nk+1|k

(8.12)

is the spatial distribution of the appearing targets. 4. Target-spawning PHD: bk+1|k (x|x′ ). This is the PHD of the multitarget Spawn distribution fk+1|k (X|x′ )—that is, of the probability (density) that a target ′ with state x at time tk will spawn a set X of new targets at time tk+1 .

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The quantity B Nk+1|k (x′ ) =

∫

bk+1|k (x|x′ )dx

(8.13)

is the target-spawning rate (the expected number of targets spawned by the target with state x′ ); and ′ sB k+1|k (x|x ) =

bk+1|k (x|x′ ) B Nk+1|k (x′ )

(8.14)

is the spatial distribution of the targets spawned by the target with state x′ . 8.2.2

General PHD Filter: Predictor

The following time-update equations apply not only to the general PHD filter but also to the classical PHD filter (Section 8.4.1) and its generalization to arbitrary clutter processes (Section 8.3.1). Suppose that we already have in hand the PHD Dk|k (x) and the expected number of targets Nk|k . We are to determine the predicted PHD Dk+1|k (x) and the predicted expected number of targets Nk+1|k . The predicted PHD is given by the exact (not approximate) equation1 Dk+1|k (x) = bk+1|k (x) +

∫

Fk+1|k (x|x′ ) · Dk|k (x′ )dx′

(8.15)

where the PHD filter pseudo-Markov density is Fk+1|k (x|x′ ) = pS (x′ ) · fk+1|k (x|x′ ) + bk+1|k (x|x′ ).

(8.16)

The expected number of predicted targets is, therefore, Nk+1|k

= =

1

∫

Dk+1|k (x)dx (8.17) ∫ ( ) B B Nk+1|k + pS (x′ ) + Nk+1|k (x′ ) · Dk|k (x′ )dx′ . (8.18)

Note: No special assumption is made regarding the nature of the prior multitarget distribution fk|k (X|Z (k) ). In particular, it is not presumed to be Poisson.

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Remark 18 (Special cases of the time-update equation) For the special case in which no target-spawning occurs, these formulas simplify to: Dk+1|k (x)

=

Nk+1|k

=

∫ bk+1|k (x) + pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′ (8.19) ∫ B Nk+1|k + pS (x′ ) · Dk|k (x′ )dx′ . (8.20)

If in addition no targets appear at all, they further simplify to Dk+1|k (x) Nk+1|k

=

∫

pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′

(8.21)

=

∫

pS (x′ ) · Dk|k (x′ )dx′ .

(8.22)

If in addition no targets disappear, they finally simplify to Dk+1|k (x) Nk+1|k 8.2.3

=

∫

fk+1|k (x|x′ ) · Dk|k (x′ )dx′

(8.23)

=

∫

Dk|k (x′ )dx′ = Nk|k .

(8.24)

General PHD Filter: Measurement Modeling

The time-update formulas for the general PHD filter require the following models: • Multi-measurement, single-target likelihood function. This is the multitarget probability density function of Υk+1 (x): LZ (x) abbr. = fk+1 (Z|x) =

δGxk+1 [0] δZ

(8.25)

where Gxk+1 [g] is the p.g.fl. of Υk+1 (x). • Generalized probability of detection. This is the probability that at least one measurement is collected from a target with state x, p˜D (x) abbr. = p˜D,k+1 (x) = 1 − fk+1 (∅|x).

(8.26)

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• Log-clutter density. This is the multiobject density function of the clutter log-p.g.fl. log Gκk+1 [g], where Gκk+1 [g] is the p.g.fl. of the clutter RFS Ck+1 : δ log Gκk+1 κZ = [0]. (8.27) δZ • Nonubiquity of clutter. In order to ensure that log Gκk+1 [g] is well defined, we must assume that it is possible for no clutter measurements to be collected: pκk+1 (0) > 0.

(8.28)

(Note that this assumption is automatically true for Poisson clutter.) As a consequence, Gκk+1 [g] > 0 for all g and thus log Gκk+1 [g] is well defined. 8.2.4

General PHD Filter: Corrector

This equation is a consequence of Clark’s general chain rule, (3.91): Dk+1|k+1 (x) = LZk+1 (x) · Dk+1|k (x)

(8.29)

where the PHD pseudolikelihood is given by ([47], Eqs. (27-29)) LZk+1 (x) = 1 − p˜D (x) +

∑ P⊟Zk+1

ωP

∑

LW (x) . κW + τW

(8.30)

W ∈P

Here, the summation is taken over all partitions P of the measurement set Zk+1 and ∫ τW = LW (x) · Dk+1|k (x)dx (8.31) ∏ W ∈P (κW + τW ) ∏ ωP = ∑ . (8.32) Q⊟Zk+1 V ∈Q (κV + τV ) For a proof, see [47], Section IV. See Appendix D for an introduction to the theory of partitions. Remark 19 (Computational complexity of general PHD filter) The practical utility of (8.30) might be questioned because of the combinatorial sum. However, similar combinatorial sums occur in the corrector equation for the extended-target PHD

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filter of [174], to be described in Chapter 21. Yet serviceable approximations are being developed despite this fact [226], [285]—see Section 21.4.3. It should be also be remembered that the “ideal” multihypothesis tracker (MHT) is also inherently combinatorial. Nevertheless, numerous techniques have been developed to make it computationally practical.

8.3

ARBITRARY-CLUTTER PHD FILTER

An immediate consequence of (8.29) through (8.32) is that: • The measurement-update for the classical PHD filter (Section 8.4.3) can be generalized to include arbitrary clutter processes, but without increasing computational complexity. This filter is described in this section. 8.3.1

Time Update Equations for the Arbitrary-Clutter Classical PHD Filter

The motion models and time-update equations for this filter are the same as those for the general PHD filter—see Sections 8.2.1 and 8.3.1. 8.3.2

Measurement Modeling for the Arbitrary-Clutter Classical PHD Filter

This filter employs the following models. Because of (5.42), the generalized probability of detection p˜D (x) of (8.26) reduces to the conventional probability of detection: p˜D (x) = pD (x). (8.33) Similarly, because of (5.49) the general single-target likelihood function LZ (x), (8.25), reduces to  if Z=∅  1 − pD (x) δGxk+1 pD (x) · Lz (x) if Z = {z} . LZ (x) = [0] = (8.34)  δZ 0 if |Z| > 1 Thus we end up with the following models: 1. Sensor probability of detection: pD (x) abbr. = pD,k+1 (x). This is the probability that a target with state x at time tk+1 will generate some measurement.2 2

The generalized probability of detection p˜D (x) in (8.26) reduces to this: p˜D (x) = pD (x).

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2. Sensor likelihood function: Lz (x) abbr. = fk+1 (z|x). This is the probability (density) that, if a target with state x at time tk+1 does generate a measurement, it will generate measurement z. 3. Arbitrary clutter distribution κk+1 (Z), of which only two aspects—κk+1 ({z}) and κk+1 (∅) with κk+1 (∅) > 0—need be known a priori. 8.3.3

Arbitrary-Clutter PHD Filter: Corrector

Under these assumptions, (8.30) reduces to ∑

LZk+1 (x) = 1 − pD (x) +

pD (x) · Lz (x) κ ˜ k+1 (z) + τk+1 (z)

(8.35)

z∈Zk+1

where (8.31) reduces to τk+1 (z) =

∫

pD (x) · Lz (x) · Dk+1|k (x)dx

(8.36)

and where the clutter “pseudointensity” function is, because of (8.27), κ ˜ k+1 (z) =

κk+1 ({z}) . pκk+1 (0)

(8.37)

Remark 20 (Derivation of arbitrary-clutter PHD filter) Because of (8.34), LW = 0 whenever |W | > 1. Thus the only term in (8.30) that survives is the one corresponding to that single partition P of Zk+1 whose cells are the |Zk+1 | singleton subsets of Zk+1 . Thus (8.30) reduces to ∑

L{z} (x) κ{z} + τ{z}

(8.38)

pD (x) · Lz (x) · Dk+1|k (x)dx

(8.39)

LZk+1 (x) = 1 − pD (x) +

z∈Zk+1

where from (8.31) τ{z} =

∫

L{z} (x) · Dk+1|k (x)dx =

∫

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and where from (8.27) κ{z}

= =

[ ] δ log Gκk+1 δ log Gκk+1 [0] = [g] δz δz g=0 [ ] κ δGk+1 1 [g] Gκk+1 [g] δz g=0 1 κk+1 ({z}) · κk+1 ({z}) = κ . Gκk+1 [0] pk+1 (0)

=

(8.40) (8.41) (8.42)

Thus (8.38) becomes (8.35). Remark 21 (Cautionary note) The form of (8.35), namely LZk+1 (x) = 1 − pD (x) +

∑

pD (x) · Lz (x) , κ ˜ k+1 (z) + τk+1 (x)

(8.43)

z∈Zk+1

might lead one to assert that κ ˜ k+1 (z) is the PHD (intensity function) κk+1 (z) =

δGκk+1 δ log Gκk+1 [1] = [1] δz δz

(8.44)

of the clutter RFS Ck+1 . However, this is clearly not true since, in general, δ log Gκk+1 δ log Gκk+1 [0] ̸= [1]. δz δz

(8.45)

Thus, just because one sees a clutter intensity-like function κ ˜ k+1 (z) in an equation such as (8.43), this does not necessarily mean that κ ˜ k+1 (z) is actually a clutter intensity function. A separate proof is required to demonstrate the fact. We will have occasion to revisit this issue with regard to “clutter agnostic” PHD and CPHD filters in Section 18.5.5.

8.4

CLASSICAL PHD FILTER

The classical PHD filter arises when we assume in (8.35) that the clutter is Poisson: κk+1 (Z) = e−λk+1 · κZ k+1 ,

(8.46)

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in which case (8.37) reduces to

κ ˜ k+1 (z) =

κk+1 ({z}) e−λk+1 · κk+1 (z) = = κk+1 (z). κ pk+1 (0) e−λk+1

(8.47)

The purpose of this section is to describe the classical PHD filter and its major characteristics. The section is organized as follows: 1. Section 8.4.1: The time-update equations for the classical PHD filter. 2. Section 8.4.2: Measurement-modeling assumptions for the classical PHD filter. 3. Section 8.4.3: The measurement-update equations for the classical PHD filter. 4. Section 8.4.4: Multitarget state estimation for the classical PHD filter. 5. Section 8.4.5: Multitarget uncertainty estimation for the classical PHD filter. 6. Section 8.4.6: Characteristics of the classical PHD filter. 8.4.1

Classical PHD Filter: Predictor

The motion models and time-update equations for the classical PHD filter are the same as those for the general PHD filter (see Section 8.2.2). 8.4.2

Classical PHD Filter: Measurement Modeling

The PHD filter measurement-update formulas in Section 8.4.3 require the following models (originally defined in Section 7.2): 1. Sensor probability of detection: pD (x) abbr. = pD,k+1 (x). This is the probability that a target with state x at time tk+1 will generate some measurement. 2. Sensor likelihood function: Lz (x) abbr. = fk+1 (z|x). This is the probability (density) that, if a target with state x at time tk+1 does generate a measurement, it will generate measurement z. 3. Clutter intensity function (also known as clutter PHD) κk+1 (z). This is the PHD of the multiobject distribution κk+1 (Z), which, in turn, is the probability (density) that a set Z of clutter measurements will be generated

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at time tk+1 . The quantity λk+1 =

∫

κk+1 (z)dz

(8.48)

is the clutter rate (expected number of clutter measurements); and ck+1 (z) =

κk+1 (z) λk+1

(8.49)

is the clutter spatial distribution. 8.4.3

Classical PHD Filter: Corrector

Suppose that we have: • A new measurement set Zk+1 = {z1 , ..., zm } with |Zk+1 | = m. • The predicted PHD Dk+1|k (x). • The predicted expected number of targets Nk+1|k . We are to determine the measurement-updated PHD Dk+1|k+1 (x) and the measurement-updated expected number of targets Nk+1|k+1 . To achieve closedform formulas, the following assumption is required: • The predicted multitarget distribution fk+1|k (X|Z (k) ) is Poisson. The measurement-updated PHD is given by Dk+1|k+1 (x) = LZk+1 (x) · Dk+1|k (x)

(8.50)

where the PHD filter pseudolikelihood function is LZ (x) = 1 − pD (x) +

∑

pD (x) · Lz (x) κk+1 (z) + τk+1 (z)

(8.51)

z∈Z

and where τk+1 (z) =

∫

pD (x) · Lz (x) · Dk+1|k (x)dx.

(8.52)

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The expected number of target tracks is, therefore, found by integrating Dk+1|k+1 (x): Nk+1|k+1 = Dk+1|k [1 − pD ] +

∑

τk+1 (z) κk+1 (z) + τk+1 (z)

(8.53)

z∈Zk+1

where Dk+1|k [1 − pD ] =

∫

(1 − pD (x)) · Dk+1|k (x)dx.

Remark 22 (A useful identity) Let ∫ fk+1 (Z|Z (k) ) = fk+1 (Z|X) · fk+1|k (X|Z (k) )δX

(8.54)

(8.55)

be the Bayes normalization factor and assume that fk+1|k (X|Z (k) ) is Poisson. Then the following identity is sometimes useful ([165], Eq. (116)): ∏ fk+1 (Z|Z (k) ) = e−λk+1 −Dk+1|k [pD ] (κk+1 (z) + τk+1 (z)) (8.56) z∈Z

where Dk+1|k [pD ] = 8.4.4

∫

pD (x) · Dk+1|k (x|Z (k) )dx.

(8.57)

Classical PHD Filter: State Estimation

The following procedure is commonly employed to estimate the current number of targets and their states. 1. The quantity Nk+1|k+1 in (8.53) is the expected number of targets. Round it off to the nearest integer n. 2. Look for the n largest local suprema x1 , ..., xn of Dk+1|k+1 (x|Z (k+1) )— that is, those x corresponding to its n largest “peaks.” 3. Take the x1 , ..., xn to be the estimates of the states of the target tracks. 4. In the event that there are fewer than n local suprema, the states corresponding to the actual number n′ of local suprema are taken to be the state estimates. In this case, it is implicitly understood that some targets are so close together that they correspond to a single peak of the PHD. Additionally, if the clutter rate is large then it is possible for the states corresponding to clutter-induced peaks to be selected as estimated target states.

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195

Classical PHD Filter: Uncertainty Estimation

The PHD filter is a “first order” approximation of the multitarget Bayes filter, in a point process sense. This does not mean, however, that the PHD filter cannot supply second-order information in the conventional sense—that is, track covariances. Let x1 , ..., xn be the track state estimates. Then for a given xi , the basic idea is to determine the covariance matrix Pi such that the Gaussian NPi (x − xi ) is, according to some criterion, a best fit to Dk+1|k+1 (x) at x = xi . For Gaussian mixture (GM) implementations of the PHD filter, determination of Pi is simple. It is more difficult for sequential Monte Carlo (SMC) implementations, because data-clustering techniques are required. See Sections 9.5 and 9.6 for more details. 8.4.6

Classical PHD Filter: Characteristics

The purpose of this section is to summarize some of the more important points regarding the classical PHD filter. These are: 1. Section 8.4.6.1: Consistency property of the PHD filter. 2. Section 8.4.6.2: Computational complexity of the PHD filter. 3. Section 8.4.6.3: “Target-like” measurements versus “clutter-like” measurements. 4. Section 8.4.6.4: The “self-gating” property of the PHD filter. 5. Section 8.4.6.5: The linearizing effect of the PHD filter. 6. Section 8.4.6.6: Window-averaging to achieve better target-number estimates. 7. Section 8.4.6.7: A generalization of the PHD filter in which the Poisson approximation is relaxed to a Gauss-Poisson approximation. 8. Section 8.4.6.8: Alternative mathematical derivations of the PHD filter. 8.4.6.1

Classical PHD Filter: Consistency Property

Assume that the scenario is single-sensor, single-target. That is, suppose that: (1) there is a single target, (2) there is a single sensor, (3) there are no missed detections,

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and (4) there are no false alarms/clutter. Given these assumptions, the PHD filter reduces to the single-sensor, single-target Bayes filter. To see why, write Dk+1|k (x) = fk+1|k (x) and Dk|k (x) = fk|k (x) where fk+1|k (x) and fk|k (x) are probability density functions. Under our assumptions, (8.23) reduces to: ∫ fk+1|k (x) = fk+1|k (x|x′ ) · fk|k (x′ )dx′ . (8.58) This is the time-update equation for the single-sensor, single-target Bayes filter, (2.25). Under our assumptions, the measurement-update for Zk+1 = {z1 }, (8.50) and (8.51), reduce to: fk+1|k+1 (x)  =

= =

(8.59) 

m ∑

pD (x) · Lzj (x)  · fk+1|k (x) κ (z ) + τ (z ) k+1 j k+1 j j=1 ( ) 1 · Lz1 (x) 1−1+ · fk+1|k (x) fk+1|k [1 · Lz1 ]

1 − pD (x) +

Lz1 (x) · fk+1|k (x) . fk+1|k [Lz1 ]

(8.60) (8.61)

This is the measurement-update equation for the single-sensor, single-target Bayes filter—that is, Bayes’ rule, (2.26). 8.4.6.2

Classical PHD Filter: Computational Complexity

The PHD filter has favorable computational characteristics. Inspection of (8.51) reveals that the PHD filter measurement-update step has computational order O(mn), where m is the current number of measurements and n is the current number of tracks. For, suppose that Dk+1|k (x) has the form

Dk+1|k (x) =

n ∑ i=1

wi · fi (x)

(8.62)

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where fi (x) is the probability distribution of a target track. Then the targetdetection part of Dk+1|k+1 (x) has the form n ∑ m ∑ pD (x) · Lzj (x) · wi · fi (x)

.

(8.63)

κk+1 (zj ) + τk+1 (zj )

i=1 j=1

Another way of looking at the computational complexity of the PHD filter is as follows. Suppose that probability of detection pD is constant. If there are n tracks currently, then on average these tracks will generate pD n measurements. ∫ If λ = κk+1 (z)dz is the current clutter rate, then the total current number of measurements is, on average, pD n + λ. Thus the computational order of the PHD filter will be O(n · (pD n + λ)) = O(pD n2 + λn). (8.64) Thus if pD and n are large, the computational load will also be large. For this reason, in practice it may be advantageous, if possible, to partition the targets into statistically noninteracting clusters, and then apply a separate PHD filter to each cluster. This issue is discussed in more detail in Section 9.2. 8.4.6.3

“Target-Like” versus “Clutter-Like” Measurements

Equations (8.50) and (8.51) can be rewritten in the form Dk+1|k+1 (x)

=

(1 − pD (x)) · Dk+1|k (x) ∑ + ωk+1|k (z) · sk+1|k+1 (x|z)

(8.65)

z∈Zk+1

where ωk+1|k (z)

=

sk+1|k+1 (x|z)

=

τk+1 (z) κk+1 (z) + τk+1 (z) pD (x) · Lz (x) · Dk+1|k (x) . τk+1 (z)

(8.66) (8.67)

The first term of (8.65) is the PHD corresponding to undetected targets, and its integral Dk+1|k [1 − pD ] is the expected number of undetected targets. The second term—the part of the PHD corresponding to the detected targets—is a weighted sum of probability distributions sk+1|k+1 (x|z).

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The weight 0 ≤ ω(z) ≤ 1 determines the degree to which sk+1|k+1 (x|z), and therefore z, contributes to the detected-target PHD. For each z, if ωk+1|k (z) > then the measurement z target. If

1 2

is target-like—that is, it was probably generated by a

1 2 then z is clutter-like—that is, it was probably not generated by a target. ωk+1|k (z) n—see (2.1)—it follows that pk+1|k+1 (n) = 0 for all n < m. 8.6.1

Comparison of the PHD and ZFA-CPHD Filters

Multiplying both sides of (8.134) by Nk+1|k+1 and using (8.52) to note that τk+1 (z) = Nk+1|k · τˆk+1 (z), we get (m+1)

m Dk+1|k+1 (x) 1 − pD (x) Gk+1|k (ϕk ) ∑ pD (x) · Lzj (x) = · (m) + . Dk+1|k (x) Nk+1|k τk+1 (zj ) Gk+1|k (ϕk ) j=1

(8.142)

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Note that the PHD filter with no clutter has the form (see (8.50) and (8.51)) m ∑ Dk+1|k+1 (x) pD (x) · Lzj (x) = 1 − pD (x) + . Dk+1|k (x) τk+1 (zj ) j=1

(8.143)

From this we may conclude that the ZFA-CPHD filter will behave much like the PHD filter—except that the nondetection term will be more accurate because of the factor (m+1) Gk+1|k (ϕk ) 1 · (m) . Nk+1|k G (ϕk ) k+1|k

This in part should help produce more accurate and stable instantaneous estimates of target number—see the example that follows. Nevertheless, if pD (x) = 1 identically then the ZFA-CPHD filter reduces to the PHD filter. Example 5 (ZFA-CPHD filter one target) Assume that the scene contains at most a single target, in which case Gk+1|k (x) = 1 − Nk+1|k + Nk+1|k · x with Nk+1|k ≤ 1. Assume also that ϕk = 1 − pD and (8.142) becomes

(8.144)

pD is constant and that m = 0. Then

(1)

1 − pD Gk+1|k (ϕk ) · · Dk+1|k (x) Nk+1|k Gk+1|k (ϕk ) Nk+1|k 1 − pD · · Dk+1|k (x) Nk+1|k 1 − Nk+1|k + Nk+1|k · ϕk 1 − pD · Dk+1|k (x) 1 − Nk+1|k · pD

Dk+1|k+1 (x) = = =

(8.145) (8.146) (8.147)

from which we get Nk+1|k+1 =

(1 − pD ) · Nk+1|k . 1 − Nk+1|k · pD

(8.148)

Comparing this to (8.73), we conclude: When there is no clutter, the ZFA-CPHD filter should have better performance than the PHD filter.

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8.7

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The standard multitarget measurement model was given in (7.7): Σk+1 = Υk+1 (x1 ) ∪ ... ∪ Υk+1 (xn ) ∪ Ck+1 ,

(8.149)

where the target-measurement RFS Υk+1 (x) is Bernoulli for every x; where the clutter RFS Ck+1 is Poisson; and where Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 are independent. In many applications, clutter Ck+1 is not independent of the targets x1 , ..., xn . One example is multipath propagation, in which a radar or sonar pulse is reflected from various surfaces, thus creating multiple returns at the receiver. Another example is radio frequency (RF) spoofing by aircraft. In RF spoofing, electronic countermeasure (ECM) devices on the aircraft receive impinging radar pulses from an air defense system. These devices then retransmit delayed versions of these pulses back to the radar. The effect is to create one or more false targets. Vo, Vo, and Cantoni have devised a generalization of the Bernoulli filter that detects and tracks a single target in Poisson state-dependent clutter [309]. The purpose of this section, however, is to propose a PHD filter for detecting and tracking multiple targets in target-dependent Poisson clutter. It employs a measurement model first considered in [179], pp. 424-426. Remark 25 (Erratum) In [173], Mahler proposed a PHD filter for target-dependent Poisson clutter. The reader is warned that the derivation of this filter appears to be incorrect. The PHD filter proposed in what follows therefore replaces the one proposed in [173]. Consider a clutter RFS that has the form 0 Ck+1 = Ck+1 (x1 ) ∪ ... ∪ Ck+1 (xn ) ∪ Ck+1

(8.150)

where 0 • Ck+1 is independent Poisson background clutter, with intensity function ∫ 0 κk+1 (z) and clutter rate λ0k+1 = κ0k+1 (z)dz.

• Ck+1 (x) is Poisson clutter associated with a target with ∫ state x, with intensity function κk+1 (z|x) and clutter rate λk+1 (x) = κk+1 (z|x)dz. 0 • Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 (x1 ), ..., Ck+1 (xn ), Ck+1 are independent.

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Then the total measurement RFS is 0 ˜ k+1 (x1 ) ∪ ... ∪ Υ ˜ k+1 (xn ) ∪ Ck+1 Σk+1 = Υ

(8.151)

˜ k+1 (x) = Υk+1 (x) ∪ Ck+1 (x). Υ

(8.152)

where We can then apply the measurement-update formula for the general PHD filter of Section 8.2 to get the following: • Measurement update equations for PHD filter for target-dependent Poisson clutter: This is (8.153)

Dk+1|k+1 (x) = LZk+1 (x) · Dk+1|k (x) where the PHD pseudolikelihood is LZk+1 (x) = e−λk+1 (x) ·(1− pD (x))+

∑ P⊟Zk+1

ωP

∑

LW (x) (8.154) κW + τW

W ∈P

where the summation is taken over all partitions P of Zk+1 , and where LW (x)

=

e

−λk+1 (x)

·

(

∏

κk+1 (z|x)

)

(8.155)

z∈W

[

κW

=

τW

=

∑ pD (x) · Lz (x) · 1 − pD (x) + κk+1 (z|x) z∈W  0  e−λk+1 if W =∅ κ0k+1 (z) if W = {z}  0 if |W | ≥ 2 ∫ LW (x) · Dk+1|k (x)dx

and where ωP = ∑

∏

W ∈P (κW

Q⊟Zk+1

∏

+ τW ) . V ∈Q (κV + τV )

The derivation can be found in Section K.18.

]

(8.156)

(8.157)

(8.158)

Chapter 9 Implementing Classical PHD/CPHD Filters 9.1

INTRODUCTION

The purpose of this chapter is to describe the major methods and issues associated with practical implementation of the classical PHD and CPHD filters. The filtering equations for both filters involve multidimensional integrals, and so further approximation is necessary to create computationally tractable algorithms. Two approaches have gained currency: Gaussian-mixture (GM) implementation and sequential Monte Carlo (SMC, also known as “particle”) implementation. SMC implementation was independently proposed in 2003 by Sidenbladh [271], Zajic and Mahler [330], and Vo, Singh, and Doucet [306]; while GM implementation was introduced in 2005 by Vo and Ma [299]. Both approaches have subsequently been extended and refined in several directions, to be described shortly. A second major development in the field has been an increasingly deeper understanding of the practical behavior of both filters. The most notable development has been analysis of the “spooky action at a distance” phenomenon, first pointed out by Fr¨anken and Ulmke [293], [88]. 9.1.1

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter: • PHD and CPHD filters can be implemented using Gaussian mixture methods, including the following variants: extended Kalman filter (EKF), unscented

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Kalman filter (UKF), cubature Kalman filter (CKF), and Gaussian particle filter (GPF). See Sections 9.5.4 and 9.5.5. • PHD and CPHD filters can be implemented using sequential Monte Carlo (SMC, also known as “particle”) methods. See Section 9.6. • Because of the phenomenon known as “spooky action at a distance,” PHD and CPHD filters tend to shift probability mass from undetected tracks to detected ones. See Section 9.2. • Because of this “spookiness,” a scenario should be partitioned into statistically noninteracting target-clusters, and a separate PHD or CPHD filter applied to each cluster. See Section 9.2. • Typical SMC implementations of PHD and CPHD filters require a complicated and computationally demanding “clustering” step in order to estimate the number and states of the targets. A new “measurement-driven” implementation approach, due to Ristic, Clark, Vo, and Vo, avoids this step as well as other difficulties. See Section 9.6.4. • A similar approach results in efficient selection of the target-birth process in Gaussian-mixture implementations of PHD and CPHD filters. See Section 9.5.7. • Gaussian mixture implementations of PHD and CPHD filters can be extended to include target type and therefore target classification. See Section 9.5.6. • The same is true of sequential Monte Carlo implementations of these filters. See Section 9.6.5. 9.1.2

Organization of the Chapter

The chapter is organized as follows: 1. Section 9.2: “Spooky action at a distance”—referring to the tendency of the PHD and CPHD filters to shift probability mass from undetected tracks to detected ones. 2. Section 9.3: Cluster-merging and cluster-splitting for PHD filters. 3. Section 9.4: Cluster-merging and cluster-splitting for CPHD filters. 4. Section 9.5: Gaussian-mixture implementation of PHD and CPHD filters.

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5. Section 9.6: Sequential Monte Carlo (SMC, also known as “particle”) implementation of PHD and CPHD filters.

9.2

“SPOOKY ACTION AT A DISTANCE”

The PHD and CPHD filters are, at their core, cluster trackers. That is, when confronted with a multiple-target scenario, both filters first try to interpret it as a target cluster. Subsequently, they attempt to resolve individual targets only as the quantity and quality of measurements permit this to be accomplished. In [293] and [88], Fr¨anken and Ulmke noted that the cluster-tracker nature of PHD and CPHD filters produces the following two counterintuitive behaviors: 1. “Spooky action at a distance”: The classical PHD and CPHD filters both shift PHD mass away from undetected tracks to detected tracks—even if these tracks are so distant from each another (with respect to sensor resolution) that they are statistically noninteracting. 2. Violation of superposition: Applying CPHD filters separately to widelyseparated target-clusters does not produce the same result as applying a single CPHD filter to the entire scene. The same is not true for the PHD filter, however. Consequently, for the purpose of practical implementation: • A multitarget scene should first be partitioned into statistically noninteracting target-clusters, with CPHD filters applied separately to each of the clusters. • As was discussed in Section 8.4.6.2, this approach is routinely applied in multihypothesis trackers (MHTs) to reduce computational complexity [245], [23], [24]. It must be applied to PHD and CPHD filters for both theoretical and practical reasons. Points 1 and 2 above can be demonstrated analytically using a simple example. Assume a scenario with the following assumptions: 1. There are at most two tracks, which are well separated with respect to sensor resolution. 2. Both tracks have track probability 0 < a ≤ 1. 3. Both targets are static, and their initial track distributions are f1 (x) and f2 (x).

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4. There are no false alarms and the probability of detection pD is constant. 5. A measurement z1 is collected from the first track, but the second track is not detected. Under these assumptions, the prior multitarget distribution is a multi-Bernoulli distribution of degree two:  2 (1 − a) if X=∅     a(1 − a) · (f1 (x) + f2 (x)) if X = {x}  f0|0 (X) = a2 · (f1 (x1 ) · f2 (x2 ) + f1 (x2 ) · f2 (x1 )) if X = {x1 , x2 },   |X| = 2    0 if otherwise (9.1) The prior p.g.fl., p.g.f., cardinality distribution, PHD, and expected number of targets are, respectively, G0|0 [h]

=

(1 − a + a · f1 [h]) · (1 − a + a · f2 [h]) 2

G0|0 (x)

=

(9.3)

(1 − a + a · x) n

(9.2)

2−n

p0|0 (n)

=

C2,n · a (1 − a)

(9.4)

D0|0 (x) N0|0

= =

a · f1 (x) + a · f2 (x) 2a

(9.5) (9.6)

where C2,n was defined in (2.1). In Section K.14, formulas for the measurement-updated PHD D1|1 (x) are derived using three filters—the PHD filter, the CPHD filter, and the multitarget recursive Bayes (MRB) filter. The results are as follows: PHD filter:

CPHD filter:

MRB filter:

D1|1 (x) = [1 + a(1 − pD )] · f1 (x) +a(1 − pD ) · f2 (x) ( ) (1 − pD ) a D1|1 (x) = 1 + · f1 (x) 2 (1 − apD ) (1 − pD ) a + · f2 (x) 2 (1 − apD ) (1 − pD )a D1|1 (x) = f1 (x) + · f2 (x). 1 − apD

These results can be interpreted as follows:

(9.7)

(9.8)

(9.9)

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• Multitarget Bayes filter: The result for this filter is exact. As one would expect, it is “superpositional,” in the sense that the two-target PHD is what would result if one applied the multitarget Bayes filter to the two tracks separately and then summed their respective PHDs (see Section K.14.3). Since the first track was detected, it definitely exists and so its track weight must be unity. The weight of the second track has decreased since it was not detected: (1 − pD ) · a a ?→ ≤ a. 1 − apD • CPHD filter: The result for the CPHD filter is not superpositional, since the two-target PHD is not the sum of the two single-target PHDs. Moreover, the weight of the undetected track is half of what it should be. The “missing half” has been shifted to the weight of the detected track, making that weight larger than it should be. This seeming “entanglement” between the weights of arbitrarily distant tracks is what Fr¨anken and Ulmke dubbed “spooky action at a distance.” • PHD filter: The result for the PHD filter is superpositional, since the twotarget PHD is the sum of the two single-target PHDs (see Section K.14.1). However, this result is even “spookier” than the CPHD filter result. That is, the weight of the undetected track is even smaller than it should be, and this missing weight has been shifted to the detected track.

9.3

MERGING AND SPLITTING FOR PHD FILTERS

As was noted in Section 8.4.6.2, it will often be desirable to partition the targets in a scene into statistically noninteracting clusters, and then apply a separate PHD filter to each cluster. In this case it will often become necessary to merge the PHDs (if multiple clusters join together) or split the PHDs (if a cluster separates into multiple clusters). The purpose of this section is to describe how this is accomplished. 9.3.1

Merging for PHD Filters

Suppose that two PHD filters are tracking two target groups, and that these groups move close enough that we should apply a single PHD filter to both of them. Let 1

D k|k (x),

2

D k|k (x)

(9.10)

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be the PHDs produced by the two PHD filters at time tk . Assuming that the two target groups are approximately independent, the two PHDs are merged via superposition: 1

2

(9.11)

Dk|k (x) = D k|k (x) + D k|k (x). 9.3.2

Splitting for PHD Filters

Suppose that a single PHD filter is tracking a target group that splits into two subgroups. How do we split the PHD into two new ones, one for each of the subgroups? The intuitively obvious approach is as follows. Suppose that we can partition the PHD into two separated groups of weighted track distributions Group 2

Group 1

?? ? ? ?? ? ? Dk|k (x) = w1 · f1 (x) + ... + wn · fn (x) + w ˜1 · f˜1 (x) + ... + w ˜n˜ · f˜n˜ (x) (9.12) where 0 < wi , w ˜j ≤ 1 and where Group 1 is in some region T and thus Group 2 is in region T c . Then the PHDs for the separating groups are as follows: 1

D k|k (x)

(9.13)

=

1T (x) · Dk|k (x) = w1 · f1 (x) + ... + wn · fn (x)

=

1T c (x) · Dk|k (x) = w ˜1 · f˜1 (x) + ... + w ˜n˜ · f˜n˜ (x). (9.14)

2

D k|k (x)

Remark 26 These heuristic approaches for merging and splitting are actually 1

2

theoretically justifiable. In the case of merging, let Ξk|k and Ξk|k be the statistically independent multitarget RFSs that are to be merged, and let their 1

2

1

respective PHDs be D k|k (x), D k|k (x). Then the merged RFS is Ξk|k = Ξk|k ∪ 2

1

2

Ξk|k and it is easily shown that its PHD is Dk|k (x) = D k|k (x) + D k|k (x). In 1

2

the case of splitting, it is being assumed that Ξk|k = Ξk|k ∪ Ξk|k and that there 1

2

is a region T such that Ξk|k = Ξk|k ∩ T and Ξk|k = Ξk|k ∩ T c . Since Ξk|k 1

2

is assumed to be Poisson, Ξk|k and Ξk|k are independent—see Remark 12 of Section 4.3.1. Thus one can conclude that Dk|k (x) can be split into the sum 1

2

1

2

1

D k|k (x) + D k|k (x) where D k|k (x), D k|k (x) are the respective PHDs of Ξk|k 2

and Ξk|k .

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223

MERGING AND SPLITTING FOR CPHD FILTERS

CPHD filtering of target clusters is not superpositional. That is, if we apply a CPHD filter to each group and then add the respective PHDs, this will not result in the same PHD that would result if a single CPHD filter were applied to the entire scene. Thus, as already noted, CPHD filters must be applied separately to noninteracting target groups. Such prepartitioning is already common practice for conventional multitarget trackers such as MHT, in order to reduce computational complexity. Pre-partitioning would similarly be necessary for CPHD filters even if it were not a theoretical necessity. The purpose of this section is to address the issues associated with merging and splitting CPHD filters. 9.4.1

Merging for CPHD Filters

Suppose that two CPHD filters are tracking two target groups, and that these groups move close enough that we should apply a single CPHD filter to both of them. How do we merge the two CPHD filters? Let 1

2 1

2

D k|k (x), pk|k (n),

(9.15)

D k|k (x), pk|k (n)

be the PHDs and cardinality distributions produced by the two CPHD filters at time tk . Assuming that the two target groups are approximately independent, the two filters can be merged as follows: 1

Dk|k (x)

=

2

1

pk|k (n)

=

(9.16)

D k|k (x) + D k|k (x) 2

(9.17)

(pk|k ∗ pk|k )(n)

where

∑

(p1 ∗ p2 )(n) =

(9.18)

p1 (i) · p2 (j)

i+j=n

denotes convolution of the discrete probability distributions p1 (n) and p2 (n). 1

This approach for merging CPHD filters is not heuristic. If

Ξk|k and

2

Ξk|k are the multitarget RFSs that are to be merged, then the merged RFS is 1

2

1

2

Ξk|k = Ξk|k ∪ Ξk|k . Because Ξk|k , Ξk|k are independent, the p.g.fl. of Ξk|k factors as 1 2 Gk|k [h] = Gk|k [h] · Gk|k [h] (9.19)

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1

2

1

2

where Gk|k [h], Gk|k [h] are the p.g.fl.’s of Ξk|k , Ξk|k . Thus 1

Dk|k (x)

=

2

1

Gk|k (x)

=

(9.20)

D k|k (x) + D k|k (x) 2

Gk|k (x) · Gk|k (x) =

∑

1

2

(pk|k ∗ pk|k )(n) · xn .

(9.21)

n≥0

9.4.2

Splitting for CPHD Filters

Suppose that a single CPHD filter is tracking a target group that splits into two subgroups. How do we split the CPHD filter into two new ones, one for each of the subgroups? The theoretical approach described in Section 9.3.2 cannot be applied, because the partitioned RFSs Ξk|k ∩ T and Ξk|k ∩ T c of Remark 26 in Section 9.3.2 are not statistically independent (since, in general, Ξk|k is not Poisson). From a purely theoretical point of view, Clark’s deconvolution formula, (4.20), could be applied—but the result would not be computationally feasible in general. It therefore appears that only heuristic approaches are possible. The approach described here was proposed by Petetin, Clark, Ristic, and Maltese [237], [238] and appears to work reasonably well.1 Consider the PHD Dk|k (x) first. As with the PHD filter in (9.12), partition it into two separated groups of weighted track distributions Group 1

Group 2

?? ? ? ?? ? ? Dk|k (x) = w1 · f1 (x) + ... + wn · fn (x) + w ˜1 · f˜1 (x) + ... + w ˜n˜ · f˜n˜ (x) (9.22) where 0 < wi , w ˜j ≤ 1. Then as with the PHD filter, specify PHDs for the separating groups: 1

D k|k (x)

=

w1 · f1 (x) + ... + wn · fn (x)

(9.23)

=

w ˜1 · f˜1 (x) + ... + w ˜n˜ · f˜n˜ (x).

(9.24)

2

D k|k (x)

Now consider the cardinality distribution pk|k (i). We must specify distribu1 1

2

2

tions pk|k (i) and pk|k (i), with corresponding expected values N k|k and N k|k , 1

Note that these authors employed it not with a classical CPHD filter, but with a CPHD filter that was hybridized with data association techniques.

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which have the following properties: 1

2

(pk|k ∗ pk|k )(i)

=

pk|k (i)

(9.25)

=

w1 + ... + wn

(9.26)

=

w ˜1 + ... + w ˜n˜ .

(9.27)

1

N k|k 2

N k|k

Assume that the target RFS is approximately multi-Bernoulli—that is, its p.g.fl. approximately has the form Gk|k [h]

(1 − w1 + w1 · f1 [h]) · · · (1 − wn1 + wn · fn [h]) ·(1 − w ˜1 + w ˜1 · f˜1 [h]) · · · (1 − w ˜n˜ + w ˜n˜ · f˜n˜ [h]).

=

(9.28)

2 Note that this assumption is not valid unless the variance σk|k of the cardinality distribution pk|k (n) is smaller than its expected value Nk|k —see Section 4.3.1. Given this and given the properties of multi-Bernoulli RFSs (see Section 4.3.4), it follows that the p.g.f., cardinality distribution, and PHD of the target RFS are

Gk|k (x)

=

pk|k (i)

=

(9.29)

(1 − w1 + w1 · x) · · · (1 − wn + wn · x) ·(1 − w ˜1 + w ˜1 · x) · · · (1 − w ˜n˜ + w ˜n˜ · x) ( n ) ( n˜ ) ∏ ∏ (1 − wi ) (1 − w ˜i ) i=1

(9.30)

i=1

(

Dk|k (x)

=

w1 wn w ˜1 w ˜n˜ , ..., , , ..., ·σn+˜n,i 1 − w1 1 − wn 1 − w ˜1 1−w ˜n˜ w1 · f1 (x) + ... + wn · fn (x) +w ˜1 · f˜1 (x) + ... + w ˜n˜ · f˜n˜ (x)

) (9.31)

where σN,n (x1 , ..., xN ) is the elementary symmetric function of degree n in N variables. (When i > n + n ˜ , pk|k (i) = 0.) Define 1

pk|k (i)

=

(

n ∏

(1 − wi )

)

· σn,i

(

· σn˜ ,i

(

i=1 2

pk|k (i)

=

(

n ˜ ∏ i=1

(1 − w ˜i )

)

w1 wn , ..., 1 − w1 1 − wn w ˜1 w ˜n˜ , ..., 1−w ˜1 1−w ˜n˜

)

(9.32)

)

(9.33)

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with corresponding expected values and p.g.f.’s 1

=

w1 + ... + wn

(9.34)

N k|k

=

w ˜1 + ... + w ˜n˜

(9.35)

Gk|k (x)

=

(1 − w1 + w1 · x) · · · (1 − wn + wn · x)

(9.36)

=

(1 − w ˜1 + w ˜1 · x) · · · (1 − w ˜n˜ + w ˜n˜ · x).

(9.37)

N k|k 2

1

2

Gk|k (x) Then

1

2

Gk|k (x) · Gk|k (x) = Gk|k (x)

(9.38)

from which (9.25) through (9.27) follow. One potential limitation of this approach is its relative computational com1 plexity. Because of the elementary symmetric functions, the distributions pk|k (i) 2 and pk|k (i) have computational order O(n2 ) and O(˜ n2 ), respectively.

9.5

GAUSSIAN MIXTURE (GM) IMPLEMENTATION

The exact closed-form Gaussian mixture implementation of PHD filters was introduced in 2005 in a seminal paper by Vo and Ma [299], [300]. It was extended by Clark, Panta, and Vo to include track labels, and its convergence properties were established in [46], [48]. In 2007 it was generalized to the CPHD filter by Vo, Vo, and Cantoni [308] and by Ulmke, Erdinc, and Willett [292]. The GM approximation depends on the assumption that the probability of detection is constant. In [292], Ulmke et al. proposed an approximation that relaxes this by presuming that the probability of detection is nonconstant but spatially slowly varying (see Section 9.5.6). Because the GM-PHD and GM-CPHD filters approximate the PHD using Gaussian sums, they can be implemented as a bank of extended Kalman filters (EKFs). Since EKFs are appropriate only for mild nonlinearities, Vo and Ma proposed that the bank of EKFs be replaced by a bank of unscented Kalman filters (UKFs) [300]. UKF implementations are usually sufficient for the degree of nonlinearity associated with range-bearing measurement models. For scenarios with still greater nonlinearity, Macagnano and de Abreu [149] proposed the cubature Kalman filter (CKF) of Arasaratnam and Haykin [9] as a replacement for the UKF in the GM-PHD filter. Clark, Vo, and Vo [49] have

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similarly proposed the Gaussian particle filter (GPF) of Kotecha and Djuric [138] as a replacement. The purpose of this section is to summarize the basic concepts underlying GM approximation. It is organized as follows: 1. Section 9.5.1: The standard GM approximation. 2. Section 9.5.2: Pruning Gaussian mixtures. 3. Section 9.5.3: Merging Gaussian mixtures. 4. Section 9.5.4: The GM-PHD filter. 5. Section 9.5.5: The GM-CPHD filter. 6. Section 9.5.6: GM approximation with nonconstant probability of detection. 7. Section 9.5.7: GM approximation with partially uniform target appearances. 8. Section 9.5.8: GM approximation with target identity. 9.5.1

Standard GM Implementation

The Gaussian mixture approximation exploits the fact that Gaussian distributions are algebraically closed under multiplication (see (2.3) and [179], pp. 699-703): NP1 (x − x1 ) · NP2 (x − x2 ) E −1

= =

NP1 +P2 (x2 − x1 ) · NE (x − e) (9.39) P1−1 + P2−1 (9.40)

E −1 e

=

P1−1 x1 + P2−1 x2 .

(9.41)

It follows that if the PHD is approximated as a Gaussian mixture, the time-update and measurement-update equations for the PHD and CPHD filters can be evaluated in closed form, provided that a few restrictions are imposed (see Sections 9.5.4.1 and 9.5.5.1). The most severe restriction is that the probability of detection is constant, pD (x) = pD (although one must also assume that the probability of target survival is constant as well, pS (x′ ) = pS ).

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In GM approximation, the prior and predicted PHDs are assumed to be, at least approximately, Gaussian mixtures νk|k

Dk|k (x)

∑

=

k|k

k|k

(9.42)

· NP k|k (x − xi )

wi

i

i=1 νk+1|k

Dk+1|k (x)

∑

=

k+1|k

k+1|k

· NP k+1|k (x − xi

wi

)

(9.43)

i

i=1

where the expected numbers of targets are given by νk|k

Nk|k =

∑

νk+1|k k|k wi ,

Nk+1|k =

i=1

∑

k+1|k

wi

.

(9.44)

i=1

It follows that the Time propagation of the PHDs is equivalent to the Time propagation of families of the form k|k

k|k

k|k

(ℓi , wi , Pi

k|k ν

k|k , xi )i=1

k|k

where, in addition to the other items in the family, ℓi GM component.

is the track label of the ith

Remark 27 (Track management in GM-PHD/CPHD filters) The track management scheme described in the following sections is based on simple rules for propagating the track labels of Gaussian components. It is easily implemented, but performance tends to suffer when targets cross or when the clutter rate is high. More effective label management requires track-to-track association techniques, such as those described in [146], [229], [231], [230].

9.5.2

Pruning Gaussian Components

As time progresses, the number of components in the GM approximation of a PHD tends to increase without bound. Thus techniques must be employed to merge similar components and prune less important components ([179], p. 630). From a purely logical point of view, it would seem that components should be merged before they are pruned. However, it is better to prune before merging. Doing so avoids the computational cost associated with merging GM components that will end up being pruned anyway.

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Suppose that we are to prune components from the measurement-updated GM system k+1|k+1 k+1|k+1 k+1|k+1 νk+1|k+1 (wi , Pi , xi )i=1 ∑νk+1|k+1 k+1|k+1 with Nk+1|k+1 = i=1 wi . Set a pruning threshold τprune , identify those components for which k+1|k+1

wi

< τprune

(9.45)

and then eliminate them. This results in a pruned system k+1|k+1

(w ˇi

ˇk+1|k+1 k+1|k+1 k+1|k+1 ν , Pˇi ,x ˇi )i=1

with νˇk+1|k+1 components. Let ν ˇk+1|k+1

w ˇ k+1|k+1 =

∑

k+1|k+1

w ˇi

(9.46)

i=1

be the combined weight of all components that remain. Define the renormalized weights k+1|k+1 w ˇ k+1|k+1 w ˆi (9.47) = Nk+1|k+1 · ik+1|k+1 w ˇ for all i = 1, ..., νˇk+1|k+1 . Then k+1|k+1

(w ˆi

ˇk+1|k+1 k+1|k+1 k+1|k+1 ν , Pˇi ,x ˇi )i=1

is the pruned GM system. 9.5.3

Merging Gaussian Components

Suppose that two Gaussian components w1 · NP1 (x − x1 ) + w2 · NP2 (x − x2 )

(9.48)

are to be merged into a single component w0 · NP0 (x − x0 ).

(9.49)

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We must first have some criterion for determining whether or not they should be merged. The Hellinger distance, (6.35), leads to the following measure of overlap between NP1 (x − x1 ) and NP2 (x − x2 ): √ det P1 P2 I = −2 log + (x1 − x2 )T (P1 + P2 )−1 (x1 − x2 ). (9.50) 1 det 2 (P1 + P2 ) Note that when the two track distributions are identical, I = 0. For computational purposes, (9.50) can be approximated as the Mahalanobis distance I˜ = (x1 − x2 )T (P1 + P2 )−1 (x1 − x2 ).

(9.51)

When I˜ < τmerge for some merging threshold τmerge , we conclude that the two components are sufficiently similar that they should be merged. Suppose, then, that we have determined that n components D(x) =

n ∑

wi · NPi (x − xi )

(9.52)

i=1

should be merged into a single component w0 · NP0 (x − x0 ).

(9.53)

In Section K.16 it is shown that the Gaussian component that has the same mean and covariance as the sum in (9.52) is w0

x0

P0

=

=

=

=

n ∑

wi

(9.54)

i=1 n ∑

w ˆ i · xi

(9.55)

i=1 n ∑

w ˆ i · Pi +

∑

w ˆi · w ˆj · (xi − xj )(xi − xj )T

(9.56)

n ∑

w ˆi · xi xTi

(9.57)

(i = 1, ..., n).

(9.58)

i=1

1≤i 6. In [148], Macagnano and de Abreu modified the CKF-PHD filter to include an adaptive measurement-gating scheme. The authors concluded that their adaptive gating approach achieved significant performance improvements, compared to standard elliptical gating. 9.5.4.7

GM-PHD Filter—Gaussian Particle Filter Variant

In this approach, proposed by Clark, Vo, and Vo in 2007 [49], the EKF or UKF in a GM-PHD filter is replaced by the Gaussian particle filter (GPF) of Kotecha and Djuric [138]. The Markov transition density and likelihood function are allowed to be nonlinear-Gaussian fk+1|k (x|x′ )

=

NQk (x − φk (x′ ))

(9.124)

fk+1 (z|x)

=

NRk+1 (z − ηk+1 (x))

(9.125)

where the functions φk (x′ ) and ηk+1 (x) are arbitrary. As usual, the PHDs are approximated as Gaussian mixtures. However, the filtering equations will now

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involve products of the form k|k

NQk (x − φk (x′ )) · NP k|k (x − xi ) i

k+1|k

NRk+1 (x − ηk+1 (x)) · NP k+1|k (x − xi

).

i

form

Applying (9.124) to persisting targets, we will end up with integrals of the ∫ k|k NQk (x − φk (x′ )) · NP k|k (x − xi )dx. i

These can be numerically evaluated via Monte Carlo integration, as follows. First, k|k for fixed i draw ν samples ui,1 , ..., ui,ν from NP k|k (∗ − xi ). Second and i again for fixed i, draw a single sample vi,j from NRk+1 (∗ − ηk+1 (ui,j )). Then by the law of large numbers,

−→

1∑ NRk+1 (x − ηk+1 (ui,j ) ν j=1 ∫ k|k NRk+1 (x − ηk+1 (x′ )) · NP k|k (x − xi )dx i

almost surely as ν → ∞. So the left side can be taken as an approximation of the right side. Third, for fixed i compute the sample mean and covariance =

ν 1∑ ui,j ν j=1

(9.126)

=

ν 1∑ k+1|k k+1|k T (ui,j − xi )(ui,j − xi ) . ν j=1

(9.127)

k+1|k

xi k+1|k

Pi

Fourth, the predicted PHD for persisting targets can be approximated as νk|k

pS

∑

k+1|k

k|k

wi

· NP k+1|k (x − xi

).

i

i=1

Applying (9.125) to detected targets, we must evaluate terms of the form k+1|k

pD · NRk+1 (zj − ηk+1 (x)) · NP k+1|k (x − xi i

κk+1 (zj ) +

i τk+1 (zj )

)

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243

where i τk+1 (zj )

= pD

∫

k+1|k

NRk+1 (zj − ηk+1 (x)) · NP k+1|k (x − xi

)dx.

(9.128)

i

The following procedure is employed. First, for fixed i, j, draw samples wi,j,1 , ..., wi,j,ν from some importance-sampling function π(x). (Clark suggests that this function k+1|k could be NP k+1|k (x − xi ) or the EKF/UKF measurement-update of this i density using zj .) Second, compute the weight corresponding to each sample for l = 1, ..., ν: k+1|k

NRk+1 (zj − ηk+1 (wi,j,l )) · NP k+1|k (wi,j,l − xi

)

i

wi,j,l =

.

(9.129)

π(wi,j,l ) Third, note that ν 1∑ i wi,j,l −→ τk+1 (zj ) ν l=1

as ν → ∞, and so the left side can be taken as an approximation of the right side. Fourth, compute the sample means and covariances and weights k+1|k+1

xi,j

=

∑ν w · wi,j,l l=1 ∑ν i,j,l w ′ l =1 i,j,l′

(9.130)

k+1|k+1

= k+1|k+1

wi,j

=

(9.131)

Pi,j ∑ν

k+1|k+1 k+1|k+1 T )(wi,j,l − xi,j ) l=1 wi,j,l · (wi,j,l − xi,j ∑ν ′ l′ =1 wi,j,l ∑ν k+1|k 1 wi · pD · ν l=1 wi,j,l . (9.132) ∑νk+1|k k+1|k 1 ∑ν κk+1 (zj ) + pD i=1 wi · ν l=1 wi,j,l

Fifth, approximate the detected-target PHD as mk+1 νk+1|k

∑ ∑ j=1

i=1

k+1|k+1

wi,j

k+1|k+1

· NP k+1|k+1 (x − xi,j i,j

).

(9.133)

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GPF-PHD/CPHD Filter Performance Results: Clark et al. implemented the GPF approach for both GM-PHD and GM-CPHD filters. They tested it in simulations involving a range-bearing sensor and five appearing and disappearing targets [49]. The target state was assumed to be (x, y, vx , vy , ω) where ω is the turn rate of a nonlinear coordinated-turn motion model. Clutter was assumed to be spatially uniform and Poisson with clutter rate λ = 25; and probability of detection was pD = 0.98. The GPF-PHD and GPF-CPHD filters both accurately tracked the targets, even when they crossed, with the GPF-CPHD filter more accurately estimating target number than the GPF-PHD filter. 9.5.5

GM-CPHD Filter

This section summarizes the exact closed-form Gaussian mixture (GM) implementation of the CPHD filter ([179], pp. 646-649). It is organized as follows: 1. Section 9.5.5.1: Modeling assumptions for the GM-CPHD filter. 2. Section 9.5.5.2: Time update equations for the GM-CPHD filter. 3. Section 9.5.5.3: Measurement update equations for the GM-CPHD filter. 4. Section 9.5.5.4: Multitarget state estimation for the GM-CPHD filter. 5. Section 9.5.5.5: The unscented Kalman filter (UKF) variant of the GMCPHD filter. Remark 31 The time-update and measurement-update formulas presented here for the GM-CPHD filter are slightly different than those given in [179], pp. 646-649. They are nevertheless equivalent to them. 9.5.5.1

GM-CPHD Filter Models

The GM implementation of the CPHD filter requires the following models: • Cardinality distributions—are finite, that is, pk|k (n) = 0 for n ≥ nmax . • Probability of target survival—is constant, pS,k+1|k (x) = pS,k+1|k abbr. = pS . • Single-target Markov density—is linear-Gaussian:5 fk+1|k (x|x′ ) = NQk (x − Fk x′ ). 5

(9.134)

This assumption can be relaxed to allow fk+1|k (x|x′ ) to be a Gaussian mixture, but at the expense of increasing the computational burden.

Implementing Classical PHD/CPHD Filters

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• Target–appearance PHD—is a Gaussian mixture: B νk+1|k

bk+1|k (x) =

∑

k+1|k

k+1|k

· NB k+1|k (x − bi

bi

)

(9.135)

i

i=1

and so the expected number of appearing targets is B νk+1|k

B Nk+1|k =

∑

k+1|k

bi

.

(9.136)

i=1

• Target-appearance cardinality distribution—is finite, that is, pB k+1|k (n) = 0 for sufficiently large n; and it must be the case that ∑

B Nk+1|k =

n · pB k+1|k (n).

(9.137)

n≥1

• Probability of detection—is constant, pD,k+1 (x) = pD,k+1 abbr. = pD (this assumption can be removed using the approximation described in Section 9.5.6). • Sensor likelihood function—is linear-Gaussian:6 Lz (x) = fk+1 (z|x) = NRk+1 (z − Hk+1 x).

(9.138)

• Clutter cardinality distribution—pκk+1 (m) is arbitrary; or equivalently, the clutter p.g.f. Gκk+1 (z) is arbitrary. • Clutter spatial distribution: ck+1 (z) is arbitrary. 9.5.5.2

GM-CPHD Filter Time Update

We are given a cardinality distribution and system of Gaussian components for the PHD, k|k k|k k|k k|k νk|k pk|k (n), (ℓi , wi , Pi , xi )i=1 , 6

This assumption can be relaxed to allow fk+1 (z|x) to be a Gaussian mixture, but at the expense of increasing the computational burden.

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with νk|k

∑

Nk|k =

k|k

wi

=

n=0

n∑ max

(9.139)

n · pk|k (n).

n=0

We are to determine formulas for the predicted cardinality distribution and the predicted system of Gaussian components, k+1|k

pk+1|k (n),

(ℓi

k+1|k

, wi

k+1|k νk+1|k )i=1 .

k+1|k

, Pi

, xi

These are determined as follows: • Time updated cardinality distribution: pk+1|k (n) =

n∑ max

pk+1|k (n|n′ ) · pk|k (n′ )

(9.140)

n′ =0

where min{n,n′ } ′

pk+1|k (n|n ) =

∑

′

i n −i pB . (9.141) k+1|k (n − i) · Cn′ ,i · pS,k (1 − pS,k )

i=0

and where Cn′ ,i was defined in (2.1). • Time updated number of GM components for the PHD: B νk+1|k = νk|k + νk+1|k .

(9.142)

Here, there are νk|k components corresponding to persisting targets, and B νk+1|k components corresponding to newly appearing targets. The timeupdated components are indexed as follows: i i

= =

1, ..., νk|k νk+1 + 1, ..., νk+1 +

B νk+1|k

(persisting)

(9.143)

(appearing).

(9.144)

• Persisting-target GM components, for i = 1, ..., νk|k : k+1|k

ℓi k+1|k wi k+1|k xi k+1|k Pi

k|k

= = = =

ℓi

(9.145)

k|k pS · w i k|k Fk xi k|k Fk Pi FkT

(9.146) (9.147) + Qk .

(9.148)

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B • Appearing-target GM components, for i = νk|k + 1, ..., νk|k + νk+1|k : k+1|k

ℓi k+1|k wi

=

new label

(9.149)

=

k+1|k bi−νk|k

(9.150)

k+1|k

=

bi−νk|k

(9.151)

k+1|k

=

Bi−νk|k .

(9.152)

k+1|k

xi k+1|k

Pi 9.5.5.3

GM-CPHD Filter Measurement Update

We are given the predicted cardinality distribution and the predicted system of Gaussian components, k+1|k

pk+1|k (n),

(ℓi

k+1|k

, wi

k+1|k

, Pi

k+1|k νk+1|k )i=1 ,

, xi

with νk+1|k

∑

Nk+1|k =

k+1|k

wi

=

n=0

n∑ max

n · pk+1|k (n).

(9.153)

n=0

We are also given a new measurement set Zk+1 = {z1 , ..., zmk+1 } with |Zk+1 | = mk+1 . We are to determine formulas for the measurement-updated cardinality distribution and the measurement-updated system of Gaussian components: k+1|k+1

pk+1|k+1 (n),

(ℓi

k+1|k+1

, wi

k+1|k+1

, Pi

k+1|k+1 νk+1|k+1 )i=1 .

, xi

The system will actually have the structure k+1|k+1 νk+1|k )i=1 , k+1|k+1 k+1|k+1 k+1|k+1 k+1|k+1 νk+1|k ,mk+1 (ℓi,j , wi,j , Pi,j , xi,j )i=1;j=1 . k+1|k+1

(ℓi

k+1|k+1

, wi

k+1|k+1

, Pi

, xi

These are determined as follows: • Measurement updated number of GM components for the PHD: νk+1|k+1 = νk+1|k + mk+1 · νk+1|k

(9.154)

where, as with the GM-PHD filter, there are νk+1|k components for undetected tracks and mk+1 · νk+1|k components for detected tracks. The

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measurement-updated components are indexed as follows: i i

= =

(undetected) (detected).

1, ..., νk+1|k 1, ..., νk+1|k ; j = 1, ..., mk+1

(9.155) (9.156)

• Measurement updated cardinality distribution: ℓZ (n) · pk+1|k (n) pk+1|k+1 (n) = ∑ k+1 l≥0 ℓZk+1 (l) · pk+1|k (l)

(9.157)

where ( ∑ min{mk+1 ,n} ℓZk+1 (n) =

(mk+1 − j)! · pκk+1 (mk+1 − j) j=0 ·j! · Cn,j · ϕn−j · σj (Zk+1 ) k ( ∑mk+1 ) κ l=0 (mk+1 − l)! · pk+1 (mk+1 − l) (l) ·σl (Zk+1 ) · Gk+1|k (ϕk )

) (9.158)

where Cn,j was defined in (2.1), and where ϕk

=

Gk+1|k (ϕk )

=

(l)

(9.159)

1 − pD,k+1 nmax ∑ pk+1|k (n) · l!Cn,l · ϕn−l k

(9.160)

n=l

σi (Zk+1 )

=

τˆk+1 (zj )

=

σmk+1 ,i

(

τˆk+1 (zmk+1 ) τˆk+1 (z1 ) , ..., ck+1 (z1 ) ck+1 (zmk+1 )

)

(9.161)

νk+1|k

∑

pD Nk+1|k ·NR

k+1|k

(9.162)

wi

l=1

k+1|k T Hk+1 k+1 +Hk+1 Pl

(zj − Hk+1 xk+1 ). l

• Measurement updated undetected-target GM components for the PHD: for i = 1, ..., vk+1|k , k+1|k+1

ℓi

k+1|k

=

(1 − pD ) · Nk+1|k

=

xi

(9.165)

=

k+1|k Pi

(9.166)

k+1|k+1

xi k+1|k+1 Pi

k+1|k wi

=

k+1|k+1

wi

(9.163)

ℓi

k+1|k

ND

· L Zk+1

(9.164)

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where ND

L Zk+1

− j)

(9.167) )

(

− l)

)

κ j=0 (mk+1 − j)! · pk+1 (mk+1 (j+1) ·σj (Zk+1 ) · Gk+1|k (ϕk ) ∑mk+1 κ l=0 (mk+1 − l)! · pk+1 (mk+1 (l) ·σl (Zk+1 ) · Gk+1|k (ϕk )

n∑ max

(j+1)

Gk+1|k (ϕk )

( ∑mk+1

=

=

(9.168)

pk+1|k (n) · (j + 1)! · Cn,j+1

n=j+1

·ϕn−j−1 . k • Measurement updated detected-target GM components for the PHD: for i = 1, ..., vk+1|k and j = 1, ..., mk+1 , k+1|k+1

ℓi,j

k+1|k

=

(9.169)

ℓi D

k+1|k

pD · w i Nk+1|k

k+1|k+1

wi,j

=

·

LZk+1 (zj ) ck+1 (zj )

(9.170) k+1|k

·NR k+1|k

xi,j

k+1|k

=

k+1|k

Pi,j

=

Kik+1

k+1|k T Hk+1 k+1 +Hk+1 Pi

=

(zj − Hk+1 xi k+1|k

xi + Kik+1 (zj − Hk+1 xi ( ) k+1|k I − Kik+1 Hk+1 Pi

(9.171)

)

(9.172)

k+1|k T Pi Hk+1

(

k+1|k

· Hk+1 Pi

)

(9.173) T Hk+1 + Rk+1

)−1

where D

LZk+1 (zj ) ( ∑m −1 k+1

i=0

=

(mk+1 − i − 1)! · pκk+1 (mk+1 − (i+1) ·σi (Zk+1 − {zj }) · Gk+1|k (ϕk )

( ∑mk+1 l=0

i − 1)

(mk+1 − l)! · pκk+1 (mk+1 − l) (l) ·σl (Zk+1 ) · Gk+1|k (ϕk )

)

(9.174) )

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and

=

σi (Zk+1 − {zj }) (9.175) ( ) ? τˆk+1 (zmk+1 ) τˆk+1 (z1 ) τˆk+1 (zj ) σmk+1 −1,i , ..., , ..., ck+1 (z1 ) ck+1 (zj ) ck+1 (zmk+1 )

and where x1 , ..., x?j , ..., xm indicates that the jth item removed from the list x1 , ..., xm : x1 , ..., xj−1 , xj+1 , ..., xm . 9.5.5.4

xj is to be

GM-CPHD Filter Multitarget State Estimation

State estimation is the same as in Section 9.5.4.4, except that the nearest integer n to Nk+1|k+1 is replaced by the MAP estimate (9.176)

n ˆ k+1|k+1 = arg sup pk+1|k+1 (n). n≥0

9.5.5.5

GM-CPHD Filter: UKF, CKF, and GPF Variants

As with the GM-PHD filter, the GM-CPHD filter can be regarded as a bank of extended Kalman filters (EKFs), which are applied separately to each Gaussian component of the PHD. The EKFs can be replaced by unscented Kalman filters (UKFs) in the same manner as described in Section 9.5.4.5. Likewise for cubature Kalman filters (Section 9.5.4.6) and Gaussian particle filters (Section 9.5.4.7). 9.5.6

Implementation with Nonconstant pD

As already noted, GM implementation requires that probability of detection be constant. The reason is that the measurement-update equations for the PHD and CPHD filter require multiplication of the components of the Gaussian mixtures by the factors pD (x) and 1 − pD (x). This would prevent closed-form solution in terms of Gaussian mixtures if pD (x) were not constant. This challenge can be addressed using an approximation first proposed by Ulmke, Erdinc, and Willett [292]. Assume that the probability of detection is approximately constant, as compared with the covariances of the target track distributions. Then we can write k+1|k

pD (x) · NP k+1|k (x − xi

(9.177)

)

i

∼ =

k+1|k

pD (xi

k+1|k

) · NP k+1|k (x − xi i

)

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and k+1|k

(1 − pD (x)) · NP k+1|k (x − xi

(9.178)

)

i

∼ =

k+1|k

(1 − pD (xi

k+1|k

)) · NP k+1|k (x − xi

)

i

for all i = 1, ..., vk+1|k . Thus νk+1|k

pD (x) · Dk+1|k (x)

∑

∼ =

k+1|k

wi

k+1|k

· pD (xi

)

(9.179)

i=1 k+1|k

·NP k+1|k (x − xi

)

i

and νk+1|k

(1 − pD (x)) · Dk+1|k (x)

∑

∼ =

k+1|k

wi

k+1|k

· (1 − pD (xi

)) (9.180)

i=1 k+1|k

·NP k+1|k (x − xi

).

i

Remark 32 (A different approach) The “pD -agnostic” beta-Gaussian mixture (BGM) approach described in Section 17.3, provides a more theoretically satisfying way of addressing nonconstant pD . In this case, pD (x) is regarded as an unknown state variable 0 ≤ a ≤ 1 and the usual state x is replaced by an augmented state ˚ x = (a, x). 9.5.7

Implementation with Partially Uniform Target Births

The PHD for the target appearance process was given in (9.135): B νk+1|k

bk+1|k (x) =

∑

k+1|k

bi

k+1|k

· NB k+1|k (x − bi

).

(9.181)

i

i=1

The most obvious way to construct this PHD is to use whatever a priori information k+1|k k+1|k is available to choose the respective bi and Bi , with the matrix norm k+1|k ∥Bi ∥ typically chosen to be large. However, doing so will tend to create a large number of birth-target components.

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k+1|k

One therefore might place the bi so as to correspond to the measurements collected at time tk+1 . This “measurement driven” approach will also tend to create a large number of birth-target components if the clutter rate is large. Moreover, it can produce a statistical bias in the target-number estimate when target number is small, similar to that identified by Ristic, Clark, and Vo in regard to SMCPHD filters (see Section 9.6.4 and [256]). A potentially more efficient approach, proposed by Beard, Vo, Vo, and Arulampalam [16], [17], is based on two innovations. The first innovation, proposed by Ristic et al., provides a means of avoiding this statistical bias [256], [257]; and will be described using a somewhat different formulation in Section 9.6.4. It consists of augmenting the state x with a binary variable o = 1, 2 where o = 0 indicates that (0, x) is the state of a persisting target and o = 1 indicates that (1, x) is the state of a newly appearing target. In this approach, the likelihood function and probability of detection for the augmented state are defined to be, respectively, Lz (o, x)

=

pD (o, x)

=

Lz (x) { pD (x) 1

(9.182) if if

o=0 . o=1

(9.183)

That is, newly appearing targets are always detected, and they generate measurements in the same way as persisting targets. This is in accordance with the intuition that no target can be asserted to exist, unless it first generates a measurement. Similarly, the Markov transition is defined to be:   fk+1|k (x|x′ ) ′ ′ f (x|x′ ) fk+1|k (o, x|o , x ) =  k+1|k 0

if if if

o = o′ = 0 o = 0, o′ = 1 . otherwise

(9.184)

That is, birth targets and persisting targets can transition only to persisting targets. Finally, the probability of target persistence and the target-birth PHD are given by pS (o′ , x′ )

=

bk+1|k (o, x)

=

pS (x′ ) {

0 bk+1|k (x)

(9.185) if if

o=0 . o=1

(9.186)

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The second innovation, proposed by Beard, Vo, Vo, and Arulampalam, is the partially uniform target appearance process.7 It is assumed that the state-vector x can be decomposed into two parts, x = (o, u)

(9.187)

with a corresponding decomposition of state spaces X = O × U with o ∈ O and u ∈ U, such that the measurement function ηk+1 (x) has the following property: ηk+1 (o, u) = ηk+1 (o).

(9.188)

That is, • The state variables in o ∈ O are at least partially observed; whereas • The state variables in u ∈ U are completely unobserved. Given this, the basic idea is to replace (9.181) by B bk+1|k (o, u) = wk+1|k ·

1O′k+1|k (o) |O′k+1|k |

2 · Nσk+1|k I (u − uk+1|k )

(9.189)

where: • O′k+1|k is an arbitrarily large but bounded region of the observed-state space O, and |O′k+1|k | is its hypervolume. • uk+1|k ∈ U and I is the identity matrix on U. B • wk+1|k is the weight of the target appearance component.

That is, target appearances are: • uniformly distributed with respect to the observed state variables, but • Gaussian-distributed with respect to the unobserved ones. Because of (9.189), the target appearance PHD will no longer be a Gaussian mixture, and thus will not result in a time-updated PHD that is a Gaussian mixture. However, Beard et al. show that one can still devise PHD and CPHD filters 7

This approach is similar to one proposed by Houssineau and Laneuville [116], who proposed B making the birth PHD uniform: bk+1|k (x) = wk+1|k · 1X0 (x).

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that are approximately Gaussian-mixture filters. This is because of the following approximations: ∫ 1 ′ Ok+1|k (o) |O′k+1|k |

∫ · NO (o − o0 )do

∼ = 1O′k+1|k (o) |O′k+1|k |

· NO (o − o0 )

O′k+1|k

=

∼ =

NO (o − o0 )do

|O′k+1|k |

(9.190)

1 (9.191)

|O′k+1|k | NO (o − o0 ) . |O′k+1|k |

(9.192)

Because of these approximations it follows that, after each execution of a timeupdate followed by a measurement-update, the PHD will be a Gaussian mixture. This fact is explained in detail in the following subsections: 1. Section 9.5.7.1: The PHD filter with a partially uniform target-birth PHD. 2. Section 9.5.7.2: The CPHD filter with a partially uniform target-birth PHD. 3. Section 9.5.7.3: Implementations of these PHD and CPHD filters. 9.5.7.1

PHD Filter with a Partially Uniform Target-Birth PHD

Time update: Assume that the single-target Markov density has the form fk+1|k (x|x′ ) = NQk (x − Fk x′ )

(9.193)

and that target survival probability is constant: pS (x′ ) = pS . Also assume that the measurement-updated PHD at time tk is a Gaussian mixture o νk|k

Dk|k (o, o, u) =

∑

k|k

k|k

k|k

wo,i · NP k|k ((o, u) − (oo,i , uo,i ))

(9.194)

o,i

i=1 k|k

where Po,i is a covariance matrix, expressed in terms of the coordinates associated with the state representation x = (o, u). Given this, the time-updated PHD is given

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255

by (see Section K.19) o νk|k

Dk+1|k (0, o, u)

=

pS

∑

k|k

k+1|k

w0,i · NP k+1|k ((o, u) − (o0,i

k+1|k

, u0,i

)) (9.195)

0,i

i=1 o νk|k

+pS

∑

k|k

k+1|k

w1,i · NR

k|k T k +Fk P1,i Fk

((o, u) − (o1,i

k+1|k

, u1,i

))

i=1

Dk+1|k (1, o, u)

=

B wk+1|k ·

1O′ (o) 2 · Nσk+1|k I (u − uk+1|k ) |O′ |

(9.196)

where k+1|k

(oo,i

k+1|k

, uo,i

k|k

)

(9.197)

Fk (oo,i , uo,i )

=

Qk + Fk Po,i FkT .

k+1|k

Po,i

k|k

=

k|k

(9.198)

Measurement update: Because of (9.188), the likelihood function has the form (9.199)

Lz (o, u) = Lz (o) = NRk+1 (z − Hk+1 o).

Also, assume that the (conventional) probability of detection is constant: pD (x) = pD and that the predicted PHD has the form of (9.195) and (9.196): νk+1|k

Dk+1|k (0, o, u)

∑

=

k+1|k

(9.200)

wi

i=1 k+1|k

·NP k+1|k ((o, u) − (oi

k+1|k

, ui

))

i

Dk+1|k (1, o, u)

=

B wk+1|k ·

1O′ (o) 2 · Nσk+1|k I (u − uk+1|k ). (9.201) |O′ |

Let νk+1|k

B wk+1|k

τk+1 (z)

= |O′ |

+ pD

∑

k+1|k

(9.202)

wi

i=1 k+1|k

T ·NRk+1 +Hk+1 Pk+1|k Hk+1 (z − Hk+1 oi

).

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Then in Section K.19 it is shown that the measurement-updated PHD for persisting targets is: (9.203)

Dk+1|k+1 (0, o, u) νk+1|k

= (1 − pD )

∑

k+1|k

wi

i=1 k+1|k

k+1|k

·NP k+1|k ((o, u) − (oi , ui i ( ∑νk+1|k mk+1

+

∑

)) k+1|k

pD · w i i=1 k+1|k ·NRk+1 +Hk+1 Pk+1|k Hk+1 (zj − Hk+1 oi )

)

κk+1 (zj ) + τk+1 (zj )

j=1 k+1|k+1

·NP k+1|k+1 ((o, u) − (oi

k+1|k+1

, ui,j

))

i

where k+1|k+1 −1

(Pi

)

=

k+1|k −1

(Pi

)

(9.204)

−1 T +Hk+1 Rk+1 Hk+1 k+1|k+1 −1

(Pi

)

k+1|k+1

(oi

k+1|k+1

, ui,j

)

=

k+1|k −1

(Pi

)

(9.205)

k+1|k k+1|k ·(oi , ui ) −1 T +Hk+1 Rk+1 zj .

On the other hand, the measurement-updated PHD for appearing targets is

Dk+1|k+1 (1, o, u)

∑

=

z∈Zk+1

B wk+1|k

|O′k+1|k |

(9.206)

2 Nσk+1|k I (u − uk+1|k ) · NRk+1 (z − Hk+1 o)

· 9.5.7.2

. κk+1 (z) + τk+1 (z)

CPHD Filter with a Partially Uniform Target-Birth PHD

While the underlying concept remains the same as for the PHD filter, the filtering equations for the CPHD filter are correspondingly more complicated and will not be further considered here. The interested reader is referred to the original papers by Beard, Vo, Vo, and Arulampalam [16], [17].

Implementing Classical PHD/CPHD Filters

9.5.7.3

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Implementations of PHD/CPHD Filters with Partially Uniform Target Births

Beard et al. have implemented their PHD and CPHD filters, and compared them with conventional GM-PHD and GM-CPHD filters whose birth PHDs consist of varying numbers of Gaussian components [16], [17]. The authors tested the approach in a challenging application: a single bearingonly sensor carried on a platform making a sinusoidal maneuver, which observes six appearing and disappearing targets in heavy clutter (clutter rate λ = 40). Despite the low-observability conditions, the authors reported good tracking performance, in comparison to conventional PHD/CPHD filters that employ a small number of Gaussian components for the target-birth PHD. They also reported that the conventional PHD/CPHD filters required significantly more computation time than the new PHD/CPHD filters—largely because they tended to cause the number of components to increase over time. The authors further reported that a large number of birth target components— 64—are required before conventional PHD/CPHD filters perform as well as the new filters. Still larger numbers of components (larger than 64) did not improve performance. 9.5.8

Implementation with Target Identity

Suppose that the target state has the form x ˜ = (τ, x) where x is the kinematic state and τ is a discrete identity variable (class, type) belonging to a finite set ˜ = X×T . T = {τ1 , ..., τN } of identity types. Thus the total target state space is X Also assume that measurements have the form ˜ z = (ϕ, z) where z is the kinematic measurement and ϕ is a feature associated with target identity.8 Then the Gaussian mixture approximation can be extended as follows: νk|k

Dk|k (τ, x) =

∑

k|k

wi

k|k

k|k

· pi (τ ) · NP k|k (x − xi )

(9.207)

i

i=1

= 8

Dk+1|k (τ, x) νk+1|k ∑ k+1|k wi i=1

(9.208) k+1|k

· pi

k+1|k

(τ ) · NP k+1|k (x − xi

)

i

The notation ‘ϕ’ for a feature should not be confused with the symbol ‘ϕk+1 ’ used in the measurement-update formulas for CPHD filters. Likewise, the notation ‘τ ’ should not be confused with the notations ‘τk+1 (z)’ and ‘ˆ τk+1 (z)’ used in PHD and CPHD filters, respectively.

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k|k

k+1|k

where pi (τ ) and pi (τ ) are probability distributions on T .9 In what follows, for the sake of conceptual clarity only the GM-PHD filter will be considered in detail, and target spawning will be neglected. In this case the PHD filter equations become ∑∫ Dk+1|k (τ, x) = bk+1|k (τ, x) + pS (τ ′ , x′ ) (9.209) τ′

·fk+1|k (τ, x|τ ′ , x′ ) · Dk|k (τ, x)dx Dk+1|k+1 (τ, x) Dk+1|k (τ, x)

=

(9.210)

1 − pD (τ, x) +

∑

pD (τ, x) · L(ϕ,z) (τ, x) κk+1 (ϕ, z) + τk+1 (ϕ, z)

(ϕ,z)∈Z

where τk+1 (ϕ, z) =

∑∫

pD (τ, x) · L(ϕ,z) (τ, x) · Dk+1|k (τ, x)dx.

(9.211)

τ

Now turn to Gaussian-mixture implementation of this PHD filter. 9.5.8.1

GM-PHD Filter Time Update with Target ID

Set ′

pS (τ ′ , x′ ) fk+1|k (τ, x|τ ′ , x′ )

= =

pτS pk+1|k (τ |τ ′ ) · NQτk (x − Fkτ x′ )

(9.212) (9.213)

(9.214)

bk+1|k (τ, x) B νk+1|k

=

∑

B,k+1|k

bτi · pi

k+1|k

(τ ) · NB k+1|k (x − bi

).

i

i=1

9

This approach uses a “flat” or single-layer target–identity taxonomy (also known as “ontology”)— that is, one in which an attempt is made to directly identify target identity. In practical application it is often preferable to employ a multilayer taxonomy—that is, one in which coarser determinations of identity are made prior to finer ones. For example, one might first determine if a ground target is a truck versus a tank, before attempting to determine what sort of truck or what sort of tank it is.

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259

Substituting (9.207) into (9.209), we get B νk+1|k

Dk+1|k (τ, x)

∑

=

B,k+1|k

bτi · pi

i=1 νk|k

+

∑

) (9.215)

i

∑

k|k

wi

k+1|k

(τ ) · NB k+1|k (x − bi ′

k|k

pτS · pi (τ ′ ) · pk+1|k (τ |τ ′ )

τ′

i=1

′

k|k

·NQτ ′ +F τ ′ P k|k (F τ ′ )T (x − Fkτ xi ). k

i

k

k

For most applications we can set pk+1|k (τ |τ ′ ) = δτ,τ ′ since, typically (but not invariably—see Remark 33 below), targets do not change identity. In this case we get B νk+1|k

Dk+1|k (τ, x)

∑

=

B,k+1|k

bτi · pi

k+1|k

(τ ) · NB k+1|k (x − bi

) (9.216)

i

i=1 νk|k

+pS

∑

k|k

wi

k|k

· pi (τ )

i=1 k|k

·NQτ +F τ P k|k (F τ )T (x − Fkτ xi ). k

k

i

k

Remark 33 (Dynamically changing target identity) It is not always the case that fk+1|k (τ |τ ′ ) = δτ,τ ′ . Possibly the most extreme example is a diesel-electric submarine. It has very different passive-acoustic phenomenologies, depending on whether it is snorkeling (and thus using its diesel engines) or submerged (and thus using its electric engines). Other examples include variable swept-wing aircraft (extended-wing versus delta-wing modes) and mobile missile launchers (launch mode versus transit mode). In such cases, a classification algorithm that presumes fk+1|k (τ |τ ′ ) = δτ,τ ′ will typically exhibit degraded performance. 9.5.8.2

GM-PHD Filter Measurement Update with Target ID

Set pD (τ, x)

=

pτD

Lϕ,z (τ, x)

=

Lϕ (τ ) · N

κk+1 (ϕ, z)

=

κk+1 (z)

(9.217) τ Rk+1

(z −

τ Hk+1 x)

(9.218) (9.219)

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where Lϕ (τ ) is the likelihood that a target of type τ will generate an identityfeature ϕ. Substituting (9.209) into (9.211), we get νk+1|k

τk+1 (ϕ, z)

∑

=

k+1|k

wi

∑

k+1|k

pτD · Lϕ (τ ) · pi

(9.220)

(τ )

τ

i=1

k+1|k

·NRτ

k+1|k τ τ (Hk+1 )T k+1 +Hk+1 Pi

τ (z − Hk+1 xi

).

Substituting (9.209) into (9.210) we get: νk+1|k

Dk+1|k+1 (τ, x)

=

(1 − pτD )

∑

k+1|k

wi

k+1|k

· pi

(τ )

(9.221)

i=1 k+1|k

·NP k+1|k (x − xi

) (

i

νk+1|k

+

∑

mk+1 k+1|k wi

i=1

∑

k+1|k

pi (τ ) · pτD · Lϕj (τ ) k+1|k+1 ·NP k+1|k+1 (x − xi,j )

)

i,j

κk+1 (zj ) + τk+1 (ϕj , zj )

j=1

where k+1|k+1

xi,j

k+1|k

=

k+1|k+1

Pi,j

= Kik+1

9.5.8.3

=

k+1|k

xi + Kik+1 (zj − Hk+1 xi ) (9.222) ( ) k+1|k k+1 I − Ki Hk+1 Pi (9.223) ( )−1 k+1|k T k+1|k T Pi Hk+1 Hk+1 Pi Hk+1 + Rk+1 . (9.224)

GM-CPHD Filter Measurement Update with Target ID

The case for the CPHD filter is more complicated but straightforward, and so will not be considered in detail here. One innovation should be pointed out, however: • It is possible to construct the cardinality distribution pτk|k (n) for the number n of targets of type τ , for any τ . Let τ Nk|k

=

τ rk|k

=

∫

Dk|k (τ, x)dx

τ Nk|k ∑ τ′ τ ′ Nk|k

(9.225) (9.226)

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261

be, respectively, the expected number of, and the fraction of targets of, type τ . Then the probability that there are n targets of type τ is pτk|k (n) =

τ (rk|k )n

n!

(n)

τ · Gk|k (1 − rk|k ).

(9.227)

Here, Gk|k (x) is the p.g.f. corresponding to the cardinality distribution pk|k (n) on the total number n of targets (that is, regardless of target type). ˜ = X × T can be Equation (9.227) is proved by noting that the joint space X rewritten as τN τ1 ˜ = X ⊎ ... ⊎ X X (9.228) τi

where X = X × {τi } is the space of targets of type τ and where ‘⊎’ denotes disjoint union (topological sum). The discussion to be presented in Section 11.6.4 then applies, with target identity τ playing the same role as the mode variable o . Given this, (9.227) immediately follows from (11.129).

9.6

SEQUENTIAL MONTE CARLO (SMC) IMPLEMENTATION

SMC approximation, better known as particle approximation or particle-system approximation, has become a standard tool for implementing RFS filters (in [179], see Section 2.5.3, Chapter 15, Section 16.5.3, and Section 16.9.2). Standard references and tutorials describing SMC methods include [11], [29], [61], [252]. For a more detailed discussion of particle implementation of PHD and CPHD filters, see the book Particle Filters for Random Set Models by Ristic [250]. The first algorithmic implementations of the PHD filter were based on SMC techniques, independently proposed in 2003 by Sidenbladh [271]; by Zajic and Mahler [330]; and by Vo, Singh, and Doucet [306]. Subsequently, the convergence properties of SMC implementations of the PHD filter were established, by Clark and Bell [44], Johansen, S. Singh, Doucet, and Vo [126]; and (for an auxiliary SMC implementation) by Whiteley, Singh, and Godsill [320], [321]. The purpose of this section is to summarize the major aspects of SMC implementation of PHD and CPHD filters, including some recent conceptual advances. For conceptual clarity, the emphasis will be on the simplest, or “bootstrap,” implementation approach, in which the Markov density (“dynamic prior”) fk+1|k (x|x′ ) is used as the importance-sampling density. The section is organized as follows:

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1. Section 9.6.1: Sequential Monte Carlo (SMC) approximation of PHD and CPHD filters. 2. Section 9.6.2: The “bootstrap” SMC-PHD filter. 3. Section 9.6.3: The “bootstrap” SMC-CPHD filter. 4. Section 9.6.4: Using measurements to select new particles: the approach of Ristic, Clark, and Vo. 5. Section 9.6.5: Particle implementation with target identity. 9.6.1

SMC Approximation

The particle approach for implementing PHD and CPHD filters is a direct generalization of the approach employed for the single-target particle filter. Intuitively speaking, it is based on approximation of the PHD as a Dirac sum: νk|k

Dk|k (x) ∼ =

∑

k|k

wi

(9.229)

· δxk|k (x) i

i=1 k|k

k|k

k|k

k|k

where x1 , ..., xνk|k are the particles and w1 , ..., wνk|k are their respective k|k

k|k

k|k

k|k

weights. More rigorously speaking, x1 , ..., xνk|k and w1 , ..., wνk|k form a particle approximation of Dk|k (x) if ∫

νk|k

θ(x) · Dk|k (x|Z k )dx ∼ =

∑

k|k

wi

k|k

· θ(xi )

(9.230)

i=1

for any unitless function θ(x) of x. Furthermore, it must be the case that, for particle approximations of arbitrarily large size, ∫

νk|k

θ(x) · Dk|k (x|Z k )dx =

lim νk|k →∞

∑

k|k

wi

k|k

· θ(xi )

(9.231)

i=1

for any unitless function θ(x) of x. The major difference between PHD particle approximation and single-target particle approximation is that the sum of weights, ν ∑ i=1

k|k

wi

∼ = Nk|k ,

(9.232)

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263

approximates the expected number of targets rather than being equal to 1. Another difference is that, once targets have been adequately localized, a k|k k|k single-target particle system x1 , ..., xνk|k consists of a single particle-cluster. The corresponding PHD-particle system, by way of contrast, consists of several particle-clusters—one for each detected target. This causes multitarget state estimation to be computationally challenging—see however, Section 9.6.4. It is common practice to assume that the particles are equally weighted: k|k wi = 1/ν for all i = 1, ..., νk|k . In this case, a particle system is conceptually like a statistical sample, with more particles located at large values of Dk|k (x) and fewer particles located at smaller ones. This convention will be followed in the sequel. 9.6.2

SMC-PHD Filter

The discussion in this section is a variant of that in [179], pp. 615-623. 9.6.2.1

SMC-PHD Filter: Time Update

For conceptual clarity, target-spawning will be neglected. Assume that the PHD Dk|k (x) at time tk has been approximated by the particle system k|k

k|k

{(w1 , x1 ), ..., (wνk|k , xνk|k )} k|k k|k with νk|k

Nk|k =

∑

k|k

(9.233)

wi

i=1

being the expected number of targets at time tk . Then we are to determine the timek+1|k k+1|k k+1|k k+1|k updated particle system {(w1 , x1 ), ..., (wνk+1|k , xνk+1|k )}. Substituting νk|k

Dk|k (x) =

∑

k|k

wi

(9.234)

· δxk|k (x) i

i=1

into the PHD filter time-update equation, (8.15) and (8.16), yields Dk+1|k (x)

=

(9.235)

bk+1|k (x) νk|k

+

∑ i=1

k|k

wi

k|k

k|k

· pS (xi ) · fk+1|k (x|xi ).

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The first and second terms correspond, respectively, to appearing-target particles and persisting-target particles. Consider each of these in turn. • Appearing-target particles: Let ∫

B Nk+1|k =

(9.236)

bk+1|k (x)dx

B B and let νk+1|k be the nearest integer to Nk+1|k . Define the probability density ˆbk+1|k (x) = bk+1|k (x) (9.237) B Nk+1|k B and draw νk+1|k samples from ˆbk+1|k (x), k+1|k

k+1|k

x1

, ..., xν B

∼ ˆbk+1|k (·),

(9.238)

k+1|k

with particles concentrated around locations where new targets are expected k+1|k k+1|k to appear. Then the particles x1 , ..., xν B represent the appearing k+1|k

targets, with corresponding weights k+1|k

wi

B = Nk+1|k ·∑

ˆbk+1|k (xk+1|k ) i . ˆbk+1|k (xk+1|k )

l=1

(9.239)

l

Remark 34 A na¨ıve approach to birth-particle selection is to place target -appearance particles in regions not currently containing targets. This approach typically requires a prohibitively large number of particles. A more sophisticated approach is to use measurements to guide the placement of new particles. However, Ristic, Clark, and Vo have shown that na¨ıve implementations of this approach lead to biased estimates of target number when the number of targets is small. They have proposed an alternative method [256], which will be described in Section 9.6.4. • Persisting-target particles: From (9.235) we see that the expected number of surviving (persisting) targets is, approximately, νk|k S Nk+1|k

=

∑ i=1

k|k

wi

k|k

· pS (xi ).

(9.240)

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265

Define the discrete probability distribution p˜S (i) on i ∈ {1, ..., νk|k } by k|k

p˜S (i) =

k|k

· pS (xi ) . S Nk+1|k

wi

(9.241)

S S S Let νk+1|k be the nearest integer to Nk+1|k and draw νk+1|k samples from pS (i): i1 , ..., iν S ∼ pS (·). (9.242) k+1|k

k|k

k|k

Then the particles xi1 , ..., xν S

are chosen to represent targets that persist

k+1|k

into time tk+1 . For each of these persisting particles, draw a single sample from the dynamic prior (the “bootstrap” approach): k+1|k

k|k

k+1|k

,....,

∼ fk+1|k (·|x1 )

x1

k|k

∼ fk+1|k (·|xν S

xν S k+1|k

k+1|k

Then the predicted particles x1 targets. 9.6.2.2

). (9.243)

k+1|k

k+1|k

, ..., xν S

represent the persisting

k+1|k

SMC-PHD Filter: Measurement Update

Assume that the predicted PHD Dk+1|k (x) has been approximated by the particle k+1|k

system {(w1

k+1|k

, x1

k+1|k

k+1|k

), ..., (wνk+1|k , xνk+1|k )} with νk+1|k

Nk+1|k =

∑

k+1|k

wi

.

(9.244)

i=1

Then we are to determine the time-updated particle system k+1|k+1

{(w1

k+1|k+1

, x1

), ..., (wνk+1|k+1 , xk+1|k+1 νk+1|k+1 )}. k+1|k

Substituting νk+1|k

Dk+1|k (x) =

∑ i=1

k+1|k

wi

· δxk+1|k (x) i

(9.245)

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into the PHD filter measurement-update equation, (8.50) and (8.51), yields νk+1|k

Dk+1|k+1 (x)

∑

=

k+1|k

k+1|k

· (1 − pD (xi

wi

)) · δxk+1|k (x)

(9.246)

i

i=1 mk+1 νk+1|k

+

∑ ∑ wik+1|k · pD (xik+1|k ) · Lzj (xik+1|k ) κk+1 (zj ) + τk+1 (zj ) j=1 i=1

·δxk+1|k (x) i

where νk+1|k

τk+1 (zj ) =

∑

k+1|k

k+1|k

· pD (xi

wi

k+1|k

) · Lzj (xi

).

(9.247)

i=1

The first and second terms correspond to the particles representing the undetected targets and the detected targets. Consider each in turn: • Undetected-target particles—for i = 1, ..., νk+1|k : k+1|k+1

xi

k+1|k

=

k+1|k+1 wi

(9.248)

xi (1 −

=

k|k pD (xi ))

·

k|k wi .

(9.249)

• Detected-target particles—for i = 1, ..., νk+1|k and j = 1, ..., mk+1 : k+1|k+1

xi,j

k+1|k

xi

=

pD (xi ) · Lzj (xi ) k|k · wi . κk+1 (zj ) + τk+1 (zj )

k|k k+1|k+1

wi,j 9.6.2.3

(9.250)

=

k|k

SMC-PHD Filter: Multitarget State Estimation

State estimation for single-target particle filters is relatively simple. If k+1|k+1

{(w1

k+1|k+1

, x1

), ..., (wνk+1|k+1 , xk+1|k+1 νk+1|k+1 )} k+1|k

(9.251)

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267

is the measurement-updated single-target particle system, then the mean and covariance of the target can be computed as νk+1|k+1

x ˆk+1|k+1

=

∑

k+1|k+1

k+1|k+1

(9.252)

· xi

wi

i=1 νk+1|k+1

Pˆk+1|k+1

=

∑

k+1|k+1

i=1 k+1|k+1 ·(xi

k+1|k+1

· (xi

wi

−x ˆk+1|k+1 )

(9.253)

−x ˆk+1|k+1 )T .

State estimation with SMC-PHD and SMC-CPHD filters, however, is typically complicated and computationally expensive. This is because a clustering algorithm (the EM algorithm, k-means, and so on) must be used to partition the multitarget particle system into separate single-target particle systems, with each partition corresponding to a hypothesized target track. Once this has been accomplished, (9.252) and (9.253) can be used to determine the means and covariances of the tracks. Fortunately, Ristic, Clark, and Vo have devised an alternative formulation of the SMC-PHD filter that does not require clustering [256]. Their approach will be described in Section 9.6.4. 9.6.3

SMC-CPHD Filter

This section is a condensation of [179], pp. 644-649. The primary point of interest is that the PHD, not the spatial distribution sk|k (x), is approximated using particle methods. 9.6.3.1

SMC-CPHD Filter: Time Update

The time-update formulas for the CPHD filter were given in (8.86) through (8.90). The particle implementation of (8.86), which is to say of Dk+1|k (x) = bk+1|k (x) +

∫

pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′ ,

(9.254)

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is identical to the time-update for the SMC-PHD filter, (9.235), except that there is no term for spawned targets: νk|k

Dk+1|k (x) = bk+1|k (x) +

∑

k|k

k|k

k|k

· pS (xi ) · fk+1|k (x|xi ).

wi

(9.255)

i=1

The update of the cardinality distribution and p.g.f. remain unchanged from (8.86) through (8.90), except that

ψk = 9.6.3.2

νk|k 1 ∑ k|k k|k w · pS (xi ). Nk|k i=1 i

(9.256)

SMC-CPHD Filter: Measurement Update

The measurement-update equations for the CPHD filter were given in (8.121) through (8.108). The equations for the cardinality distribution and p.g.f., (8.118) through (8.121), are unchanged, except that (8.113) becomes νk+1|k

1 ϕk = Nk+1|k

∑

k+1|k

k+1|k

· (1 − pD (xi

wi

)).

(9.257)

i=1

The measurement-update equation for the PHD, (8.121), becomes k+1|k

Dk+1|k+1 (x)

=

wi k+1|k · (1 − pD (xi )) Nk+1|k

(9.258)

ND

· L Zk+1 · δxk+1|k (x) i

+

k+1|k wi

Nk+1|k

k+1|k m ∑ pD (xi ) · Lzj (xk+1|k )

ck+1 (zj )

j=1

D

·LZk+1 (zj ) · δxk+1|k (x). i

Thus the particle representation for undetected targets is (for i = 1, ..., νk+1|k ) k+1|k+1

xi

k+1|k

=

k+1|k+1

wi

=

(9.259)

xi k+1|k wi

Nk+1|k

k+1|k

· (1 − pD (xi

ND

)) · L Zk+1

(9.260)

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whereas for detected targets it is (for i = 1, ..., νk+1|k and j = 1, ..., mk+1 ) k+1|k+1

xi,j

k+1|k

=

k+1|k+1

wi,j 9.6.3.3

=

(9.261)

xi k+1|k wi

Nk+1|k

·

k+1|k pD (xi )

· Lzj (xk+1|k ) D · LZk+1 (zj ). (9.262) ck+1 (zj )

SMC-CPHD Filter: Multitarget State Estimation

State estimation is accomplished in the same manner as in Section 9.6.2.3, except that the MAP estimate νk+1|k+1 = arg sup pk+1|k+1 (n)

(9.263)

n

is used instead of Nk+1|k+1 = 9.6.4

∑νk+1|k+1 i=1

wk+1|k+1 .

Using Measurements to Choose New Particles

Since SMC-PHD and SMC-CPHD filters have target appearance models, it is important to have a good methodology for choosing new particles. Since newly appearing targets will generate unanticipated measurements, one obvious approach is to use the measurements in Zk+1 at the next time tk+1 to choose the locations of the new particles. This is often described as “measurement-driven” particle placement. However, Ristic, Clark, Vo, and Vo have shown that, if this approach is applied na¨ıvely to the PHD filter, then target-number estimates will be biased downward (see [256], p. 3 and [257], p. 1659). The cause of this bias is not understood. It also appears to be significant only for smaller numbers of targets, but can have a pronounced effect when this is the case. As a remedy, in [256], [257] Ristic et al. proposed a reformulation of the SMC-PHD filter that: • Avoids this bias. • Is faster and more accurate than the conventional SMC-PHD filter. • Avoids the necessity of using ad hoc and computationally expensive clustering algorithms during the multitarget state-estimation step. Because of the final two points, the approach is advantageous even when the number of targets is not small. The purpose of this section is to describe the

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technique, which can also be applied to the SMC-CPHD filter—see [257] for more details. For the sake of conceptual simplicity, only the SMC-PHD filter version is considered here. The key idea is to replace the single-target state space X with a new state space ˜ =X⊎B X (9.264) where ‘⊎’ denotes disjoint union, and where B = X is a copy of X.10 Intuitively speaking, X is the space of “persisting targets” whereas B is the space of “birth ˜ is defined as targets.” Integration on X ∫ ∫ ∫ ˜ ˜ f (˜ x)d˜ x= f (x)dx + f˜(b)db. (9.265) X

9.6.4.1

B

Unbiased SMC-PHD Filter: Models

Given this model, any motion or measurement model that involves the original state variable x must be redefined using the new state variable x ˜, where either x ˜ = x ∈ X or x ˜ = b ∈ B. Thus Ristic et al. make the following definitions: • Probability of target survival ([256], Eq. (14))—is the same for new and persistent targets: { pS (x) if x ˜=x p˜S (˜ x) = . (9.266) pS (b) if x ˜=b • Birth-target PHD ([256], Eq. (11))—persisting targets cannot be new targets: { 0 if x ˜=x ˜bk+1|k (˜ x) = . (9.267) bk+1|k (b) if x ˜=b • Markov transition density ([256], Eq. (12))—a birth target can transition to a persisting target, but not vice versa:  ˜ = x, x ˜ ′ = x′  fk+1|k (x|x′ ) if x ′ ′ ˜ f (x|b ) if x ˜ = x, x ˜ ′ = b′ . fk+1|k (˜ x|˜ x)= (9.268)  k+1|k 0 if otherwise 10 In [256], Risti´c et al. actually defined ˜ = X × {0, 1}, X ˜ = X × B. The alternative notation has been chosen for conceptual clarity. which is equivalent to X

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271

• Probability of detection ([256], Eq. (17))—new targets are always detected: { pD (x) if x ˜=x p˜D (˜ x) = . (9.269) 1 if x ˜=b • Likelihood function ([256], Eq. (18))—new targets and persisting targets have the same measurement statistics: { Lz (x) if x ˜=x ˜ Lz (˜ x) = . (9.270) Lz (b) if x ˜=b ˜ can be equivalently expressed ˜ k|k (˜ It follows that a PHD D x) defined on X as two PHDs Dk|k (x)

=

˜ k|k (x) D

(9.271)

=

˜ k|k (b) D

(9.272)

B

D k|k (b)

defined on X and B, respectively. Consequently, the revised PHD filter will actually consist of two coupled PHD filters, one for persisting targets and one for birth targets: ... →

Dk|k (x)

... →

D k|k (b)

→ ↑

Dk+1|k (x)

→

D k+1|k (b)

B

9.6.4.2

→ ↑↓

Dk+1|k+1 (x)

→

D k+1|k+1 (b)

B

→ ...

B

→ ...

Unbiased SMC-PHD Filter: Time Update

In the following, the target-spawning model will be ignored for conceptual clarity. The time-update equation results from substituting the models in Section 9.6.4.1 into the classical PHD filter time-update equation, (8.15), and using (9.265): ∫ ˜ ˜ ˜ k+1|k (˜ Dk+1|k (˜ x) = bk+1|k (˜ x) + p˜S (˜ x) · f˜k+1|k (˜ x|˜ x′ ) · D x′ )d˜ x′ . (9.273) That is, • Persisting-target time-update ([256], Eq. (16)): ∫ ( ) B Dk+1|k (x) = pS (x) · fk+1|k (x|x′ ) · Dk+1|k (x′ ) + D k+1|k (x′ ) dx′ . (9.274)

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• Birth-target time-update ([256], Eq. (16)): B

D k+1|k (b) = bk+1|k (b). 9.6.4.3

(9.275)

Unbiased SMC-PHD Filter: Measurement Update

The measurement-update equation results from substituting the models in Section 9.6.4.1 into the classical PHD filter measurement-update equation, (8.50) and (8.51), and using (9.265): ˜ k+1|k+1 (˜ D x) ˜ Dk+1|k (˜ x) τk+1 (z)

=

∑

1 − p˜D (˜ x) +

˜ z (˜ p˜D (˜ x) · L x) κk+1 (z) + τk+1 (z)

(9.276)

z∈Zk+1

=

∫

˜ z (˜ ˜ k+1|k (˜ p˜D (˜ x) · L x) · D x)d˜ x.

(9.277)

That is, • Persisting-target measurement-update ([256], Eq. (20)): Dk+1|k+1 (x) Dk+1|k (x)

=

1 − pD (x) +

∑

(9.278)

pD (x) · Lz (x) κk+1 (z) + τk+1 (z)

z∈Zk+1

τk+1 (z)

=

∫

pD (x) · Lz (x) · Dk+1|k (x)dx ∫ + Lz (x) · bk+1|k (x)dx.

(9.279)

• Birth-target measurement-update ([256], Eq. (21)): B

D k+1|k+1 (b) =

∑ z∈Zk+1

Lz (b) · bk+1|k (b) . κk+1 (z) + τk+1 (z)

(9.280)

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9.6.4.4

273

Unbiased SMC-PHD Filter: SMC Implementation

SMC implementation consists of the following steps. Suppose that the predicted PHD is approximated as a particle system: νk+1|k

Dk+1|k (x) ∼ =

∑

k+1|k

(9.281)

· δxk+1|k (x).

wi

i

i=1

Then: 1. Step 1: Creation of birth particles: For each zj ∈ Zk+1 , generate ρ new k+1|k k+1|k particles b1,j , ..., bρ,j in such a manner that zj can be considered k+1|k

k+1|k

to be a random sample drawn from Lz (bi,j ) · bk+1|k (bi,j ). Their weights are all set to the same value for i = 1, ..., ρ · mk+1 ([256], Eq. (22)) bk+1|k

k+1|k

=

(9.282)

bi mk+1

=

∑ 1 ρ · mk+1 j=1

ρ ∑

k+1|k

(9.283)

Lzj (bl+(j−1)mk+1 ,j )

l=1

k+1|k

·bk+1|k (bl+(j−1)mk+1 ,j ). 2. Step 2: Write the approximate birth-target PHD as a particle system using these weights and these particles: bk+1|k (x) ∼ = bk+1|k

ρ m k+1 ∑ ∑

(9.284)

δbk+1|k (x). i,j

i=1 i=1

3. Step 3: Update the persisting-target PHD weights ([256], Eq. (23)): k+1|k+1

wi

k+1|k

=

(1 − pD (xi mk+1

+

∑ j=1

k+1|k

(9.285)

)) · wi

k+1|k pD (xi )

·

k+1|k ) Lzj (xi

L(zj )

·

k+1|k wi

.

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where ([256], Eq. (25)): ρ·mk+1

L(zj )

=

∑

κk+1 (zj ) +

k+1|k

(9.286)

bi

i=1 νk+1|k

+

∑

k+1|k

k+1|k

) · Lzj (xi

pD (xi

k+1|k

) · wi

.

i=1

4. Step 4: Update the birth-target PHD weights ([256], Eq. (24)): mk+1 k+1|k+1

bi

=

∑ bk+1|k i . L(z j) j=1

(9.287)

5. Step 5: Separately resample the birth-target particle set and persisting-target particle set. 6. Step 6: Define the weights j = 0, 1, ..., mk+1 ([256], Eq. (26)):

k+1|k+1

wi,j

=

  (1 − pD (xk+1|k )) · w k+1|k i i k+1|k



pD (xi

k+1|k

)·Lzj (xi L(zj )

if

j=0

if

j>0

(9.288)

k+1|k

)·wi

νk+1|k

∑

k+1|k+1

Wj

=

k+1|k+1

wi,j

(9.289)

.

i=1

7. Step 7: Define the state estimates and their covariances, without resort to clustering, as ([256], Eqs. (27,28)), for j = 1, ..., mk+1 : νk+1|k k+1|k+1 x ˆj

=

∑

k+1|k+1

k+1|k

(9.290)

· xi

wi,j

i=1 νk+1|k k+1|k+1 Pˆj

=

∑

k+1|k+1

i=1 k+1|k ·(xi

k+1|k

· (xi

wi,j

k+1|k+1

−x ˆj

)

(9.291)

k+1|k+1 T

−x ˆj

) . k+1|k+1

8. Step 8: Eliminate those estimates in Step 7 for which Wj a suitable threshold value.

is less than

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275

Ristic et al. compared this approach with a more conventional approach employing k-means clustering. It greatly outperformed the k-means approach, achieving much better localization performance, while cardinality-estimation performance was essentially the same (see [256], Figure 4). 9.6.5

Implementation with Target Identity

The SMC-PHD and SMC-CPHD filters can be extended, at least in a na¨ıve fashion, to include target identity. The single-target state will have the form (x, c) where x is the kinematic state and c is a discrete ID parameter. Consequently, a k|k k|k k|k k|k particle system will have the form (c1 , x1 ), ..., (cνk|k , xνk|k ) with weights k|k

k|k

w1 , ...., wνk|k . The resulting particle-PHD filter has computational complexity O(mnC) where m is the current number of measurements, n is the current number of tracks, and C is the number of target types.

Chapter 10 Multisensor PHD and CPHD Filters 10.1

INTRODUCTION

The measurement-update equations for the classical PHD and CPHD filters, Sections 8.4.3 and 8.5.4, apply only to the single-sensor case. The subject of this chapter is PHD and CPHD filters for multiple, independent sensors. 10.1.1

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter: • The general single-sensor PHD filter of Section 8.2 can be generalized to the multisensor case (Section 10.3). The measurement-update for this filter is combinatorial and thus computationally problematic in general. • The “iterated-corrector” PHD and CPHD filters are the most commonly used approximate multisensor PHD and CPHD filters. However, they are not theoretically satisfactory, because they depend on the order of the sensor. This order-dependence can result in degraded performance when the sensors’ probabilities of detection are significantly different. This is particularly noticeable in the case of the iterated-corrector PHD filter (Section 10.5). • Computationally tractable and theoretically satisfactory approximate multisensor PHD and CPHD filters exist: the parallel-combination approximate multisensor (PCAM) PHD and CPHD filters. There are three such filters: the PCAM-CPHD filter (Section 10.6.1), the PCAM-PHD filter (Section 10.6.2),

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and the simplified PCAM-PHD filter (Section 10.6.3). The simplified PCAMPHD filter is the conceptually and computationally simplest, but is not appropriate for scenarios containing a small number of targets. • A proposed approximate multisensor PHD filter, based on the averaging of PHD pseudolikelihoods, is conceptually and theoretically erroneous (Section 10.7). • All of these filters have been implemented and compared in simulations. Among the approximate filters, the PCAM-CPHD and PCAM-PHD filters perform the best and the averaged-pseudolikelihood PHD filter performs the worst (Section 10.8). 10.1.2

Organization of the Chapter

The chapter is organized as follows: 1. Section 10.2: The multisensor-multitarget recursive Bayes filter. 2. Section 10.3: The multisensor generalization of the general single-sensor PHD filter of Section 8.2. 3. Section 10.4: The multisensor generalization of the single-sensor classical PHD filter of Section 8.4. 4. Section 10.5: The iterated-corrector approximation for multisensor application of the classical PHD and CPHD filters. 5. Section 10.6: Principled but computationally tractable “parallel combination approximate multisensor” (PCAM) approximations for multisensor application of the classical PHD and CPHD filters. 6. Section 10.7: An erroneous approximate multisensor PHD filter based on an averaged-pseudolikelihood approach. 7. Section 10.8: Performance comparisons of the multisensor PHD and CPHD filters described in this chapter.

Multisensor PHD and CPHD Filters

10.2

279

THE MULTISENSOR-MULTITARGET BAYES FILTER

Targets have state vectors. However, sensors also have state vectors. For example, ∗ the state x of a sensor could have the form ∗

˙ α, x = (x, y, z, x, ˙ y, ˙ z, ˙ ℓ, θ, α, φ, θ, ˙ φ, ˙ µ, χ)

(10.1)

where x, y, z and x, ˙ y, ˙ z˙ are the position and velocity coordinates of the sensor˙ α, carrying platform, ℓ is its fuel level, θ, α, ϕ and θ, ˙ φ˙ are the sensor’s bodyframe coordinates and their rates, µ is the sensor mode, and χ is the current communications transmission path employed by the sensor. Consequently, a single-target measurement model actually has the form ∗

Zk+1 = ηk+1 (x) + Vk+1 abbr. = ηk+1 (x, x) + Vk+1

(10.2)

∗

where x is the current state of the sensor. Similarly, the sensor likelihood function actually has the form ∗

∗

∗

Lz (x, x) = fk+1 (z|x, x) = fVk+1 (z − ηk+1 (x, x))

(10.3)

and the multiobject probability distribution of the clutter process actually has the form ∗ κk+1 (Z) abbr. = κk+1 (Z|x). (10.4) Suppose that there are s sensors and that the jth sensor collects meaj j

surements z drawn from the measurement space Z. At any given time, let the j

measurement set collected by the jth sensor be denoted by Z. Then the joint multisensor measurement space is 1

s

(10.5)

Z = Z ⊎ ... ⊎ Z

where ‘⊎’ denotes disjoint union. At any given time, the total measurement set has the form 1 s Z = Z ⊎ ... ⊎ Z. (10.6) Let j

j

j

j

∗j

L j (X) abbr. = f k+1 (Z|X) abbr. = fk+1 (Z|X, x)

(10.7)

Z ∗j

be the multitarget likelihood function for the jth sensor, where x is the current state of the jth sensor.

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According to the discussions in Section 3.5.3 and Section 4.2.5, we can write the distribution of Z as a joint distribution: 1

s

1

s

fk+1 (Z|X) = fk+1 (Z ⊎ ... ⊎ Z|X) = fk+1 (Z, .., Z|X).

(10.8)

The sensors are conditionally independent of the multitarget state if 1

1

s

s

1

s

(10.9)

fk+1 (Z, .., Z|X) = f k+1 (Z|X) · · · f k+1 (Z|X). j

j

j

Given this, let Z (k) : Z 1 , ..., Z k be the time sequence of measurements collected by the jth sensor at time tk . Then the Bayes-optimal multisensor measurement-update is accomplished using Bayes’ rule: 1

s

fk+1|k+1 (X|Z (k+1) , ..., Z (k+1) ) 1

=

(10.10)

s

1

s

fk+1 (Z k+1 , .., Z k+1 |X) · fk+1|k (X|Z (k) , ..., Z (k) ) 1

s

1

s

fk+1 (Z k+1 , ..., Z k+1 |Z (k) , ..., Z (k) ) ( 1 ) s 1 s f k+1 (Z k+1 |X) · · · f k+1 (Z k+1 |X) 1

s

·fk+1|k (X|Z (k) , ..., Z (k) )

=

1

s

1

(10.11)

s

fk+1 (Z k+1 , ..., Z k+1 |Z (k) , ..., Z (k) ) where 1

s

1

fk+1 (Z, ..., Z|Z

∫

s

(k)

, ..., Z

(k)

)

=

1

s

1

s

f k+1 (Z|X) · · · f k+1 (Z|X) (10.12) 1

s

·fk+1|k (X|Z (k) , ..., Z (k) )δX. In abbreviated form, 1

fk+1|k+1 (X) =

s

fk+1 (Z k+1 , .., Z k+1 |X) · fk+1|k (X) 1

s

.

(10.13)

fk+1 (Z k+1 , ..., Z k+1 ) As with the single-sensor, multitarget Bayes filter, the multisensor-multitarget Bayes filter must be approximated. This is the purpose of the filters described in the remainder of the chapter.

Multisensor PHD and CPHD Filters

10.3

281

THE GENERAL MULTISENSOR PHD FILTER

The general PHD filter described in Section 8.2 can be extended to a general multisensor PHD filter, as was shown in [47], Section V. This filter is the subject of this section. 10.3.1

General Multisensor PHD Filter: Modeling

For the sake of conceptual clarity, the two-sensor case will be considered. The general multisensor case follows by extrapolation (see Remark 35 in Section 10.3.2). The measurement-update equations for the two-sensor PHD filter requires the following models, which correspond to the single-sensor models in Section 8.2: • A “undotted sensor” (first sensor) and a “dotted sensor” (second sensor), where the nomenclature refers to the fact that the models for the latter will be dotted (as in ‘p˙ D ’) whereas those for the former will not be dotted (as in ‘pD ’). • Joint measurement set: Z˜k+1 = Zk+1 ⊎ Z˙ k+1

(10.14)

where Zk+1 is the measurement set collected by the undotted sensor, and Z˙ k+1 is the measurement set collected by the dotted sensor. • Single-target likelihood functions. These are: LZ (x) =

δGx [0], δZ

δ G˙ x L˙ Z˙ (x) = [0] δ Z˙

(10.15)

where Gx [g] is the p.g.fl. of the undotted target-measurement RFS Υk+1 (x); and where G˙ x [g] ˙ is the p.g.fl. of the dotted target-measurement RFS ˙ k+1 (x). Υ • Generalized probabilities of detection. These are the probabilities that the sensors will, respectively, collect at least one measurement: πD (x) = 1 − L∅ (x),

π˙ D (x) = 1 − L˙ ∅ (x).

˙ ⊆ Z˙ k+1 , • Joint likelihood integral. For any W ⊆ Zk+1 and W ∫ τW,W˙ = LW (x) · L˙ W˙ (x) · Dk+1|k (x)dx.

(10.16)

(10.17)

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• Joint generalized probability of detection. The probability that at least one of the sensors will collect at least one measurement: π ˜D (x) = 1 − (1 − πD (x)) · (1 − π˙ D (x)) .

(10.18)

• Non-ubiquity of clutter. We must assume the following: for both sensors, it is possible that no clutter measurements will be collected at all: pκk+1 (0) > 0,

p˙ κk+1 (0) > 0

(10.19)

where pκk+1 (m) is the cardinality distribution of the undotted clutter RFS Ck+1 and p˙ κk+1 (m) is the cardinality distribution of the dotted clutter RFS C˙ k+1 . • Clutter log-distributions. These are the multiobject density functions of the respective clutter log-p.g.fl.’s: κZ =

δ log Gκk+1 [0], δZ

κ˙ Z˙ =

δ log G˙ κk+1 [0] δ Z˙

(10.20)

where Gκk+1 [g] denotes the p.g.fl. of the undotted-sensor clutter process, and G˙ κk+1 [g] ˙ denotes the p.g.fl. of the dotted-sensor clutter process. 10.3.2

General Multisensor PHD Filter: Update

This is ˜ Dk+1|k+1 (x) = L Zk+1 ,Z˙ k+1 (x) · Dk+1|k (x)

(10.21)

where ˜ L Zk+1 ,Z˙ k+1 (x)

=

1−π ˜D (x) +

∑

ωP

(10.22)

˜k+1 P⊟Z

·

∑ ˙ ∈P W ⊎W

LW (x) · L˙ W˙ (x) . δ|W˙ |,0 · κW + δ|W |,0 · κ˙ W˙ + τW,W˙

Here, the first summation is taken over all partitions P of the joint measurement set Z˜k+1 = Zk+1 ⊎ Z˙ k+1 ; and ( ) ∏ δ · κ + δ · κ ˙ + τ ˙ ˙ ˙ ˙ W |W |,0 W ⊎W ∈P |W |,0 W W,W ( ). (10.23) ωP = ∑ ∏ δ · κ + δ · κ ˙ + τ ˜ ˙ ˙ ˙ ˙ V |V |,0 Q⊟Zk+1 V ⊎V ∈Q |V |,0 V V,V

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Remark 35 (General PHD filter, more than two sensors) For three sensors (“undotted,” “dotted,” and “double-dotted”), the analog of (10.22) is ˜ L ¨k+1 (x) Zk+1 ,Z˙ k+1 ,Z

=

∑

1−π ˜D (x) +

(10.24)

ωP

˜k+1 P⊟Z

∑ ˙ ⊎W ¨ ∈P W ⊎W

¨ ¨ (x) LW (x) · L˙ W˙ (x) · L W   δ |W · δ · κ ¨ |,0 ˙ |,0 W |W  +δ|W ˙ W˙  ¨ |,0 · δ|W |,0 · κ    +δ ˙ · δ|W |,0 · κ ¨W ¨  |W |,0 +τW,W˙ ,W ¨

where π ˜D (x)

=

τW,W˙ ,W ¨

=

(10.25)

1 − (1 − pD (x)) · (1 − p˙ D (x)) · (1 − p¨D (x)) ∫ ¨ ¨ (x) · Dk+1|k (x)dx LW (x) · L˙ W˙ (x) · L W 

ωP

=

 δ |W ¨ |,0 · δ|W ˙ |,0 · κW  +δ|W ∏ ˙ W˙  ¨ |,0 · δ|W |,0 · κ   ˙ ⊎W ¨ ∈P  W ⊎W +δ|W˙ |,0 · δ|W |,0 · κ ¨W ¨  +τW,W˙ ,W ¨  δ|V¨ |,0 · δ|V˙ |,0 · κV  +δ|V¨ |,0 · δ|V |,0 · κ˙ V˙ ∑ ∏  ˜k+1 Q⊟Z V ⊎V˙ ⊎V¨ ∈Q  +δ ¨ V¨ |V˙ |,0 · δ|V |,0 · κ +τV,V˙ ,V¨

(10.26)

 .(10.27)   

For more than three sensors, the pattern is clear.

10.4

THE MULTISENSOR CLASSICAL PHD FILTER

Assume that the undotted and dotted sensors have standard multitarget measurement models: clutter is Poisson and targets generate at most a single measurement. Then

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from (10.21) through (10.23) we know that  if Z=∅  1 − pD (x) pD (x) · Lz (x) if Z = {z} LZ (x) =  if |Z| > 1 0  Z=∅  −λk+1 if κk+1 (z) if Z = {z} κZ =  0 if |Z| > 1  if Z˙ = ∅  1 − p˙ D (x) ˙ ˙ ˙ LZ˙ (x) = p˙ (x) · Lz˙ (x) if Z = {z˙ }  D ˙ >1 0 if |Z|  Z˙ = ∅  −λ˙ k+1 if κ˙ Z˙ = κ˙ (˙z) if Z˙ = {˙z} .  k+1 ˙ >1 0 if |Z|

(10.28)

(10.29)

(10.30)

(10.31)

Then we get the following result (originally established in [166]): ˜ Dk+1|k+1 (x) = L Zk+1 ,Z˙ k+1 (x) · Dk+1|k (x)

(10.32)

where ˜ L ˜D (x) + Zk+1 ,Z˙ k+1 (x) = 1 − π

∑

ωP

˜k+1 P⊟2 Z

∑

ρW ⊎W˙ .

(10.33)

˙ ∈P W ⊎W

Here, the summation is taken over all partitions P of Z˜k+1 = Zk+1 ⊎ Z˙ k+1 that are “binary” in the following sense. The partition P is binary if every cell ˙ ∈ P is binary, in that it has one of the following three forms: W ⊎W ˙ = {z}, W ⊎W

˙ = {z˙ }, W ⊎W

˙ = {z, z˙ }. W ⊎W

Also,

ρW ⊎W˙

ωP

=

      

=

∑

pD (x)·ℓz (x)·(1−p˙ D (x)) 1+τz,∅ (1−pD (x))·p˙ D (x)·ℓ˙z˙ (x) 1+τ∅,z˙ pD (x)·ℓz (x)·p˙ D (x)·ℓ˙z˙ (x) τz,z˙

∏

˙ ∈P W ⊎W

˜k+1 Q⊟Z

∏

dW,W˙

V ⊎V˙ ∈Q

dV,V˙

if

˙ =∅ W = {z}, W ˙ = {z˙ } W = ∅, W

if

˙ = {z˙ } W = {z}, W

if

(10.34)

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where

ρW ⊎W˙

dW ⊎W˙

   

=

  

pD (x)·ℓz (x)·(1−p˙ D (x)) 1+τz,∅ (1−pD (x))·p˙ D (x)·ℓ˙z˙ (x) 1+τ∅,z˙ pD (x)·ℓz (x)·p˙ D (x)·ℓ˙z˙ (x) τz,z˙

  1 + τz,∅ 1 + τ∅,z˙  τz,z˙

=

if if if

if

˙ =∅ W = {z}, W ˙ = {z˙ } W = ∅, W

if

˙ = {˙z} W = {z}, W

if

˙ =∅ W = {z}, W ˙ = {z˙ } W = ∅, W ˙ = {z˙ } W = {z}, W

and where ℓz (x)

=

τz,∅

=

τ∅,z˙

=

τz,z˙

=

Lz (x) Lz˙ (x) , ℓ˙z˙ (x) = κk+1 (z) κk+1 (˙z) ∫ pD (x) · ℓz (x) · (1 − p˙ D (x)) · Dk+1|k (x)dx ∫ (1 − pD (x)) · p˙ D (x) · ℓ˙z˙ (x) · Dk+1|k (x)dx ∫ pD (x) · ℓz (x) · p˙ D (x) · ℓ˙z˙ (x) · Dk+1|k (x)dx.

(10.35) (10.36) (10.37) (10.38)

The fact that (10.21) through (10.23) reduce to these formulas is established in Section K.20. 1 1 2 2 Example 6 (Binary partitions) Suppose that Z˜k+1 = {z1 , z2 } ∪ {z1 , z2 }. Then ˜ the binary partitions of Zk+1 are 1

P1

=

1

P2

= = = = = =

2

2

2

2

1

2

(10.41)

2

1

1

(10.40)

2

1

1

(10.39)

2

(10.42)

2

(10.43)

2

{{z1 , z1 }, {z2 , z2 }} 1

P7

1

{{z2 , z1 }, {z1 }, {z2 }} 1

P6

2

{{z1 , z2 }, {z2 }, {z1 }} 1

P5

2

{{z2 , z2 }, {z1 }, {z1 }} 1

P4

2

{{z1 , z1 }, {z2 }, {z2 }} 1

P3

1

{{z1 }, {z2 }, {z1 }, {z2 }}

(10.44)

2

{{z1 , z2 }, {z2 , z1 }}.

(10.45)

Remark 36 (Sensor-consistency gating) One possible method for reducing computational load in the multisensor classical PHD filter is “sensor-consistency gat1 2 ing” [166]. That is, discard partitions that contain any measurement-pair {z, z}

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such that 1

2

1 2 pD Lz1 pD Lz2 ∼ = 0.

(10.46)

Such pairs are less likely to have been jointly generated by the same target. This approach is, of course, based on the presumption that the clutter rates of both sensors are not too large, as compared to the corresponding sensor resolutions.

Remark 37 (Classical PHD filter, more than two sensors) The three-sensor and multisensor cases follow from (10.24) and (10.28). For according to (10.24), ∑ LZk+1 ,Z˙ k+1 ,Z¨k+1 (x) = 1 − π ˜D (x) + ωP (10.47) ˜k+1 P⊟Z

∑ ˙ ⊎W ¨ ∈P W ⊎W

¨ ¨ (x) LW (x) · L˙ W˙ (x) · L W   δ |W ¨ |,0 · δ|W ˙ |,0 · κW  +δ|W ˙ W˙  ¨ |,0 · δ|W |,0 · κ    +δ ˙ · δ|W |,0 · κ ¨¨  |W |,0

W

+τW,W˙ ,W ¨

¨ ¨ (x) vanishes unless where, according to (10.28), the product LW (x) · L˙ W˙ (x) · L W ˙ ¨ ˙ ¨ the subsets W, W , W of each cell W ⊎ W ⊎ W of the partition P have no more than a single element. Thus the only terms in the second summation that survive are those corresponding to cells that are “ternary” in the following sense: they must have one of the following seven forms: ˙ W ⊎W ˙ W ⊎W ˙ W ⊎W ˙ W ⊎W

¨ ⊎W ¨ ⊎W ¨ ⊎W ¨ ⊎W

= =

{z} {z˙ }

(10.48) (10.49)

= =

{¨ z} {z, z˙ }

(10.50) (10.51)

˙ ⊎W ¨ W ⊎W ˙ ⊎W ¨ W ⊎W ˙ ⊎W ¨ W ⊎W

= = =

{z, ¨ z} {z˙ , ¨ z} {z, z˙ , ¨ z}.

(10.52) (10.53) (10.54)

Consequently, the only terms in the first summation that survive are those corresponding to partitions P of Z˜k+1 = Zk+1 ⊎ Z˙ k+1 ⊎ Z¨k+1 that are “ternary” in the following sense: every cell in P is ternary. For more than three sensors, the pattern is clear. For s sensors, the first summation will be taken over all partitions

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P of the multisensor measurement set Z˜k+1 that are “s-ary” in the sense that every cell of P is “s-ary.” And a cell is “s-ary” if it contains no more than a single measurement from each sensor. 10.4.1

Implementations of the Exact Classical Multisensor PHD Filter

The two-sensor classical PHD filter has been implemented and compared to other approaches in [221]. See Section 10.8 for more details. Moratuwage, Vo, and Danwei Wang have successfully implemented this filter in a Simultaneous Localization and Mapping (SLAM) robotics application [205].

10.5

ITERATED-CORRECTOR MULTISENSOR PHD/CPHD FILTERS

Since the exact multisensor PHD filter (and thus also the multisensor CPHD filter) is computationally problematic, approximation techniques are necessary. The simplest and oldest of these is the obvious heuristic approach: the iterated corrector multisensor PHD/CPHD filters. For the sake of conceptual clarity, assume that we have only two sensors, with the respective sensor models 1

1

1

1

1

pD (x),

Z,

(10.55)

Lz1 (x) = f k+1 (z|x)

1 1

1

1

1

1

pκk+1 (m)

κk+1 (z) = λk+1 ck+1 (z),

(10.56)

and 2

2

2

2

2

Z,

pD (x),

(10.57)

Lz2 (x) = f k+1 (z|x)

2 2

2

2

pκk+1 (m).

κk+1 (z) = λk+1 ck+1 (z),

(10.58)

Then as the name implies, in the iterated-corrector approach one simply applies the CPHD/PHD filter corrector step in succession, once for each of the two sensors. Thus in the case of the PHD filter, one first applies the corrector step for the first sensor: first

? ?? ? 1

1

1

2

Dk+1|k+1 (x| Z (k+1) , Z (k) ) 2

Dk+1|k (x|Z (k) , Z (k) )

1

∑

1

= 1 − pD (x) + 1

1

z∈Z k+1

pD (x) · Lz1 (x) 1

1

1

1

κk+1 (z) + τ k+1 (z)

(10.59)

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where 1

1

τ k+1 (z) =

∫

1

1

2

1

pD (x) · Lz1 (x) · Dk+1|k (x|Z (k) , Z (k) )dx.

(10.60)

Then one applies the corrector step for the second sensor: first

second

? ?? ? ? ?? ? 1

2

2

2

Dk+1|k+1 (x| Z (k+1) , Z (k+1) ) 1

∑

2

= 1 − pD (x) +

2

Dk+1|k+1 (x|Z (k+1) , Z (k) )

2

2

pD (x) · Lz2 (x) 2

2

2

2

κk+1 (z) + τ k+1 (z)

z∈Z k+1

(10.61) where first 2

2

τ k+1 (z) =

∫

? ?? ?

2

1

2

2

pD (x) · Lz2 (x) · Dk+1|k+1 (x| Z (k+1) , Z (k) )dx

(10.62)

and where the notations first

second

? ?? ?

? ?? ?

1

Z (k+1) ,

2

, Z (k+1)

indicate that the first sensor is applied first and the second sensor is applied second. 10.5.1

Limitations of the Iterated-Corrector Approach

The iterated-corrector approach is conceptually simple but not entirely satisfactory, from either a theoretical or a practical point of view. This is because: • The value of the posterior PHD depends on the order in which the sensors are applied: first

second

? ?? ? ? ?? ? 1

second

2

Dk+1|k+1 (x| Z (k+1) , Z (k+1) )

1

̸=

first

? ?? ? ? ?? ? 2

Dk+1|k+1 (x| Z (k+1) , Z (k+1) ). (10.63)

• As a consequence, tracking performance can be less or more effective, depending on the sensor order.

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Specifically, Nagappa and Clark [219] have demonstrated that a significant performance deterioration results when the probabilities of detection of the two 2 1 sensors are markedly different, say pD ≪ pD . In this case, the first sensor—that is, the one with the larger pD —should be applied before the second sensor—that is, the one with the smaller pD . On the other hand, if the probabilities of detection are approximately equal then tracking performance is not significantly affected by sensor order. The ultimate source of this behavior lies in the theoretical assumptions that are implicitly being made. In the case of the PHD filter, we begin by assuming 1

2

that the predicted distribution fk+1|k (X|Z (k) , Z (k) ) is Poisson. If the first sensor 1

2

is applied first, this results in an updated distribution fk+1|k+1 (X|Z (k+1) , Z (k) ). When we apply the corrector step for the second sensor, we are additionally and implicitly assuming that this distribution is Poisson. On the other hand, if the 1

2

second sensor is applied first then the distribution fk+1|k+1 (X|Z (k) , Z (k+1) ) is implicitly assumed to be Poisson. Because the two pairs of Poisson approximations are different, they lead to a dependence on sensor order. Similar comments apply to the iterated-corrector CPHD filter.

10.6

PARALLEL COMBINATION MULTISENSOR PHD AND CPHD FILTERS

The exact multisensor CPHD or PHD filters are computationally problematic. The iterated-corrector approximation is computationally tractable, but leads to potential performance problems. So what is to be done? The following describes a theoretically principled, order-independent, and computationally tractable approximation for multisensor PHD and CPHD filters, first proposed in 2010 [151]. The fundamental idea is as follows. Suppose that there are s sensors with respective multitarget likelihood functions j

j

j

(10.64)

L j (X) = f k+1 (Z|X) Z j

where Z denotes a measurement set collected by the jth sensor. Let 1

s

fk+1|k (X) abbr. = fk+1|k (X|Z (k) , ..., Z (k) )

(10.65)

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j

j

j

be the prior distribution, where Z (k) : Z 1 , ..., Z k denotes the time sequence of 1

s

measurement sets collected by the jth sensor. Let Z k+1 , ..., Z k+1 be the new measurement sets collected by the sensors. If the sensors are independent, then the posterior distribution conditioned on the new measurements is 1

s

fk+1|k+1 (X) abbr. = fk+1|k+1 (X|Z (k) , ..., Z (k) ) 1

s

1

f k+1 (Z k+1 |X) · · · f k+1 (Z k+1 |X) · fk+1|k (X)

=

1..s

f

(10.66)

s

1

(10.67)

s

k+1 (Z k+1 , ..., Z k+1 )

where 1..s

f

1

s

k+1 (Z k+1 , ..., Z k+1 ) 1

abbr.

= =

s

1

s

f (Z , ..., Z k+1 |Z (k) , ..., Z (k) ) ∫k+1 k+1 1 s 1 s f k+1 (Z k+1 |X) · · · f k+1 (Z k+1 |X) · fk+1|k (X)δX.

(10.68) (10.69)

Equivalently, 1

s

1

s

fk+1|k+1 (X) ∝ f k+1 (Z k+1 |X) · · · f k+1 (Z k+1 |X) · fk+1|k (X).

(10.70)

Now, we can rewrite fk+1|k+1 (X) as a “Bayes parallel combination” ([179], Eq. (8.5)) of the form 1

s

fk+1|k+1 (X) ∝ fk+1|k+1 (X|Z k+1 ) · · · fk+1|k+1 (X|Z k+1 ) · fk+1|k (X)1−s (10.71) where j

1

j

s

fk+1|k+1 (X|Z k+1 ) abbr. = fk+1|k (X|Z (k) , ..., Z (k+1) , ..., Z (k) )

(10.72)

is the multitarget posterior, “singly-updated” using only the jth measurement set j

Z k+1 . Thus, for example, if s = 2 then (10.71) becomes 1

2

fk+1|k+1 (X) ∝ fk+1|k+1 (X|Z k+1 ) · fk+1|k+1 (X|Z k+1 ) · fk+1|k (X)−1 . (10.73) Given these preliminaries, three principled multisensor approximate filters result from three different assumptions about the prior fk+1|k (X), the singlyj

updated posteriors fk+1|k+1 (X|Z k+1 ), and the sensor clutter processes. The assumptions underlying these filters are as follows:

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1. Parallel Combination Approximate Multisensor (PCAM) CPHD Filter: j

fk+1|k (X) and all of the fk+1|k+1 (X|Z k+1 ) are approximately i.i.d.c.; and all of the sensor clutter processes are i.i.d.c. That is, fk+1|k (X)

∼ =

|X|! · pk+1|k (|X|) · sX k+1|k

∼ =

|X|! · pk+1|k+1 (|X|) · sX k+1|k+1 (10.75)

=

|Z|! · pκk+1 (|Z|) · cZ k+1 .

(10.74)

j

fk+1|k+1 (X|Z k+1 ) j

j

j

j

j

j

j j

κk+1 (Z)

2. PCAM-PHD Filter: fk+1|k (X)

j

(10.76)

is approximately Poisson; and all of the

j

fk+1|k+1 (X|Z k+1 ) are approximately i.i.d.c.; and all of the sensor clutter processes are Poisson. That is, fk+1|k (X)

∼ =

X e−Nk+1|k · Dk+1|k

∼ =

|X|! · pk+1|k+1 (|X|) · sX k+1|k+1 (10.78)

=

e−λk+1 · κZ k+1 .

(10.77)

j

fk+1|k+1 (X|Z k+1 )

j

j

j

j j

j j

κk+1 (Z)

3. Simplified PCAM (SPCAM) PHD Filter:

(10.79)

fk+1|k (X)

and all of the

j

fk+1|k+1 (X|Z k+1 ) are approximately Poisson; and all of the sensor clutter processes are Poisson. That is, fk+1|k (X)

∼ =

X e−Nk+1|k · Dk+1|k j

j

fk+1|k+1 (X|Z k+1 )

∼ =

j

e−N k+1|k+1 · D X k+1|k+1 j

j j

κk+1 (Z)

(10.80) (10.81)

j j

=

e−λk+1 · κZ k+1 .

(10.82)

The basic ideas underlying the parallel-combination approximation are most easily illustrated by looking at the simplest special case: the simplified PCAM-PHD

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filter. Substituting (10.80) through (10.82) into (10.71) we get: 1

s

fk+1|k+1 (X|Z k+1 ) · · · fk+1|k+1 (X|Z k+1 ) (10.83)

∝

fk+1|k+1 (X)

·fk+1|k (X)1−s 1

s

X X 1−s DX k+1|k+1 · · · D k+1|k+1 · (Dk+1|k )

∝

1

1−s (D k+1|k+1 · · · D k+1|k+1 · Dk+1|k )X

=

(10.84)

s

(10.85)

and so X fk+1|k+1 (X) = e−Nk+1|k+1 · Dk+1|k+1

(10.86)

where 1

Dk+1|k+1 (x)

=

Nk+1|k+1

=

s

D k+1|k+1 (x) · · · D k+1|k+1 (x) · Dk+1|k (x)1−s (10.87) ∫ 1 s D k+1|k+1 (x) · · · D k+1|k+1 (x) (10.88) ·Dk+1|k (x)1−s dx.

However, because of the measurement-update equations for the classical PHD filter, (8.50) and (8.51), we can write j

j

j

pD (x) · Lj (x) ∑ j D k+1|k+1 (x) j z = Lj (x) = 1 − pD (x) + . (10.89) j j j j Dk+1|k (x) Z k+1 κ ( z) + τ ( z) k+1 k+1 j j z∈Z k+1

So, after substitution of these equations, (10.87) and (10.88) become the measurement update equations for the simplified PCAM-PHD filter of Section 10.6.3: 1

Dk+1|k+1 (x)

=

Nk+1|k+1

=

1

L 1 (x) · · · L 1 (x) · Dk+1|k (x) (10.90) Z Z k+1 ∫ k+1 1 1 L 1 (x) · · · L 1 (x) · Dk+1|k (x)dx. (10.91) Z k+1

Z k+1

The same basic reasoning is applied to derive the measurement-update equations for the PCAM-PHD and PCAM-CPHD filters. Because the approximation assumptions are more general, the resulting equations are more complicated than (10.90) and (10.91).

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As will be pointed out shortly, the simplified PCAM-PHD filter is not entirely satisfactory, since it does not reduce to the correct Bayesian solution in the singletarget case. Nevertheless, it has been employed with some success by a few researchers, and therefore remains worth consideration. The remainder of the section is organized as follows: 1. Section 10.6.1: The PCAM-CPHD filter. 2. Section 10.6.2: The PCAM-PHD filter. 3. Section 10.6.3: The simplified PCAM-PHD filter. 10.6.1

Parallel Combination Multisensor CPHD Filter

Suppose that at time tk we are given the spatial distribution sk|k (x) and the cardinality distribution pk|k (n). Then the measurement-update of the PCAMCPHD filter consists of the following steps. First, the usual time-update equations for the CPHD filter are used to construct the predicted spatial distribution sk+1|k (x) and the predicted cardinality distribution pk+1|k (n). Second, let ∑ Gk+1|k (x) = pk+1|k (n) · xn (10.92) n≥0 (1)

Nk+1|k

=

(10.93)

Gk+1|k (1).

Third, assume that there are s sensors collecting measurements of the form j j

z ∈ Z, and that they are governed by the following models, for j = 1, ..., s: j

j

• Probabilities of detection: pD (x) def. = pD,k+1 (x). j

j

j

• Single-target likelihood functions: Lj (x) abbr. = f k+1 (z|x). z

j j

j

j j

j

• Clutter intensity functions: κk+1 (z) = λk+1 · ck+1 (z) where λk+1 is the j j clutter rate and ck+1 (z) is the clutter spatial distribution. j j

• Clutter cardinality distributions and p.g.f.’s: pκk+1 (m) and Gκk+1 (z) = ∑ j κ m m≥0 pk+1 (m) · z .

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From these, define the following intermediate parameters: ∫ j j ϕk+1 = (1 − pD (x)) · sk+1|k (x)dx ∫ j j j j τˆk+1 (z) = pD (x) · Lj (x) · sk+1|k (x)dx.

(10.94) (10.95)

z

Fourth and finally, suppose that at time 1

tk the s sensors collect the j

s

j

respective measurement sets Z k+1 , ..., Z k+1 with |Z k+1 | = m. Then we are to construct the spatial distribution sk+1|k+1 (x) and cardinality distribution 1

s

pk+1|k+1 (n), updated using all of the measurement sets Z k+1 , ..., Z k+1 . Given these preliminaries, the measurement-updated spatial distribution and cardinality distribution are ([151], Eqs. (9-21)): sk+1|k+1 (x)

=

1 ˜ 1 ·L (x) · sk+1|k (x) s Z k+1 ,...,Z k+1 Nk+1|k+1

(10.96)

pk+1|k+1 (n)

=

n p˜(n) · θk+1 ˜ k+1 ) G(θ

(10.97)

where 1

˜ L

(x)

=

N k+1|k+1

=

1

s

Z k+1 ,...,Z k+1 1..s

s

˜ (1) (θk+1 ) LZ1 k+1 (x) · · · LZs k+1 (x) G · 1 (10.98) s ˜ k+1 ) G(θ N k+1|k+1 · · · N k+1|k+1 ∫ 1 s L 1 (x) · · · L s (x) · sk+1|k (x)dx (10.99) Z k+1

Z k+1

1..s

N k+1|k+1 θk+1

= 1

s

(10.100)

N k+1|k+1 · · · N k+1|k+1 and

Nk+1|k+1

=

p˜k+1|k+1 (n)

=

˜ k+1|k+1 (x) G

=

˜ (1) G k+1|k+1 (θk+1 ) ·θ ˜ k+1|k+1 (θk+1 ) k+1 G 1

s

ℓz1 (n) · · · ℓzs (n) · pk+1|k (n) ∑ p˜k+1|k+1 (n) · xn n≥0

(10.101) (10.102) (10.103)

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295

and where j

min{n,m}

∑

j

ℓj

(n)

=

Z k+1

j

j

j

(m − l)! · pκk+1 (m − l) · l!

(10.104)

l=0 j

j j

·Cn,l · ϕn−l k+1 · σ l (Z k+1 ) j j

Lj

(x)

j

(10.105)

α0 · (1 − pD (x))

=

Z k+1 j j

j

j

∑ pD (x) · Lj (x) · α(z) z

+

j

j

j

ck+1 (z)

j

z∈Z k+1 j j j j ∑ τˆk+1 (z) · α(z)

j

j j

N k+1|k+1

α0 · ϕk+1 +

=

j

j

(10.106)

j

ck+1 (z)

j

z∈Z k+1

and where

 j j j κ ( m − l)! · p ( m − l) k+1 l=0   j j j (l+1) ·G ( ϕ ) · σ ( Z ) l k+1 k+1 j k+1|k  α0 =  j ∑m j j j κ i=0 (m − l)! · pk+1 (m − i)   j j j (i) ·Gk+1|k (ϕk+1 ) · σ i (Z k+1 )   j ∑m j j j κ l=0 (m − l − 1)! · pk+1 (m − l − 1)   

j ∑m

j

(l+1)

j

j

j

·Gk+1|k (ϕk+1 ) · σ l (Z k+1 − {z})   α(z) = j ∑m j j j κ ( m − l)! · p ( m − i) k+1 i=0   j j j (i) ·Gk+1|k (ϕk+1 ) · σ i (Z k+1 )   j j j j τ ˆ ( z ) j j k+1 j  τˆk+1 (z1 ) m  σ l (Z k+1 ) = σ j  j , ..., j . j j m,l ck+1 (z1 ) ck+1 (z j ) j

j

(10.107)

(10.108)

(10.109)

m

The following characteristics of the PCAM-CPHD filter should be pointed out:

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Remark 38 (Computability of PCAM-CPHD filter) The PCAM-CPHD filter is potentially computationally tractable, provided that we assume that, for all larger values of n, pk+1|k (n) = 0. Given this, the computational complexity appears to 1

s

be O(m3 · · · m3 · n), where n is the current number of tracks and current numbers of measurements collected by the jth sensor.

j

m is the

Remark 39 (PCAM-CPHD filter and CPHD filter) In the single-sensor case, the PCAM-CPHD filter reduces to the CPHD filter. Remark 40 (Consistency of the PCAM-CPHD filter) Suppose that there are no missed detections, no false alarms, and that the number of targets is known a priori to be one—that is, assume the single-target filtering scenario. Then the PCAMCPHD filters reduces to the multisensor, single-target Bayes filter. Remark 41 (“Spooky action at a distance”) The PCAM-CPHD filter exhibits the “spooky action at a distance” behavior described in Section 9.2 for the singlesensor CPHD filter. This means that, as with the single-sensor CPHD filter, multitarget scenarios should be broken up into statistically noninteracting clusters, and a separate filter assigned to each cluster. Remark 42 (Disjoint FoVs) Because of this “spookiness,” the PCAM-CPHD filter does not behave intuitively when the sensors have mutually disjoint fields of view. Specifically, it does not reduce to separate, noninteracting filters, one for each field of view (as is the case with the multisensor classical PHD filter described in Section 10.4). Since no multisensor fusion is possible in such situations, this should not pose a difficulty in practice. Remark 43 (Implementations of PCAM-CPHD filter) Nagappa and Clark [221] have implemented two-sensor and three-sensor versions of the PCAM-CPHD filter. See Section 10.8 for more details. 10.6.2

Parallel Combination Multisensor PHD Filter

Suppose that at time tk we are given the PHD Dk|k (x). Then the measurementupdate step for this filter consists of the following steps. First, the usual time-update equation for the PHD filter is used to construct the predicted PHD Dk+1|k (x).

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Second, let Nk+1|k

=

sk+1|k (x)

=

∫

(10.110)

Dk+1|k (x)dx

Dk+1|k (x) . Nk+1|k

(10.111)

Third, assume that there are s sensors collecting measurements of the form j

j

j

j

j

z ∈ Z. Let ϕk+1 and τˆk+1 (z) be as in (10.94) and (10.95), respectively. Then the measurement-update for the PHD is ([151], Eqs. (22-29)): Dk+1|k+1 (x) = L

1

s

Z k+1 ,...,Z k+1

(10.112)

(x) · Dk+1|k (x)

where 1

s

L1 ˜ L

Z k+1 1

(x)

s

=

Z k+1 ,...,Z k+1

χk+1 ·

(x) · · · L s

1

(x)

Z k+1

(10.113)

s

ν k+1|k+1 · · · ν k+1|k+1

χk+1

(10.114)

= ∑

1

s

1 (n n≥0 ℓz

∑

+ 1) · · · ℓzs (n + 1) · 1

s

1 (j) · · · ℓs (j) j≥0 ℓz z

·

n Nk+1|k ·θ n n!

j Nk+1|k ·θ j j!

1..s

ν θ

1

1..s

ν

k+1|k+1

k+1|k+1

= =

s

ν k+1|k+1 · · · ν k+1|k+1 ∫ 1 s 1 L 1 (x) · · · L s (x) Z k+1 Z k+1 Nk+1|k ·Dk+1|k (x)dx

(10.115) (10.116)

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and where j

min{n,m}

∑

j

ℓj (x)

=

z

j j

j

j j

n−l λm−jl k+1 · l! · Cn,l · ϕk+1 · σ l (Z k+1 )

(10.117)

l=0 j

Lj

(x)

(10.118)

=

Z k+1 j j

∑

j

1 − pD (x) + j

j

z∈Z k+1

pD (x) · Lj (x) z

j j

j

j

κk+1 (z) + Nk+1|k · τˆk+1 (z) j

∑

j j

ν k+1|k+1

=

ϕk+1 + j

j

z∈Z k+1

j

τˆk+1 (z)

(10.119) j

j

j

j

κk+1 (z) + Nk+1|k · τˆk+1 (z)

j j

and where σ l (Z k+1 ) was defined in (10.109). The following characteristics of the PCAM-PHD filter should be pointed out: Remark 44 (Computability of PCAM-PHD filter) The PCAM-PHD filter may be more computationally problematic than the PCAM-CPHD filter. This is because of the infinite sums in the definition of the quantity χk+1 . For the PCAM-PHD filter to be tractable, these series must be truncated to a suitable degree of approximation. If the series do not converge rapidly enough, this could present a computational issue. Remark 45 (PCAM-PHD filter reduces to PHD filter) The PCAM-PHD filter reduces to the classical PHD filter in the single-sensor case. Remark 46 (Consistency of the PCAM-PHD filter) Suppose that there- are no missed detections, no false alarms, and that the number of targets is known a priori to be one—that is, assume the single-target filtering scenario. Then the PCAMPHD filter reduces to the multisensor, single-target Bayes filter. Remark 47 (“Spooky action at a distance”) Like the PCAM-CPHD filter, the PCAM-PHD filter exhibits the “spooky action at a distance” behavior described in Section 9.2. As with the PCAM-CPHD filter, scenarios should be broken up into statistically noninteracting clusters, and a separate filter assigned to each cluster. Remark 48 (Implementations of the PCAM-PHD filter) Nagappa and Clark [221] have implemented two-sensor and three-sensor versions of the PCAM-PHD filter. See Section 10.8 for more details.

Multisensor PHD and CPHD Filters

10.6.3

299

Simplified PCAM-PHD Filter

This, the simplest of the approximate multisensor PHD filters, was first proposed without proof in [165], Eqs. (105-107). It has the following measurement-update equation: Dk+1|k+1 (x) = L

1

Z k+1

(x) · · · L

s

Z k+1

(10.120)

(x) · Dk+1|k (x)

where j j j

Lj

(x)

1 − pD (x) +

=

pD (x) · Lj (x)

∑

j

Z k+1 j

z

j

j

j

(10.121) j

κk+1 (z) + τ k+1 (z)

j

z∈Z k+1 j

∫

j

τ k+1 (z)

=

j j

pD (x) · Lj (x) · Dk+1|k (x)dx.

(10.122)

z

The following characteristics of this filter should be noted: Remark 49 (Computability of simple PCAM-PHD filter) The complexity of the j 1 s filter is O(m · · · m · n) where n is the current number of tracks and m is the current number of measurements collected by the jth sensor. Remark 50 (Behavior with small target number) The simplified PCAM-PHD filter is unlikely to perform well when the number of targets is small. This is because it does not reduce to the correct formula in the single-sensor, single-target special case—that is, to the multisensor version of Bayes’ rule. For in this case (10.121) reduces to j Lj (x) j z Lj (x) = j k+1j (10.123) zk+1 τ k+1 (zk+1 ) where j

j

τ k+1 (z) =

∫

j

Lj (x) · fk+1|k (x)dx.

(10.124)

z

Thus (10.120) reduces to 1

L z1 fk+1|k+1 (x) =

1

s k+1 1

(x) · · · L zs k+1 (x) s

s

τ k+1 (zk+1 ) · · · τ k+1 (zk+1 )

· fk+1|k (x)

(10.125)

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whereas the correct Bayesian solution is 1

s

L z1 fk+1|k+1 (x) =

k+1

(x) · · · L zs k+1 (x) 1

1..s

τ

· fk+1|k (x)

s

(10.126)

k+1 (zk+1 , ..., zk+1 )

where 1..s

τ

1

s

k+1 (zk+1 , ..., zk+1 ) =

∫

1

s

L z1

k+1

(x) · · · L zs k+1 (x) · fk+1|k (x)dx.

(10.127)

Remark 51 (Implementations) Despite this last fact, some researchers have applied the simplified PCAM-PHD filter with success. Lian, Han, Liu, and Chen have applied it to the problem of joint tracking and sensor-bias estimation ([145], Eqs. (11-13)). They tested a particle implementation of their filter in simulations involving four appearing and disappearing targets, a range-bearing sensor, a rangeonly sensor, and a bearing-only sensor. They demonstrated that their filter successfully estimated the translational biases in the three sensors while also successfully tracking the targets. Also, Delande, Duflos, Heurguier, and Vanheeghie applied a partitioning method to lower computational cost [63]. They tested a particle implementation of their filter in simulations involving 11 maneuvering, appearing, and j disappearing targets, and five Gaussian sensors with different choices of pD (x), j

j j

λk+1 , and f k+1 (z|x). They demonstrated that the filter performed effectively.

10.7

AN ERRONEOUS “AVERAGED” MULTISENSOR PHD FILTER

All of the multisensor PHD and CPHD filters just described have one thing in common: they rely on multiplication of the pseudolikelihood functions of the sensors. For example, the simplest approximate PHD filter described in Section 10.6.3 is based on the product ×

1

L1

s

Z,...,Z

s

(10.128)

(x) = L 1 (x) · · · L s (x) Z

Z

of the pseudolikelihoods for the different sensors: j j

∑

j j

L j (x) = 1 − pD (x) + Z j

j

z∈Z k+1

pD (x) · Lj (x) z

j

j

j

(10.129) j

κk+1 (z) + τ k+1 (z)

Multisensor PHD and CPHD Filters

where j

j

τ k+1 (z) =

∫

301

j j

pD (x) · Lj (x) · Dk+1|k (x)dx.

(10.130)

z

The use of product-based approaches is intuitively and theoretically compelling. Statistical independence typically leads to products of fundamental descriptors; and products of probability distributions typically lead to smaller targetlocalization error-covariance matrices as long as the sources are not too greatly in conflict. However and to the contrary, it has been proposed [280] that the proper “Bayesian” approach for a multisensor PHD filter should be based on the average of the pseudolikelihoods:1 +

L1

s

(x) =

Z,...,Z

) s 1 (1 L 1 (x) + ... + L s (x) . Z Z s

(10.131)

As has been pointed out in [167] and [181], such an approach is questionable for an obvious reason: it tends to decrease (rather than increase) target-localization accuracy. The purpose of this section is to demonstrate this fact using simple examples. Section 10.8 reports simulation results leading to the same conclusion. Let us consider the simplest possible special case: a single-target with no clutter and no missed detections. The predicted and updated PHDs are then probability density functions: Dk+1|k (x) = fk+1|k (x) and Dk+1|k+1 (x) = fk+1|k+1 (x); and (10.131) reduces to +

Lz,..., 1 s (x) = z 1



1

s

Lz1 (x)

Lzs (x)



1 . + ... + s s 1 s τ1 k+1 (z) τ k+1 (z)

(10.132)

In [282], the author of [280] appears to retreat from this claim, stating that the filter described in [280] was actually a “traffic filter” that “was misidentified there as a multisensor target filter” ([282], Section I). That is, the claimed “multisensor multitarget intensity filter” in [280] is erroneous. However, in [282] the author goes on to claim the validity of a “traffic filter” generalization of the averaged-pseudolikelihood approach, in which the average is replaced by a weighted average using ∑s ℓ functions β ℓ (x) such that ℓ=1 β (x) = 1 identically. This claim is also not true—see Remark 10 in Section 4.2.5.2.

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Given this, the multisensor measurement update using the averaged likelihood is +

=

=

= j

(10.133)

f k+1|k+1 (x)  1  s Lzs (x) 1  Lz1 (x)  · fk+1|k (x) + ... + s s 1 s τ1 k+1 (z) τ k+1 (z) 1  s Lzs (x) · fk+1|k (x) 1  Lz1 (x) · fk+1|k (x)  + ... + s s 1 1 s τ k+1 (z) τ k+1 (z) ) 1( 1 s fk+1|k+1 (x|z) + ... + fk+1|k+1 (x|z) s j

(10.134)

(10.135) j

where the τ k+1 (z) were defined in (10.130) and where fk+1|k+1 (x|z) is the posterior distribution updated using the jth measurement alone. The following difficulties are apparent: +

1. Lz,..., is not a proper likelihood function for the single-target case, 1 s (x) z because it is unitless and integrates to infinity rather than to unity: ∫ + 1 s Lz,..., (10.136) 1 s (x)dz · · · dz = ∞. z +

2. f k+1|k+1 (x) is not equal to the Bayes-optimal solution, namely the multisensor, single-target version of Bayes’ rule: 1

fk+1|k+1 (x) =

s

Lz1 (x) · · · Lzs (x) · fk+1|k (x) 1

s

(10.137)

τk+1 (z, ..., z) where 1

s

τk+1 (z, ..., z) =

∫

1

s

Lz1 (x) · · · Lzs (x) · fk+1|k (x)dx.

(10.138)

+

3. Since the track distribution f k+1|k+1 (x) is a mixture distribution, it will exhibit increased track uncertainty (compared to the single-sensor track j distributions fk+1|k+1 (x|z))—rather than, as should be the case, decreased track uncertainty.

Multisensor PHD and CPHD Filters

Figure 10.1

303

The graph of (10.139).

The third point is most easily demonstrated using a simple example. Assume that there are two bearing-only sensors in the plane, with respective likelihood functions ( ) 1 1 (z − x)2 Lz (x, y) = Nσ2 (z − x) = √ · exp − (10.139) 2σ 2 2πσ ( ) 2 1 (z − y)2 Lz (x, y) = Nσ2 (z − y) = √ · exp − . (10.140) 2σ 2 2πσ That is, the sensors are oriented so as to triangulate the position of a target located at (x, y). The graphs of these functions are plotted in Figures 10.1 and 10.2, respectively. For the sake of conceptual clarity, further assume that the prior distribution is fk+1|k (x, y) = Nσ02 (x − x0 ) · Nσ02 (y − y0 ). (10.141) Then =

∫

Lz (x, y) · fk+1|k (x, y)dxdy

=

∫

Nσ2 (z − x) · Nσ02 (x − x0 ) · Nσ02 (y − y0 )dxdy (10.143)

=

Nσ2 +σ02 (z − x0 )

1

τ k+1 (z)

1

(10.142)

(10.144)

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Figure 10.2

The graph of (10.140).

and similarly =

∫

=

Nσ2 +σ02 (z − y0 ).

2

τ k+1 (z)

2

(10.145)

Lz (x, y) · fk+1|k (x, y)dxdy

(10.146)

Thus with averaging, the measurement-updated track distribution is

+

1 f k+1|k+1 (x, y) = 2

=

=

(

1

2

Lz1 (x,y)

Lz2 (x, y)

+

1

2

τ k+1 (z1 )

τ k+1 (z2 )

)

·fk+1|k (x, y) ( ) 1 Nσ2 (z1 − x) Nσ2 (z2 − y) + 2 Nσ2 +σ02 (z1 − x0 ) Nσ2 +σ02 (z2 − y0 ) ·Nσ02 (x − x0 ) · Nσ02 (y − y0 ) ( ) Nω2 (x − p0 ) · Nσ02 (y − y0 ) 1 +Nσ02 (x − x0 ) · Nω2 (y − q0 ) 2

(10.147)

(10.148)

(10.149)

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305

where ω 2 , p20 , and q02 are given by 1 1 1 = 2 + 2, ω2 σ σ0

p0 x0 z1 = 2 + 2, ω2 σ0 σ

q0 y0 z2 = 2 + 2. ω2 σ0 σ

(10.150)

+

It may be verified that the mean and variance of f k+1|k+1 (x, y) are ∫ + + µ = (x, y) · f k+1|k+1 (x, y)dxdy =

1 (x0 + p0 , y0 + q0 ) 2

(10.151) (10.152)

and ∫

=

1 1 − (x0 + p0 )2 − (y0 + q0 )2 + 4 4

=

·f k+1|k+1 (x, y)dxdy ) 1 1( 2 ω 2 + σ02 + p + q02 + y02 + x20 − (x0 + p0 )2 2 0 4 1 − (y0 + q0 )2 . 4

+

σ2

(x2 + y 2 )

(10.153)

+

(10.154)

Now let σ0 → ∞, so that the prior distribution is effectively uniform. Then ω 2 → σ 2 , p20 → z12 , and q02 → z22 , and so +

µ +2

σ

→ →

1 (x0 + z1 , y0 + z2 ) 2 σ02 → ∞.

(10.155) (10.156)

That is, averaging leads to an arbitrarily large uncertainty in the target localization. By way of contrast, the track distribution computed using Bayes’ rule is 1 ×

f k+1|k+1 (x, y)

=

2

Lz1 (x,y) · Lz2 (x, y)

· fk+1|k (x, y) τ k+1 (z1 , z2 ) Nω2 (x − p0 ) · Nω2 (y − q0 ). 12

=

(10.157) (10.158)

In this case the mean and variance can be shown to be ×

µ ×2

σ

= =

(p0 , q0 ) −→ (z1 , z2 ) 2

2ω −→ 2σ

2

(10.159) (10.160)

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Figure 10.3 (10.158).

The graph of the track distribution for the Bayes product likelihood,

where the limit is taken as σ0 → ∞. That is, Bayes’ rule leads to a triangulated target position with finite localization accuracy. These analytical conclusions have been verified in simulations—see [221] and Section 10.8. The graph of the Bayesian solution is displayed in Figure 10.3 whereas the graph of the averaged solution, (10.149), is displayed in Figure 10.4. The difference between the Bayesian and the averaged-likelihood approaches is even more pronounced when the number of sensors increases. Suppose that additional bearing-only sensors are applied, all with orientations different than the first two and different than each other. Then the variance of the averaged filter increases with the number of averaged sensors—whereas the variance greatly decreases if Bayes’ rule is used instead.

10.8

PERFORMANCE COMPARISONS

Nagappa and Clark [221] have conducted simulations comparing the multisensor PHD and CPHD filters described in this chapter. In a first set of simulations, they compared two-sensor versions of the following multisensor PHD filters:

Multisensor PHD and CPHD Filters

Figure 10.4 (10.149).

307

The graph of the track distribution for the averaged likelihood,

• Iterated-corrector multisensor classical PHD filter (Section 10.5). • PCAM-PHD filter (Section 10.6.2). • Multisensor classical PHD filter (Section 10.4). • Averaged pseudolikelihood (APL) PHD filter (Section 10.7). Both sensors had pD = 0.95 and clutter rate λ = 10. The evaluations were made using the following two versions of the optimal subpattern assignment (OSPA) metric (Section 6.2.3): • E-OSPA—the OSPA metric, with the Euclidean metric as the base metric, (6.15). This metric measures the error in both target number and target localization, but not the error in track uncertainty (that is, error in the covariance). • H-OSPA—the OSPA metric, but with the Hellinger distance as the base metric, (6.35). Unlike the E-OSPA metric, this metric does measure the error in track uncertainty. The results were as follows ([221], Figure 4): • E-OSPA: The multisensor classical PHD filter performed significantly better than the alternatives—as would be expected because it is the most accurate

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PHD filter solution. The performance of the other three was roughly comparable with each other, except that the APL-PHD filter performed somewhat worse. • H-OSPA: In this case the differences were much more pronounced. Once again, the multisensor classical PHD filter performed significantly better than the others. The PCAM-PHD filter and iterated-corrector PHD filter were roughly comparable, with the PCAM-PHD filter performing slightly better. The APL-PHD filter performed considerably worse than the others. This is to be expected, given this filter’s inherent tendency to increase rather than decrease track covariance (as described in Section 10.7). In a second set of related simulations, Nagappa and Clark compared twosensor versions of the following multisensor CPHD filters: • Iterated-corrector CPHD filter. • PCAM-CPHD filter (Section 10.6.1). The results were as follows ([221], Figure 5): • E-OSPA and H-OSPA The PCAM-CPHD filter performed significantly better than the iterated-corrector CPHD filter, especially towards the end of the scenario. In a third set of simulations, three-sensor versions of the following algorithms were compared: • Iterated-PHD filter. • PCAM-PHD filter. • Iterated CPHD filter. • PCAM-CPHD filter. • Averaged-pseudolikelihood (APL) PHD filter. In this case, two sensors had pD = 0.95 and the third pD = 0.9. The results were as follows ([221], Figure 6): • E-OSPA: In decreasing order of performance, the filters ranked as follows: PCAM-CPHD, PCAM-PHD, iterated CPHD, APL-PHD, iterated PHD. The

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poor performance of the iterated-PHD filter is attributable to its susceptibility to differing probabilities of detection (as discussed in [219] and in Section 10.5.1). Nevertheless, the iterated-CPHD filter still significantly outperformed the APL-PHD filter. • H-OSPA: In decreasing order of performance: PCAM-CPHD, PCAM-PHD, iterated CPHD, iterated PHD, APL-PHD. The performance of the APL-PHD filter was particularly bad, with the iterated PHD filter having an intermediate position. Similar results were observed when the probability of detection of the third sensor was decreased from pD = 0.9 to pD = 0.85 and again to pD = 0.7 ([221], Table V).

Chapter 11 Jump-Markov PHD/CPHD Filters 11.1

INTRODUCTION

Rapidly maneuvering targets severely challenge the capabilities of conventional Bayes tracking filters—including all of the RFS filters considered up to this point. This is because the time-update step of such filters depends on a single a priori model of target motion. For the single-sensor, single-target Bayes filter, this model consists of a single item: the Markov transition density fk+1|k (x|x′ )—which, in turn, is usually based on a statistical motion model such as Xk+1|k = φk (x′ ) + Wk . According to this model, if a target has state x′ at time tk then it must have state φk (x′ ) at time tk+1 —except for a degree of uncertainty modeled by the plant noise Wk . In the multitarget case, the motion model will include additional items: the target probability of survival, pS,k+1|k (x′ ); the target appearance RFS; and, perhaps, also a target-spawning RFS. Such models have a common limitation: only a single model of motion can be addressed at any given moment. The Markov density fk+1|k (x|x′ ) models a single type of target trajectory; pS (x′ ) models a single type of target disappearance; and so on. In single-target tracking, the most well-known algorithms for addressing maneuvering targets are the multiple motion model filters, such as the popular interacting multiple-model (IMM) filter [202]. Such filters typically employ a “library” of motion models and apply some methodology to select, on-the-fly, the motion model (or the amalgam of motion models) that best accounts for observed target motion.

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The conceptually simplest such algorithms run multiple filters in parallel (a “filter bank,” that is, one filter for each motion model), gradually winnowing out the least explanatory filters. The filter bank approach is computationally intensive. Thus more sophisticated filters have been devised that select the motion model using statistical methods. Jump-Markov models provide the most well-known theoretical foundation for this kind of filter. The basic idea is simple: append a discrete state variable o (the “mode variable” or “jump variable”) to the kinematic state x, resulting in an augmented state x ¨ = (o, x). A jump-Markov filter is just a Bayes filter defined on this augmented state. The ultimate goal of the filter is to continually select o so that the observed target motion is most accurately modeled. To address the problem of multiple and independent rapidly-maneuvering targets, it is necessary to generalize the jump-Markov approach to the multitarget realm. The optimal generalization, the jump-Markov multitarget Bayes filter, is computationally intractable in general. Thus one must devise principled approximations, for example: jump-Markov generalizations of the classical PHD and CPHD filters. That is the purpose of this chapter, which is drawn from [170]. (JumpMarkov versions of the multi-Bernoulli filter will be addressed in Section 13.5 of Chapter 13). Following the general philosophy outlined in Chapter 7, the following methodology will be employed: 1. Begin with the multitarget Bayes filter as the conceptual starting point; 2. Generalize it to a multitarget jump-Markov Bayes filter. 3. Derive PHD and CPHD filter equations from this generalized filter. This procedure ensures that the resulting PHD/CPHD filters adhere as closely as possible to the jump-Markov theoretical framework. It is to be contrasted with the following, more ad hoc, approach: 1. Begin with the classical PHD (or CPHD filter) as the conceptual starting point. 2. Try to draw an analogy between this filter and the conventional single-target jump-Markov Bayes filter. 3. On the basis of this analogy, propose filtering equations for a jump-Markov PHD or CPHD filter.

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The major drawback of this approach is that it is excessively subjective. What if someone else applies a different analogy, and thereby arrives at a different filter? How does one determine if either analogy is accurate? The end result is a potential “Tower of Babel” of jump-Markov PHD/CPHD filters, based on various heuristically-based analogies. What reason is there to believe that any of these filters are actually PHD or CPHD filters in the general sense, as defined in Sections 5.10.4 and 5.10.5? 11.1.1

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter: • The single-sensor, single-target jump-Markov Bayes filter is a conventional single-sensor, single-target Bayes filter, but defined on an augmented state space of the form ¨ = {1, ..., O} × X X (11.1) where X is the kinematic state space and o = 1, ..., O are the indices of the model modes. Thus the single-sensor, single-target jump-Markov filter propagates hybrid discrete-continuous probability distributions of the form fk|k (o, x|Z k ). See Section 11.2. • The na¨ıve generalization of this to the multitarget realm is not correct. By this is meant a Bayes filter on {1, ..., O} × X∞ , where X∞ is the class of all finite subsets of X. This approach is not correct because it rests on a hidden assumption: that all targets move in the same manner according to a single common model. See Section 11.3.1. • The correct multitarget jump-Markov filter is a multiobject Bayes filter on ¨ ∞ of all finite subsets of the single-target the multiobject state space X ¨ augmented space X. Thus it propagates multiobject probability distributions ¨ (k) ) where of the form fk|k (X|Z ¨ = {(o1 , x1 ), ..., (on , xn )}. X

(11.2)

See Section 11.3.2. • To achieve computationally attractive jump-Markov PHD and CPHD filters, one must assume that the clutter multiobject probability distribution κk+1 (Z) is independent of the mode: κk+1 (Z|o) = κk+1 (Z). See Section 11.4.1.

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• As a consequence, both the jump-Markov PHD filter and jump-Markov CPHD filter are classical PHD and CPHD filters, but defined on the aug¨ The jump-Markov PHD filter propagates mented single-target state space X. (k) PHDs of the form Dk|k (o, x|Z ); and the jump-Markov CPHD filter propagates, in addition, cardinality distributions pk|k (n|Z (k) ) where n is the total number of all targets (that is, regardless of what mode they are in). See Sections 11.4 and 11.5. • The jump-Markov PHD and CPHD filters can be implemented using both Gaussian mixture and sequential Monte Carlo (SMC) techniques. See Section 11.7. • The usual jump-Markov approach is based on an implicit assumption: that the target state space for each motion model is always the same. This assumption is often not valid in practical application. However, Chen, McDonald, and Kirubarajan have shown that the multiple-model approach can be generalized to include model-dependent state spaces. See Section 11.6. 11.1.2

Organization of the Chapter

The chapter is organized as follows: 1. Section 11.2: A review of the single-sensor, single-target jump-Markov Bayes filter. 2. Section 11.3: The general multitarget jump-Markov Bayes recursive filter. 3. Section 11.4: A jump-Markov version of the classical PHD filter. 4. Section 11.5: A jump-Markov version of the classical CPHD filter. 5. Section 11.6: Jump-Markov CPHD filter with mode dependent target-state spaces. 6. Section 11.7: Implementation of jump-Markov PHD and CPHD filters using Gaussian mixture and sequential Monte Carlo methods. 7. Section 11.8: Implemented jump-Markov PHD and CPHD filters.

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11.2

315

JUMP-MARKOV FILTERS: A REVIEW

A jump-Markov system results when we append a discrete state variable o (the “mode variable” or “jump variable”) to the single-target kinematic state x, with a finite number of possible values o = 1, ..., O. This results in an augmented state of the form x ¨ = (o, x) where x ∈ X. Note that an implicit assumption is being made: • The target-state space for every model o is the same: X. The case when this assumption is not true is addressed in Section 11.6. The integral on the augmented state space is defined by ∫

O ∫ ∑

f (o, x)dx,

(11.3)

which hereafter will be abbreviated as ∫ ∑∫ f (¨ x)d¨ x= f (o, x)dx.

(11.4)

f (¨ x)d¨ x=

o=1

o

Given this, the likelihood function and Markov transition density for the system must have the form Lz (o, x) = fk+1 (z|o, x),

fk+1|k (o, x|o′ , x′ ).

(11.5)

Because of Bayes’ rule, the Markov transition function factors as fk+1|k (o, x|o′ , x′ ) = fk+1|k (o|o′ , x′ ) · fk+1|k (x|o, o′ , x′ ).

(11.6)

It is typically assumed that the new mode is independent of the previous kinematic state, and that the new kinematic state is independent of the new mode: fk+1|k (o|o′ , x′ ) ′

′

fk+1|k (x|o, o , x )

= =

fk+1|k (o|o′ ) = χo,o′ ′

′

fk+1|k (x|o , x ).

(11.7) (11.8)

In summary, fk+1|k (o, x|o′ , x′ ) = χo,o′ · fk+1|k (x|o′ , x′ )

(11.9)

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where χo,o′ is the Markov transition matrix for the Markov chain defined by the mode variable o = 1, ..., O; and where fk+1|k (x|o′ , x′ ) is the Markov transition density (motion model) corresponding to the mode variable o′ . Also, for the purposes of maneuvering-target tracking, the likelihood function is usually assumed to be independent of the mode: fk+1 (z|o, x) = fk+1 (z|x).

(11.10)

In this case, the problem of determining the best motion model is the same as the problem of determining the value of o′ that corresponds to the most accurate motion model fk+1|k (x|o′ , x′ ). When fk+1 (z|o, x) and fk+1|k (x|o′ , x′ ) are linear-Gaussian for each choice of o, x (respectively each choice of o′ , x′ ), then the jump-Markov system is called a jump-Markov linear system (JMLS). 11.2.1

The Jump-Markov Bayes Recursive Filter

The single-sensor, single-target Bayes filter was described in Section 2.2.7. The Time evolution of a jump-Markov system is described by the Bayes recursive filter defined on the augmented state x ¨ = (o, x): ... → fk|k (o, x|Z k ) → fk+1|k (o, x|Z k ) → fk+1|k+1 (o, x|Z k+1 ) → ... where fk|k (o, x|Z k ) is the probability (density) that the target has kinematic state x and is subject to the model corresponding to the mode o. This filter is defined by the equations fk+1|k (¨ x|Z k )

=

fk+1|k+1 (¨ x|Z k+1 )

=

fk+1 (zk+1 |Z k )

=

∫

fk+1|k (¨ x|¨ x′ ) · fk|k (¨ x′ |Z k )d¨ x′

fk+1 (zk+1 |¨ x) · fk+1|k (¨ x|Z k ) fk+1 (zk+1 |Z k ) ∫ fk+1 (zk+1 |¨ x) · fk+1|k (¨ x|Z k )d¨ x

(11.11) (11.12) (11.13)

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or, equivalently, by fk+1|k (o, x|Z k )

=

∑

χo,o′

∫

fk+1|k (x|o′ , x′ )

(11.14)

o′

·fk|k (o′ , x′ |Z k )dx′ fk+1|k+1 (o, x|Z k+1 )

=

fk+1 (zk+1 |Z k )

=

fk+1 (zk+1 |o, x) · fk+1|k (o, x|Z k ) fk+1 (zk+1 |Z k ) ∫ ∑ fk+1 (zk+1 |o, x)

(11.15) (11.16)

o

·fk+1|k (o, x|Z k )dx. 11.2.2

State Estimation for Jump-Markov Filters

What is the best way of determining the state of a jump-Markov system? This question has been addressed by Boers and Driessen [26]. Among others, they considered the following state estimators: • Classical MAP estimator: (ˆ ok|k , x ˆk|k ) = arg sup fk|k (o, x|Z k ).

(11.17)

o,x

This is the most probable estimate. The target is estimated to have kinematic state x = x ˆk|k and is, simultaneously, estimated to be governed by mode o = oˆk|k . • Marginally maximal target-state corresponding to the marginally maximal mode: ∫ oˆk|k = arg max fk|k (o, x|Z k )dx (11.18) o

x ˆk|k

=

arg sup fk|k (ˆ ok|k , x|Z k ).

(11.19)

x

That is, marginalize out the kinematic state and determine the marginally most probable mode; then determine the most probable kinematic state that has this mode. • Marginally maximal target state: x ˆk|k = arg sup

∑

x o

fk|k (o, x|Z k ).

(11.20)

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This estimator is well-suited for situations in which one wishes to estimate the target state, but determination of the jump mode is not necessary. The mode is integrated out as a nuisance variable, and the target state estimate is the MAP estimator applied to the marginal target-state distribution.

11.3

MULTITARGET JUMP-MARKOV SYSTEMS

The purpose of this section is to extend the theory of jump-Markov systems to the multitarget realm. The following two issues are addressed: 1. Section 11.3.1: What is the proper definition of a multitarget jump-Markov system? 2. Section 11.3.2: The multitarget jump-Markov Bayes recursive filter. 11.3.1

What Is a Multitarget Jump-Markov System?

A certain degree of care is required for the proper definition of a multitarget jumpMarkov system. Given the definition of a jump-Markov system in Section 11.2, one might be tempted to define it as follows: • Na¨ıve definition of a multitarget jump-Markov system: Append the jump variable o to the multitarget state X, resulting in an augmented state of the form (o, X). This approach can be expected to result in deteriorated performance. The reason is that all targets in X = {x1 , ..., xn } are presumed to be moving in the same way—that is, to have the motion model corresponding to the value of o. To better understand why this is the case ([170], Section III-B), consider the simplest multitarget motion model: independent target motions and no target disappearance and target appearance. Let the state sets at times k and k + 1 be, respectively, X ′ = {x′1 , ..., x′n }, X = {x1 , ..., xn } (11.21) with |X ′ | = n′ and |X| = n. Also, by (7.71), the multitarget Markov density must have the form

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=

319

fk+1 (o, X|o′ , X ′ ) (11.22) ∑ ′ ′ ′ ′ δn,n′ · χo,o′ fk+1|k (xπ1 |o , x1 ) · · · fk+1|k (xπn |o , xn ) π

where the summation is taken over all permutations π on 1, ..., n. Suppose now that the optimal mode has been determined to be oˆk|k . Then we are implicitly stating that the optimally inferred motion-mode of each and every target is the motion model corresponding to oˆk|k . As a more concrete example, suppose that oˆk|k is the mode value corresponding to a right-hand turn. Then a right-hand turn is being presumed to optimally describe the motion of all of the targets in the scene. Clearly, this is problematic. It is also clear that the summation in the right side of (11.22) should actually have the form ∑ δn,n′ χoπ1 ,o′1 · fk+1|k (xπ1 |o′1 , x′1 ) · · · χoπn ,o′n · fk+1|k (xπn |o′n , x′n ). (11.23) π

This would allow each target to be governed by its own motion model. A better modeling approach permits this. The jump variable o should be appended not to the multitarget state X, but (as with single-target jump-Markov systems) to the single-target state, resulting in single-target states of the form x ¨ = (o, x). Given this, a multitarget state must be a finite set of such augmented states: not (o, X) but, rather, ¨ = {¨ X x1 , ..., x ¨n } = {(o1 , x1 ), ..., (on , xn )}.

(11.24)

That is, target x1 is subject to mode o1 , target x2 is subject to mode o2 , and so on. In this case, (7.71) gives us

=

¨ X ¨ ′) fk+1 (X| ∑ δn,n′ fk+1|k (¨ xπ1 |¨ x′1 ) · · · fk+1|k (¨ xπn |¨ x′n )

(11.25)

π

=

δn,n′

∑

χoπ1 ,o′1 · fk+1|k (xπ1 |o′1 , x′1 )

π

· · · χoπn ,o′n · fk+1|k (xπn |o′n , x′n ), which has the right form. The following fact is an immediate consequence of this reasoning:

(11.26)

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• The following statistics must be mode-independent: the p.g.fl. Gk|k [g], the p.g.f. Gk|k (x), and the cardinality distribution pk|k (n). For, suppose that they had the form Gk|k [g|o], Gk|k (x|o), pk|k (n|o). Then it would follow that a single value of the mode o is being imposed on all targets simultaneously. 11.3.2

The Multitarget Jump-Markov Filter

Given the discussion in the previous section, the Time evolution of the multitarget jump-Markov system is described by the Bayes filter ¨ (k) ) → fk+1|k (X|Z ¨ (k) ) → fk+1|k+1 (X|Z ¨ (k+1) ) → ... ... → fk|k (X|Z where, by (5.1) through (5.3), ¨ (k) ) fk+1|k (X|Z

=

¨ (k+1) ) fk+1|k+1 (X|Z

=

fk+1 (Zk+1 |Z (k) )

=

∫

¨ X ¨ ′ ) · fk|k (X ¨ ′ |Z (k) )δ X ¨′ fk+1|k (X|

(11.27)

¨ · fk+1|k (X|Z ¨ (k) ) fk+1 (Zk+1 |X) (11.28) (k) fk+1 (Zk+1 |Z ) ∫ ¨ · fk+1|k (X|Z ¨ (k) )δ X.(11.29) ¨ fk+1 (Zk+1 |X)

Here, the integrals are set integrals, but they must be defined in terms of the integral on the augmented single-target state space, (11.4): ∫ ¨ X ¨ f (X)δ (11.30) ∫ ∑ 1 = f ({¨ x1 , ..., x ¨n })d¨ x1 · · · d¨ xn n! n≥0 ∑ 1 ∑ ∫ = f ({(o1 , x1 ), ..., (on , xn )})dx1 · · · dxn . (11.31) n! o ,...,o n≥0

11.4

1

n

JUMP-MARKOV PHD FILTER

The purpose of this section is to apply this top-down perspective to the derivation of a jump-Markov PHD filter. This filter was first proposed by Pasha, Vo, Tuan, and

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Ma [234], [232], [233], [297]. In their approach, the jump-Markov PHD filter is the classical PHD filter, but defined on augmented states x ¨ = (o, x). Thus its time- and measurement-update equations can be obtained by substituting (o, x) whenever x occurs in the classical PHD filter equations in Sections 8.4.1 and 8.4.3. The following topics are considered: 1. Section 11.4.1: Modeling assumptions for the jump-Markov PHD filter. 2. Section 11.4.2: Time update equations for the jump-Markov PHD filter. 3. Section 11.4.3: Measurement update equations for the jump-Markov PHD filter. 4. Section 11.4.4: Multitarget state estimation for the jump-Markov PHD filter. 11.4.1

Jump-Markov PHD Filter: Models

The jump-Markov PHD filter is based on the following motion and measurement models: • Jump-Markov state transition density: fk+1|k (o, x|o′ , x′ ) = χo,o′ · fk+1|k (x|o′ , x′ )

(11.32)

where fk+1|k (x|o′ , x′ ) is the Markov transition density corresponding to mode o′ ; and where χo,o′ is the transition matrix for the mode. • Mode-dependent target probability of survival: pS (o′ , x′ ) abbr. = pS,k+1|k (o′ , x′ ). • Mode-dependent PHD for target appearances: bk+1|k (o, x). • Mode-dependent PHD for target spawning: bk+1|k (o, x|o′ , x′ ). • Mode-dependent likelihood function: Lz (o, x) abbr. = fk+1 (z|o, x). • Mode-dependent probability of detection: pD (o, x) abbr. = pD,k+1 (o, x). We must also consider the Poisson clutter process. It turns out that, if we are to achieve a computationally reasonable PHD filter, it cannot be mode dependent. For, assume to the contrary that the clutter intensity function is mode dependent: κok+1 (z) = λok+1 · cok+1 (z).

(11.33)

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Then expressed in more complete notation, clutter will be dependent upon the single-target state variable x ¨ = (o, x): κk+1 (z|o, x) = κok+1 (z). State-dependent Poisson clutter models for PHD filters were considered in [179], pp. 424-426. The PHD filter equations corresponding to this model were given in Section 8.7. There it was shown that the measurement-update equation for this filter involves a combinatorial sum over all partitions of the current measurement set. Consequently, if we want a jump-Markov PHD filter that is both theoretically rigorous and computationally practical, the Poisson clutter RFS cannot be allowed to depend on the jump variable o. Thus we have one more model: • Mode-independent clutter intensity function: κk+1 (z) = λk+1 · ck+1 (z)

(11.34)

where, as usual, λk+1 is the clutter rate and ck+1 (z) is the clutter spatial distribution. These considerations have the following consequence: the jump Markov PHD filter is just an ordinary PHD filter, but defined on augmented states x ¨ = (o, x) rather than purely kinematic states x. Thus its time- and measurement-update equations can be obtained by substituting (o, x) whenever x occurs in the conventional PHD filter equations in Sections 8.2.2 and 8.4.3. 11.4.2

Jump-Markov PHD Filter: Time Update

The time-update equations for the jump-Markov PHD filter are: Dk+1|k (o, x) = bk+1|k (o, x)+

∑∫

Fk+1|k (o, x|o′ , x′ )·Dk|k (o′ , x′ )dx′ (11.35)

o′

where the PHD filter pseudo-Markov density is Fk+1|k (o, x|o′ , x′ ) = bk+1|k (o, x|o′ , x′ ) + χo,o′ · pS (o′ , x′ ) · fk+1|k (x|o′ , x′ ). (11.36)

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The expected number of predicted targets is, therefore,

=

(11.37)

Nk+1|k ∑∫ Dk+1|k (o, x)dx o

=

B Nk+1|k

+

∑∫ (

B Nk+1|k (x′ ) + pS (o′ , x′ )

)

(11.38)

o′ ′

·Dk|k (o′ , x )dx′ where B Nk+1|k =

∑∫

bk+1|k (o, x)dx

(11.39)

o

is the expected number of appearing targets and where ∑∫ B ′ ′ Nk+1|k (o , x ) = bk+1|k (o, x|o′ , x′ )dx

(11.40)

o

is the expected number of targets spawned by a target with joint state (o′ , x′ ). 11.4.3

Jump–Markov PHD Filter: Measurement Update

The measurement-update equations for the jump-Markov PHD filter are: ∑ pD (o, x) · Lz (o, x) Dk+1|k+1 (o, x) = 1 − pD (o, x) + Dk+1|k (o, x) κk+1 (z) + τk+1 (z)

(11.41)

z∈Zk+1

where τk+1 (z) =

∑∫

pD (o, x) · Lz (o, x) · Dk+1|k (o, x)dx.

(11.42)

o

The expected number of updated target tracks is ∑

τk+1 (z) κk+1 (z) + τk+1 (z)

(11.43)

(1 − pD (o, x)) · Dk+1|k (o, x)dx.

(11.44)

Nk+1|k+1 = Dk+1|k [1 − pD ] +

z∈Zk+1

where Dk+1|k [1 − pD ] =

∑∫ o

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Remark 52 (Computational complexity) The measurement-update step consists of conventional PHD filter measurement updates for each PHD Dk+1|k+1 (o, x), for o = 1, ..., O. Thus the computational complexity is O(mnO) where m is the current number of measurements and n is the current number of tracks. 11.4.4

Jump-Markov PHD Filter: State Estimation

State-estimation for the jump-Markov PHD filter combines single-target jumpMarkov state estimation, as described in Section 11.2.2, with PHD filter state estimation, as considered in Section 8.4.4. The purpose of state estimation for jump-Markov systems is, usually, to more accurately estimate target states using multiple motion models. It is usually not necessary to determine which model is applicable to which target at any given time. Therefore, the jump variable o can be integrated out as a nuisance variable to get the PHD for targets alone:

=

(11.45)

Dk+1|k+1 (x) ∑ Dk+1|k+1 (o, x) o

=

∑ o



1 − pD (o, x) +

∑ z∈Zk+1

 pD (o, x) · Lz (o, x)  κk+1 (zj ) + τk+1 (z)

(11.46)

·Dk+1|k (o, x). Given this, one can employ the usual PHD estimation process. That is, determine the expected number of targets, Nk+1|k+1 as in (11.43). Then round Nk+1|k+1 off to the nearest integer ν, and find states x1 , ..., xν corresponding to the ν largest suprema of Dk+1|k+1 (x).

11.5

JUMP-MARKOV CPHD FILTER

In this section, the top-down perspective of Section 11.3 is applied to the derivation of a jump-Markov CPHD filter, as proposed in [170]. The following topics are considered: 1. Section 11.5.1: Modeling assumptions for the jump-Markov CPHD filter. 2. Section 11.5.2: Time update equations for the jump-Markov CPHD filter.

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3. Section 11.5.3: Measurement update equations for the jump-Markov CPHD filter. 4. Section 11.5.4: Multitarget state estimation for the jump-Markov CPHD filter. 11.5.1

Jump-Markov CPHD Filter: Modeling

The motion and measurement models for the jump-Markov CPHD filter are the same as those for the jump-Markov PHD filter (Section 11.4.1), with the following exceptions: • As with the classical CPHD filter, there is no target-spawning model. • Mode-independent cardinality distribution for target appearance: pB k+1|k (n), where it must be the case that the two different ways of computing the targetbirth rate are the same: ∑ ∑∫ B Nk+1|k = n · pB (n) = bk+1|k (o, x)dx. (11.47) k+1|k o

n≥0

• Mode-independent cardinality distribution for clutter: pκk+1 (m) and modeindependent clutter intensity function κk+1 (z), where it must be the case that the two different ways of computing the clutter rate are the same: λk+1 =

∑

m · pκk+1 (m) =

∫

κk+1 (z)dz.

(11.48)

m≥0

As was explained in Section 11.4.1, the clutter RFS cannot be mode dependent if we are to achieve a computationally tractable filter. Thus (11.48) must be independent of the mode variable o. 11.5.2

Jump-Markov CPHD Filter: Time Update

As was the case with the jump-Markov PHD filter, the jump Markov CPHD filter is just an ordinary CPHD filter, but defined on augmented states x ¨ = (o, x). Thus its time- and measurement-update equations can be obtained by substituting (o, x) whenever x occurs in the classical CPHD filter equations in Sections 8.5.2 and 8.5.4.

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We are given a spatial distribution sk|k (o, x) and a cardinality distribution pk|k (n). We are to determine the predicted spatial distribution sk+1|k (o, x), the predicted expected number of targets Nk+1|k , and either the predicted cardinality distribution pk+1|k (n) or the predicted p.g.f. Gk+1|k (x). The predicted spatial distribution is given by ∫ ( ) ∑ bk+1|k (o, x) + Nk|k o′ χo,o′ pS (o′ , x′ ) ·fk+1|k (x|o′ , x′ ) · sk|k (o′ , x′ )dx′ sk+1|k (o, x) = (11.49) Bk+1|k + Nk|k · ψk or, alternatively, the predicted PHD is given by Dk+1|k (o, x)

=

bk+1|k (o, x) +

∑

χo,o′

∫

pS (o′ , x′ )

(11.50)

o′

·fk+1|k (x|o′ , x′ ) · Dk|k (o′ , x′ )dx′ where ψk = sk|k [pS ] =

∑∫

pS (o, x) · sk|k (o, x)dx.

(11.51)

o

The predicted cardinality distribution and its corresponding p.g.f. are given by Gk+1|k (x)

=

pk+1|k (n)

=

GB k+1 (x) · Gk|k (1 − ψk + ψk · x) ∑ pk+1|k (n|n′ ) · pk|k (n′ )

(11.52) (11.53)

n′ ≥0

where pk+1|k (n|n′ ) =

n ∑

′

i n −i pB . k+1|k (n − i) · Cn′ ,i · ψk (1 − ψk )

(11.54)

i=0

The predicted expected number of targets is B Nk+1|k = Nk+1|k + Nk|k · ψk .

11.5.3

(11.55)

Jump-Markov CPHD Filter: Measurement Update

We are given a predicted spatial distribution sk+1|k (o, x) and a predicted cardinality distribution pk+1|k (n), such that the two different ways of computing the

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expected number of targets are the same: Nk+1|k =

∑∫

sk+1|k (o, x)dx =

o

∑

(11.56)

n · pk+1|k (n).

n≥0

We are to determine the measurement-updated spatial distribution sk+1|k+1 (o, x), the measurement-updated expected number of targets Nk+1|k+1 , and either the measurement-updated cardinality distribution pk+1|k+1 (n) or the measurementupdated p.g.f. Gk+1|k+1 (x). The spatial distribution is given by (11.57)

sk+1|k+1 (o, x) = LZk+1 (o, x) · sk+1|k (o, x) where the CPHD filter pseudolikelihood function is 

1 LZk+1 (o, x) = Nk+1|k+1



ND

∑

+

(1 − pD (o, x)) · L Zk+1 z∈Zk+1

pD (o,x)·Lz (o,x) ck+1 (z)

D

· LZk+1 (z)



 (11.58)

where ( ∑m

) − j)! · pκk+1 (m − j) (j+1) ·σj (Zk+1 ) · Gk+1|k (ϕk ) ( ∑m ) (11.59) κ l=0 (m − l)! · pk+1 (m − l) (l) ·σl (Zk+1 ) · Gk+1|k (ϕk ) ( ∑m−1 ) κ (m − i − 1)! · p (m − i − 1) k+1 i=0 (i+1) ·σi (Zk+1 − {zj }) · Gk+1|k (ϕk ) ( ∑m ) (11.60) κ l=0 (m − l)! · pk+1 (m − l) (l) ·σl (Zk+1 ) · Gk+1|k (ϕk ) ∑ pk+1|k (n) · l! · Cn,l · ϕn−l (11.61) k j=0 (m

ND

L Zk+1

=

LZk+1 (zj )

=

D

(l)

Gk+1|k (ϕk )

=

n≥l (j+1) Gk+1|k (ϕk )

=

∑ n≥j+1

pk+1|k (n) · (j + 1)! · Cn,j+1 · ϕn−j−1 k

(11.62)

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and where ϕk

= =

sk+1|k [1 − pD ] ∑∫ (1 − pD (o, x)) · sk+1|k (o, x)dx

(11.63) (11.64)

o

σi (Zk+1 )

=

σi (Zk+1 − {zj })

=

(

) τˆk+1 (z1 ) τˆk+1 (zm ) σm,i , ..., ck+1 (z1 ) ck+1 (zm )   τˆ? τˆk+1 (z1 ) k+1 (zj ) , ..., ck+1 (zj )  σm−1,i  ck+1 (z1 ) τˆ (z . m) k+1 , ..., ck+1 (zm )

(11.65)

(11.66)

Also, the measurement-updated number of targets is ND

Nk+1|k+1 = ϕk · L Zk+1 +

∑ τˆk+1 (z) D · LZk+1 (z) ck+1 (z)

(11.67)

z∈Zk+1

where τˆk+1 (z) =

∑∫

pD (o, x) · Lz (o, x) · sk+1|k (o, x)dx.

(11.68)

o

The measurement-updated cardinality distribution and p.g.f. are, respectively, pk+1|k+1 (n)

=

Gk+1|k+1 (x)

=

ℓZ (n) · pk+1|k (n) ∑ k+1 l≥0 ℓZk+1 (l) · pk+1|k (l) ( ∑m j ) κ j=0 x · (m − j)! · pk+1 (m − j) ·G(j) (x · ϕk ) · σj (Zk+1 ) ( ∑m ) κ i=0 (m − i)! · pk+1 (m − i) ·G(i) (ϕk ) · σi (Zk+1 )

where the cardinality pseudolikelihood function is ( ∑ ) min{m,n} κ (m − j)! · p (m − j) · j! · C n,j k+1 j=0 ·ϕn−j · σj (Zk+1 ) k ) . ℓZk+1 (n) = ( ∑m κ l=0 (m − l)! · pk+1 (m − l) · σl (Zk+1 ) (l) ·Gk+1|k (ϕk )

(11.69)

(11.70)

(11.71)

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Remark 53 (Computational complexity) The measurement-update step requires a conventional CPHD filter measurement-update for each of the spatial distributions sk+1|k+1 (o, x), for o = 1, ..., O. Thus the computational complexity is O(m3 nO) where m is the current number of measurements and n is the current number of tracks and O is the number of jump modes. 11.5.4

Jump-Markov CPHD Filter: State Estimation

State estimation for the jump-Markov CPHD filter is the same as for the jumpMarkov PHD filter (Section 8.4.4), except that the target-number estimate is the MAP estimate of the cardinality distribution.

11.6

VARIABLE STATE SPACE JUMP-MARKOV CPHD FILTERS

The jump-Markov CPHD filter just described is based on an implicit assumption: the target state space for every motion model must be the same. However, this will not necessarily be adequate for many practical applications. The following is a simple example of two common single-target motion models with different singletarget state spaces: 1. Constant-velocity (CV) motion model: During the time-interval ∆t = tk+1 − tk , the target is assumed to move along a straight line path from its current position p = (x, y)T , with its current velocity v = (vx , vy )T . The state space consists of all vectors (x, y, vx , vy )T ∈ R4 . The state transition function is defined as      x 1 0 ∆t 0 x  y   0 1 0 ∆t   y     . φk+1|k  (11.72)  vx  =  0 0 1 0   vx  vy 0 0 0 1 vy 2. Coordinated-turn (CT) motion model: The target is assumed to move from its current position p = (x, y)T to the left or the right along a circular arc with unknown angular turn rate ω (radians per second). The state space

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consists of all vectors (ω, x, y, vx , vy )T ∈ R5 . The state transition model is     ω ω  x   x + vx · sin ω∆t − vy · 1−cos ω∆t  ω ω     1−cos ω∆t   φk+1|k  (11.73) + vy · sin ωω∆t   y  =  y + vx · . ω  vx    vx · cos ω∆t − vy · sin ω∆t vy vx · sin ω∆t + vy · cos ω∆t Chen Xin, McDonald, and Kirubarajan have devised a means of addressing such problems [36]. The purpose of this section is to describe their approach. The basic idea is as follows. Let o = 1, ..., O be the model modes and let o X be the target state space corresponding to the mode o. Chen Xin et al. define the multimode state space1 1 O ¨ = X ⊎ ... ⊎ X, X (11.74) where as usual ‘⊎’ denotes disjoint union. They then construct a conventional ¨ can have any of O CPHD filter on this state space.2 An element x ¨ of X o o o possible forms: x ¨ =x for x ∈ X and o = 1, ..., O. Thus the integral of a ¨ has the form function f¨(¨ x) on X ∫ ∫ ∫ ∑∫ o o 1 1 O O ¨(x)d ¨(x)d f¨(¨ x)d¨ x= f x = f x + ... + f¨(x)dx. (11.75) o 1 O o

X

X

X

Chen et al.’s approach raises the following issues, which will be addressed in the subsections that follow: ¨ probability of • Modeling: the following functions must be defined on X: target survival p¨S (¨ x); Markov transition density f¨k+1|k (¨ x|¨ x′ ); target appearance PHD ¨bk+1|k (¨ x); probability of detection p¨D (¨ x); and likelihood ¨ z (¨ function L x) = f¨k+1 (z|¨ x). • Multimode PHDs: There will be a PHD o

o o (k) ¨ k|k (x|Z D k|k (x|Z (k) ) def. =D )

(11.76)

o

defined on a different state space X for each choice o = 1, ..., O. This PHD describes the density of the targets that are in mode o at time tk . 1 2

¨ is being used differently than earlier in the chapter, where previously Note that here the symbol X ¨ = {1, ..., O} × X. The intended meaning in this section will be clear from context. X This approach is related to that used for the “clutter agnostic” CPHD filters of Chapter 18—which also involves a disjoint union of state spaces.

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• Cardinality estimation: The CPHD filter cardinality distribution has the form 1 O o p¨k|k (¨ n|Z (k) ) where n ¨ = n + ... + n, where n is the number of targets that are in mode o, for o = 1, ..., O. Thus the MAP estimate n ¨ k|k = arg sup p¨k|k (¨ n|Z (k) )

(11.77)

n ¨ o

provides an estimate of n ¨ but not of the individual n. Consequently, it is necessary to determine a formula for the following item: the cardinality o o distribution pk|k (n|Z (k) ) for the number of targets that are in mode o. Given this, one could then take o

o

o

nk|k = arg sup pk|k (n|Z (k) )

(11.78)

o

n

to be the MAP estimate of the number of targets in state o. This issue is addressed in Section 11.6.4. The section is organized as follows: 1. Section 11.6.1: Modeling for the variable state–space CPHD filter. 2. Section 11.6.2: The time-update equations for the variable state space CPHD filter. 3. Section 11.6.3: The measurement-update equations for the variable state space CPHD filter. 4. Section 11.6.4: Multitarget state estimation for the variable state space CPHD filter. 11.6.1

Variable State Space CPHD Filters: Modeling

The following models are assumed: • Probability of target survival: o

o

o

o

o

p¨S (x) def. = pS (x) abbr. = pS,k+1|k (x) o

(11.79)

o

where pS (x) is the probability of target survival for a target with mode o o and state x.

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• Markov transition density: o,o′

o o′ f¨k+1|k (x| x ′ ) def. = χo,o′ · f

o o′ ′

k+1|k (x| x

)

(11.80)

where χo,o′ is the probability that a target in mode o′ will transition to o,o′

mode o; and where f

o o′ ′

k+1|k (x| x o′ ′

′

) is the probability (density) that a target o

that had mode o and state x at time tk will have the state x transitions from mode o′ to mode o.

if it

• i.i.d.c. target appearance process: the PHD of the target appearance process is o o o def. ¨bk+1|k (x) = bk+1|k (x) (11.81) o o

where bk+1|k (x) is the PHD of the target appearance process for mode o; where ∑ 1 1 OB O p¨B n) def. = pB (11.82) k+1|k (¨ k+1|k (n) · · · p k+1|k (n); 1

O

n+...+ n=¨ n o

o

o

and where pB k+1|k (n) appear.

is the probability that n targets in mode o will

• Probability of detection: o

o

o

o

o

p¨D (x) def. = pD (x) abbr. = pD,k+1 (x)

(11.83)

o

o

where pD (x) is the probability of target detection for a target with mode o o and state x. • Likelihood function: o

o

o o o f¨k+1 (z|x) def. = Lz (x) abbr. = f k+1 (z|x)

(11.84)

o o

where f k+1 (z|x) is the probability (density) that a target with mode o o and state x will generate measurement z, given that the target has been detected. • i.i.d.c. clutter process: As with a conventional CPHD filter, the spatial distribution of the clutter is ck+1 (z) and the probability that m clutter measurements will be generated is pκk+1 (m).

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Given these preliminaries, the variable state space CPHD filter consists of O + 1 tightly coupled filters: ... →

p¨k|k (¨ n|Z (k) )

→

p¨k+1|k (¨ n|Z (k) )

... →

D k|k (x|Z (k) )

→

D k+1|k (x|Z (k) )

.. .

.. .

.. .

.. .

1

1

1

→ ↑↓

p¨k+1|k+1 (¨ n|Z (k+1) )

→ ↑↓ .. .

D k+1|k+1 (x|Z (k+1) )

1

1

→ ...

1

→ ...

.. .

↑↓ O

... →

O

O

D k|k (x|Z (k) )

→

O

O

D k+1|k (x|Z (k) )

→

O

D k+1|k+1 (x|Z (k+1) )

→ ...

Here, the top row is a filter on the cardinality distribution p¨k|k (¨ n|Z (k) ) on the total o

o

number n ¨ of targets. The other rows are filters for the PHDs D k|k (x|Z (k) ) of o those targets that are in mode o and have state-variable x. 11.6.2

Variable State Space CPHD Filters: Time Update

From Section 8.5.2 we know that the time-update equations for the CPHD filter ¨ are (suppressing dependence on the measurementdefined on the state space X (k) history Z ): p¨k+1|k (¨ n)

=

∑

p¨k+1|k (¨ n|¨ n′ ) · p¨k|k (¨ n′ )

n ¨ ′ ≥0

¨ k+1|k (¨ D x)

= ¨bk+1|k (¨ x) +

∫

¨ k|k (¨ ·D x′ )d¨ x′

p¨S (¨ x′ ) · f¨k+1|k (¨ x|¨ x′ )

(11.85)

(11.86)

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where ψ¨k

=

p¨k+1|k (¨ n|¨ n′ )

=

∫

(11.87)

p¨S (¨ x) · s¨k|k (¨ x)d¨ x

n ¨ ∑

′ p¨B n − i) · Cn¨ ′ ,i · ψ¨ki (1 − ψ¨k )n¨ −i k+1|k (¨

(11.88)

i=0

s¨k|k (¨ x)

=

¨k|k N

=

¨ k|k (¨ D x) ¨k|k N ∫ ∑ ¨ k|k (¨ n ¨ · p¨k|k (¨ n) = D x)d¨ x.

(11.89) (11.90)

n ¨ ≥0

Abbreviate o

o o (k) ¨ k|k (x|Z D k|k (x) abbr. = D ) o

o

(11.91)

o

¨ k+1|k (x|Z (k) ) D k+1|k (x) abbr. = D

(11.92)

to be the PHDs for targets that are in mode o. Then given the modeling assumptions in Section 11.6.1, the time-update equations are: ∑ p¨k+1|k (¨ n) = p¨k+1|k (¨ n|¨ n′ ) · p¨k|k (¨ n′ ) (11.93) n ¨ ′ ≥0 o

o o

o

D k+1|k (x)

=

bk+1|k (x) +

∑

χ

o,o′

∫

o′

o′

o,o′

p S (x′ ) · f

o o′ ′

k+1|k (x| x

) (11.94)

o′ o

o′

o′

·D k|k ( x ′ )d x ′ where ψ¨k

=

∫ 1 ∑ o o o o o pS (x) · D k|k (x)dx ¨k|k N

(11.95)

n ¨ ∑

(11.96)

o

p¨k+1|k (¨ n|¨ n′ )

=

′ p¨B n − i) · Cn¨ ′ ,i · ψ¨ki (1 − ψ¨k )n¨ −i k+1|k (¨

i=0 1

¨k|k N

=

o

N k|k

=

O

N k|k + ... + N k|k ∫ o o o N k|k (x)dx.

(11.97) (11.98)

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11.6.3

335

Variable State Space CPHD Filters: Measurement Update

From Section 8.5.4 we know that the measurement-update equations for the CPHD ¨ are filter defined on the state space X

p¨k+1|k+1 (¨ n)

=

¨ k+1|k+1 (¨ D x)

=

ℓ¨Zk+1 (¨ n) · p¨k+1|k (¨ n) ∑ ¨ ¨k+1|k (l) l≥0 ℓZk+1 (l) · p ¨ Z (¨ ¨ k+1|k (¨ L x) · D x) k+1

(11.99) (11.100)

where ℓ¨Zk+1 (¨ n)

= ( ∑ min{m,¨ n}

(m − j)! · pκk+1 (m − j) · j! · Cn,j ·ϕ¨n−j ·σ ¨j (Zk+1 ) k ( ∑m ) κ ¨l (Zk+1 ) l=0 (m − l)! · pk+1 (m − l) · σ ¨ (l) (ϕ¨k ) ·G k+1|k   ND (1 − p¨D (¨ x)) · L Zk+1 1  ∑  (11.102) ¨ z (¨ x)·L x) D m p¨D (¨ ¨k+1|k N + j=1 ck+1 (zjj ) · LZk+1 (zj ) j=0

¨ Z (¨ L x) k+1

=

(11.101) )

and where ( ∑m

) − j)! · pκk+1 (m − j) ¨ (j+1) (ϕ¨k ) ·¨ σj (Zk+1 ) · G k+1|k ( ∑m ) (11.103) κ (m − l)! · p k+1 (m − l) l=0 ¨ (l) (ϕ¨k ) ·¨ σl (Zk+1 ) · G k+1|k ( ∑m−1 ) κ i=0 (m − i − 1)! · pk+1 (m − i − 1) ¨ (i+1) (ϕ¨k ) ·¨ σi (Zk+1 − {zj }) · G k+1|k ( ∑m ) (11.104) κ l=0 (m − l)! · pk+1 (m − l) ¨ (l) (ϕ¨k ) ·¨ σl (Zk+1 ) · G k+1|k j=0 (m

ND

L Zk+1

=

LZk+1 (zj )

=

D

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¨ (l) (ϕ¨k ) G k+1|k

=

∑

p¨k+1|k (¨ n) · l! · Cn¨ ,l · ϕ¨nk¨ −l

(11.105)

n ¨ ≥l

¨ (j+1) (ϕ¨k ) G k+1|k

∑

=

p¨k+1|k (¨ n) · (j + 1)! · Cn¨ ,j+1 · ϕ¨nk¨ −j−1 (11.106)

n ¨ ≥j+1

and where ∫

1

ϕ¨k

=

σ ¨i (Zk+1 )

=

σ ¨i (Zk+1 − {zj })

=

τ¨k+1 (z)

=

¨ k+1|k (¨ (1 − p¨D (¨ x)) · D x)d¨ x (11.107) ¨k+|k N ( ) τ¨k+1 (z1 ) τ¨k+1 (zm ) σm,i , ..., (11.108) ck+1 (z1 ) ck+1 (zm )   τ¨? τ¨k+1 (z1 ) k+1 (zj ) , ..., ck+1 (zj )  σm−1,i  ck+1 (z1 ) τ¨ (z (11.109) m) k+1 , ..., ck+1 (zm ) ∫ 1 ¨ z (¨ ¨ k+1|k (¨ p¨D (¨ x) · L x) · D x)d¨ x.(11.110) ¨k+|k N

Also, ND

¨k+1|k+1 = ϕ¨k · L Z N + k+1

∑ τ¨k+1 (z) D · LZk+1 (z). ck+1 (z)

(11.111)

z∈Zk+1

Given the modeling assumptions in Section 11.6.1, these equations take the form p¨k+1|k+1 (¨ n) o

=

ℓ¨Zk+1 (¨ n) · p¨k+1|k (¨ n) ∑ ¨ ¨k+1|k (l) l≥0 ℓZk+1 (l) · p

=

LZk+1 (x) · D k+1|k (x)

o

o

D k+1|k+1 (x)

o

o

(11.112)

o

(11.113)

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where ℓ¨Zk+1 (¨ n)

(11.114) )

= ( ∑ min{m,¨ n}

(m − j)! · pκk+1 (m − j) · j! · Cn¨ ,j ·ϕ¨kn−j · σ ¨j (Zk+1 ) ( ∑m ) κ ¨l (Zk+1 ) l=0 (m − l)! · pk+1 (m − l) · σ ¨ (l) (ϕ¨k ) ·G k+1|k   ND o o (1 − pD (x)) · L Zk+1 1   (11.115) o o o ∑m po (x)· Lz (x) D ¨k+1|k N + j=1 Dck+1 (zjj ) · LZk+1 (zj ) j=0

o

o

LZk+1 (x)

=

and where 1

¨k+1|k N

=

O

(11.116)

N k+1|k + ... + N k+1|k ND

(11.117)

L Zk+1 ∑m

j=0 (m

=

∑m

l=0 (m

− j)! ·

pκk+1 (m

−

− l)! · pκk+1 (m −

¨ (j+1) (ϕ¨k ) j) · σ ¨j (Zk+1 ) · G k+1|k (l) ¨ l) · σ ¨l (Zk+1 ) · Gk+1|k (ϕ¨k )

( ∑m−1

κ i=0 (m − i − 1)! · pk+1 (m − i − 1) ... ¨ (i+1) (ϕ¨k ) · σ i (Zk+1 − {zj }) · G k+1|k ( ∑m ) κ l=0 (m − l)! · pk+1 (m − l) ¨ (l) (ϕ¨k ) ·¨ σl (Zk+1 ) · G k+1|k

D

LZk+1 (zj )

=

¨ (l) (ϕ¨k ) G k+1|k

=

∑

p¨k+1|k (¨ n) · l! · Cn¨ ,l · ϕ¨nk¨ −l

) (11.118)

(11.119)

n ¨ ≥l

¨ (j+1) (ϕ¨k ) G k+1|k

=

∑ n ¨ ≥j+1

p¨k+1|k (¨ n) · (j + 1)! · Cn¨ ,j+1 · ϕ¨nk¨ −j−1 (11.120)

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and where ϕ¨k

(11.121)

= ∑∫

σ ¨i (Zk+1 )

=

σ ¨i (Zk+1 − {zj })

=

1 o o o o (1 − pD (x)) · D k+1|k (x)dx ¨k+1|k N o ( ) τ¨k+1 (zm ) τ¨k+1 (z1 ) σm,i , ..., (11.122) ck+1 (z1 ) ck+1 (zm )   τ¨? τ¨k+1 (z1 ) k+1 (zj ) , ..., ck+1 (zj )  σm−1,i  ck+1 (z1 ) τ¨ (z (11.123) m) k+1 , ..., ck+1 (zm )

1 τ¨k+1 (z)

= ¨k+1|k N o

o

∑∫

o

o

o

o

(11.124)

pD (x) · Lz (x)

o o

o

·D k+1|k (x)dx. Also, for the next time-update step we need to know ND

¨k+1|k+1 = ϕ¨k · L Z N + k+1

∑ τ¨k+1 (z) D · LZk+1 (z). ck+1 (z)

(11.125)

z∈Zk+1

11.6.4

Variable State Space CPHD Filters: State Estimation

State estimation for the variable state space CPHD filter is the same as for the conventional CPHD filter (see Section 8.5.5). First, estimate the total number of targets (irrespective of the mode that they are in) using a MAP estimator: ν = arg sup p¨k|k (¨ n|Z (k) ).

(11.126)

n ¨ o1

oν

Given this, the states of the targets can be estimated by finding the states x 1 , ..., x ν ¨ k|k (¨ corresponding to the ν largest peaks of D x|Z (k) ). To accomplish this, one must determine the peaks of the mode-specific PHDs 1

1

O

O

D k|k (x|Z (k) ), ..., D k|k (x|Z (k) ) and then find the ν largest of those. This requires that we first estimate the number of targets that are in any given mode o.

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Towards this end, Chen, McDonald, and Kirubarajan have shown that it is possible to estimate the number of targets that are in a given mode o. Let ¨ k|k (x) be p¨k|k (¨ n|Z (k) ) be the cardinality distribution at time tk and let G its p.g.f. Let ∫ o

o

o

o

D k|k (x|Z (k) )dx

N k|k =

(11.127)

be the expected number of targets that are in mode o, and let o

N k|k

o

r k|k =

(11.128) 1

O

N k|k + ... + N k|k be the fraction of targets that are in mode o. Then the cardinality distribution for those targets that are in mode o is: oo

o

o

pk|k (n|Z

(k)

r nk|k )=

o

n!

o

(n)

o

¨ (1 − r k|k ). ·G k|k

(11.129)

Given this, the MAP estimate of the number of targets that have mode o is o

o

o

nk|k = arg sup pk|k (n|Z (k) ).

(11.130)

o

n

¨ k|k be the random joint state set for To see why these equations are true, let Ξ ¨ k|k all targets, irrespective of mode. Then the (random) number of elements of Ξ that are in mode o is o ¨ k|k ∩ X|. |Ξ (11.131) o

¨ k|k ∩ X is the cardinality distribution of the number The cardinality distribution of Ξ o ¨ k|k ∩ X is of actual targets in mode o. According to (4.137), the p.g.f. of Ξ G¨

o

Ξk|k ∩X

(11.132)

(x) = GΞ¨ k|k [1 − 1 o + x · 1 o ]. X

X

¨ k+1|k+1 is i.i.d.c., this becomes Since Ξ G¨

o

(x)

=

Ξk|k ∩X

¨ k|k (N ¨ −1 D ¨ o o G k|k k|k [1 − 1 + x · 1 ])

(11.133)

¨ k|k (1 − N ¨ −1 D ¨ o G k|k k|k [1 ]

(11.134)

X

=

X

X

+x · =

¨ −1 D ¨ o N k|k k|k [1X ])

¨ k|k (1 − ro k|k + x · ro k|k ). G

(11.135)

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The last equation follows because ∫ ∫ o o o ¨ k|k (¨ o D x)d¨ x D k|k (x)dx −1 X ¨ D ¨ k|k [1 o ] = N = (11.136) 1 O 1 O k|k X N k|k + ... + N k|k N k|k + ... + N k|k o

N k|k

o

= 1

O

(11.137)

= r k|k .

N k|k + ... + N k|k Thus the cardinality distribution for the number of targets in mode o is: [ ] o 1 dn ¨ o o o o pk|k (n) = (11.138) o o Gk|k (1 − r k|k + x · r k|k ) n! dxn x=0  o  o o r nk|k ¨ (n) (1 − ro k|k + x · ro k|k ) =  o ·G (11.139) k|k n! x=0 o on

r k|k =

o

n! 11.7

o

¨ (n) (1 − ro k|k ). ·G k|k

(11.140)

IMPLEMENTING JUMP-MARKOV PHD/CPHD FILTERS

The purpose of this section is to briefly describe the practical implementation of jump-Markov PHD and CPHD filters. Gaussian mixture implementation is addressed in Section 11.7.1, and sequential Monte Carlo (SMC) implementation is addressed in Section 11.7.2. 11.7.1

Gaussian Mixture Jump-Markov PHD/CPHD Filters

Gaussian mixture implementation of jump-Markov PHD and CPHD filters is a direct generalization of Gaussian mixture implementation of the classical PHD and CPHD filters, as described in Sections 9.5.4 and 9.5.5. For the sake of conceptual clarity, only the jump-Markov PHD filter will be addressed. Gaussian-mixture implementation of the jump-Markov CPHD filter can be accomplished similarly. 11.7.1.1

Jump-Markov GM-PHD Filter Models

The GM implementation of the PHD filter requires the following modeling assumptions:

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• Probability of target survival: does not depend on the target kinematic state but does depend on the mode, pS,k+1 (o, x) = pS,k+1 (o) abbr. = poS . • Single-target Markov transition density—is linear-Gaussian and mode dependent: fk+1|k (x|o, x′ ) = NQok (x − Fko x′ ). (11.141) • Target-appearance PHD—is mode dependent, and is a Gaussian mixture for each o: B νk+1|k

bk+1|k (o, x) =

∑

k+1|k

k+1|k

· NB k+1|k (x − bi,o

bi,o

(11.142)

)

i,o

i=1

in which case the expected number of appearing targets in mode o is B νk+1|k

B,o Nk+1|k

=

∑

k+1|k

bi,o

(11.143)

,

i=1

whereas the total expected number of appearing targets is νB

B Nk+1|k =

O k+1|k ∑ ∑

k+1|k

bi,o

(11.144)

.

o=1 i=1

• Target-spawning PHD—is mode dependent and is a Gaussian mixture for each o, o′ : S νk+1|k

′

′

bk+1|k (o, x|o , x ) =

∑

k+1|k

k+1|k ′

ej,o,o′ · NGk+1|k (x − Ej,o′

x ).

(11.145)

j,o′

j=1

• Probability of detection—is independent of the target kinematic state but dependent on the mode, pD,k+1 (o, x) = pD,k+1 (o) abbr. = poD . (This assumption can be removed using the approximation described in Section 9.5.6.) • Sensor likelihood function: is possibly mode dependent and linear-Gaussian: o o Lz (o, x) = fk+1 (z|o, x) = NRk+1 (z − Hk+1 x).

(11.146)

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Advances in Statistical Multisource-Multitarget Information Fusion

• Clutter intensity function: is mode-independent, κk+1 (z) = λk+1 · ck+1 (z) where λk+1 is the clutter rate and ck+1 (z) is the clutter spatial distribution. The Gaussian mixture implementation of the jump-Markov PHD filter is based on the assumption that, for each o and all k ≥ 0, Dk|k (o, x) and Dk+1|k (o, x) can be represented as Gaussian mixtures: νk|k

Dk|k (o, x)

∑

=

k|k

k|k

(11.147)

wi,o · NP k|k (x − xi,o ) i,o

i=1 νk+1|k

Dk+1|k (o, x)

∑

=

k+1|k

k+1|k

· NP k+1|k (x − xi,o

wi,o

).

(11.148)

i,o

i=1

Here, o Nk|k =

∫

νk|k

Dk|k (o, x)dx =

∑

k|k

wi,o

(11.149)

i=1

is the expected number of targets that are in mode o, and

Nk|k =

k|k O ν ∑ ∑

k|k

(11.150)

wi,o

o=1 i=1

is the total expected number of targets, regardless of mode. This means that the Gaussian-mixture representation of a PHD can be equivalently replaced by a system of Gaussian components k|k

k|k

k|k

k|k ν

,O

k|k (ℓi,o , wi,o , Pi,o , xi,o )i=1;o=1

k|k

where ℓi,o is the track label associated with each component. 11.7.1.2

Jump-Markov GM-PHD Filter Time Update

We are given a system of Gaussian components k|k

k|k

k|k

k|k ν

,O

k|k (ℓi,o , wi,o , Pi,o , xi,o )i=1;o=1

We are to determine formulas for the predicted system of Gaussian components k+1|k

(ℓi,o

k+1|k

, wi,o

k+1|k

, Pi,o

k+1|k νk+1|k ,O )i=1;o=1 .

, xi,o

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These will actually have the structure k+1|k

k+1|k

B k+1|k νk|k +ν

k+1|k

,O

(ℓi.o,o′ , wi,o,o′ , Pi,o,o′ , xi,o,o′ )i=1;o,ok+1|k , ′ =1 k+1|k

k+1|k

k+1|k

k+1|k

νk|k ,ν S

,O

k+1|k (ℓi,j,o,o′ , wi,j,o,o′ , Pi,j,o,o′ , xi,j,o,o′ )i=1;j=1;o,o ′ =1

as defined by the following equations (as demonstrated in Section K.17): • Time updated number of GM components: B S νk+1|k = νk|k · O 2 + νk+1|k · O 2 + νk|k · vk+1|k · O2 .

(11.151)

Here, there are νk|k · O 2 components corresponding to persisting targets, B νk+1|k ·O 2 components corresponding to newly appearing targets, and S νk|k · vk+1|k · O 2 components corresponding to spawned targets. The timeupdated components are indexed as follows: i

=

1, ..., νk|k ; o = 1, ..., O

i

=

νk|k + 1, ..., νk|k +

i

=

1, ..., νk|k ;

B νk+1|k ;

(11.152) ′

o, o = 1, ..., O

(11.153)

o, o′ = 1, ..., O. (11.154)

S j = 1, ..., vk+1|k ;

• Persisting-target GM components, for i = 1, ..., νk|k , o, o′ = 1, ..., O: k+1|k

k|k

ℓi,o,o′

=

k+1|k

(11.155)

ℓi,o′ ′

k|k

wi,o,o′

=

wi,o′ · χo,o′ · poS

(11.156)

′

k+1|k xi,o,o′ k+1|k Pi,o,o′

=

k|k Fko xi,o′

(11.157)

=

′ Qok

(11.158)

+

′ ′ k|k Fko Pi,o′ (Fko )T .

B • Appearing-target GM components, for i = νk|k + 1, ..., νk|k + νk+1|k , o = 1, ..., O: k+1|k

ℓi,o k+1|k wi,o k+1|k xi,o

=

new label

(11.159)

=

k+1|k bi−νk+1|k,o

(11.160)

=

k+1|k bi−νk+1|k,o

(11.161)

=

Bi−νk+1|k,o .

k+1|k

Pi,o

k+1|k

(11.162)

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Advances in Statistical Multisource-Multitarget Information Fusion

S • Spawned-target GM components, for i = 1, ..., νk|k and j = 1, ..., vk+1|k ′ and o, o = 1, ..., O: k+1|k

ℓi,j,o,o′

=

new label

=

ej,o,o′ · wi,o′

(11.164)

=

k+1|k k|k Ej,o′ xi,o′

(11.165)

=

k+1|k Gj,o′

(11.166)

k|k

k+1|k

wi,o,o′ ,j k+1|k xi,o,o′ ,j k+1|k Pi,o,o′ ,j

11.7.1.3

(11.163) k|k

+

k+1|k k|k k+1|k Ej,o′ Pi,o′ (Ej,o′ )T .

Jump-Markov GM-PHD Filter Measurement Update

We are given the predicted system of Gaussian components: k+1|k

(ℓi,o

k+1|k

, wi,o

k+1|k νk+1|k ,O )i=1;o=1 .

k+1|k

, Pi,o

, xi,o

We are also given a new measurement set Zk+1 = {z1 , ..., zmk+1 } with |Zk+1 | = mk+1 . We are to determine formulas for the measurement-updated system of Gaussian components k+1|k+1

(ℓi,o

k+1|k+1

, wi,o

k+1|k+1

, Pi,o

k+1|k+1 νk+1|k+1,O )i=1;o=1 .

, xi,o

This will actually have the structure k+1|k+1 νk+1|k ,O )i=1;o=1 , k+1|k+1 k+1|k+1 k+1|k+1 k+1|k+1 νk+1|k ,O,mk+1 (ℓi,o,j , wi,o,j , Pi,o,j , xi,o,j )i=1;o=1;j=1 k+1|k+1

(ℓi,o

k+1|k+1

, wi,o

k+1|k+1

, Pi,o

, xi,o

as defined by the following equations (as shown in Section K.17): • Measurement updated number of GM components for the PHD: νk+1|k+1 = νk+1|k · O + mk+1 · νk+1|k · O

(11.167)

where there are νk+1|k · O components for undetected tracks and mk+1 · νk+1|k · O components for detected tracks. The measurement-update components are indexed as follows: i

=

1, ..., νk+1|k ; o = 1, ..., O

(11.168)

i

=

1, ..., νk+1|k ; o = 1, ..., O; j = 1, ..., mk+1 .

(11.169)

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• Measurement updated nondetection components: for i = 1, ..., νk+1|k and o = 1, ..., O, k+1|k+1

k+1|k

ℓi,o

(11.170)

=

ℓi,o

=

(1 − poD ) · wi,o

(11.171)

=

k+1|k xi,o

(11.172)

=

k+1|k Pi,o .

(11.173)

• Measurement updated detection components: o = 1, ..., O and j = 1, ..., mk+1 ,

for i = 1, ..., νk+1|k and

k+1|k+1

k+1|k

wi,o k+1|k+1 xi,o k+1|k Pi,o

k+1|k+1

k+1|k

ℓi,o,j

=

(11.174)

ℓi νk+1|k

τk+1 (zj )

∑ ∑

=

o

k+1|k

wi,o

· poD

(11.175)

i=1 k+1|k

·NRo

k+1|k o o (Hk+1 )T k+1 +Hk+1 Pi,o

k+1|k+1

o (zj − Hk+1 xi,o

k+1|k

xi,o,j

=

k+1|k+1 Pi,o,j

=

k+1 Ki,o

k+1|k

k+1 o xi,o + Ki,o (zj − Hk+1 xi,o ( ) k+1|k k+1 o I − Ki,o Hk+1 Pi,o

(11.176)

)

(11.177)

k+1|k o Pi,o (Hk+1 )T

=

(

k+1|k

o · Hk+1 Pi,o

)

(11.178)

o o (Hk+1 )T + Rk+1

)−1

and k+1|k+1

wi,o,j (

k+1|k wi,o

·

(11.179) )

poD k+1|k

·NRo

k+1|k o o (Hk+1 )T k+1 +Hk+1 Pi,o

o (zj − Hk+1 xi,o

) .

= κk+1 (zj ) + τk+1 (zj ) 11.7.1.4

Jump-Markov GM-PHD Filter Multitarget State Estimation

State estimation for the GM-PHD filter is accomplished as indicated in Section 11.4.4. We are given the measurement-updated system k+1|k+1

(ℓi,o

k+1|k+1

, wi,o

k+1|k+1

, Pi,o

k+1|k+1 νk+1|k+1 ,O )i=1;o=1

, xi,o

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with associated PHD νk+1|k+1

Dk+1|k+1 (o, x) =

∑

k+1|k+1

k+1|k+1

· NP k+1|k+1 (x − xi,o

wi,o

).

(11.180)

i,o

i=1

The total expected number of targets is

Nk+1|k+1 =

∑ νk+1|k+1 ∑ o

k+1|k+1

wi,o

.

(11.181)

i=1

Let n be the integer nearest to Nk+1|k+1 and determine those n Gaussian k+1|k+1 k+1|k+1 components for which wi is largest. Then the associated xi are the k+1|k+1 track state estimates, and the associated Pi are their track covariances. 11.7.2

Particle Implementation of Jump-Markov PHD and CPHD Filters

SMC implementation of jump-Markov PHD and CPHD filters is essentially the same as SMC implementation of the classical PHD and CPHD filters. The primary k|k k|k k|k νk|k difference is that a particle system has the form {(oi , xi , wi )}i=1 rather k|k k|k νk|k than {(xi , wi )}i=1 .

11.8

IMPLEMENTED JUMP-MARKOV PHD/CPHD FILTERS

A handful of researchers have addressed jump-Markov PHD and CPHD filters. All have adopted a bottom-up theoretical approach. That is, they take the PHD filter or the CPHD filter as their starting point, and then attempt to generalize it to jumpMarkov systems. The purpose of this section is to summarize this work: 1. Section 11.8.1: The jump-Markov PHD filter of Pasha, Vo, Tuan, and Ma. 2. Section 11.8.2: The IMM-like jump-Markov PHD filter of Punithakumar, Kirubarajan, and Sinha. 3. Section 11.8.3: The Best-Fitting-Gaussian (BFG) PHD filter of Wenling and Yingmin. 4. Section 11.8.4: The jump-Markov CPHD filter of Georgescu and Willett.

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5. Section 11.8.5: The Current Statistical Model (CSM) PHD filter of Mengjun, Shaohua, Zhiguo, and Kangsheng. 6. Section 11.8.6: The variable state space CPHD filter of Chen Xin, McDonald, and Kirubarajan. 11.8.1

Jump-Markov PHD Filter of Pasha et al.

This approach [234], [232], [233], [297] is consistent with a top-down approach— that is, this jump-Markov PHD filter is identical to the one described in Section 11.4. The material in this section is drawn from [234]. The approach differs from that in Section 11.4 only in that the time-update equations are slightly different. In order to implement the filter using Gaussian mixture techniques, the authors examine specific target appearance and targetspawning models: bk+1|k (o, x) bk+1|k (o, x|o′ , x′ )

= =

pk+1|k (o) · bk+1|k (x) (11.182) ′ ′ ′ ′ pk+1|k (o|x, o , x ) · bk+1|k (x|o , x ). (11.183)

That is, modes of appearing targets are independent of the targets themselves. Likewise, the modes of spawned targets depend on the modes and states of the targets that generated them, but not on the targets themselves. 11.8.2

IMM-Type JM-PHD Filter of Punithakumar et al.

These authors were the first to propose a jump-Markov PHD filter, in 2004 [243], [244]. The material in this section is drawn from [244]. Their intention was to devise a jump-Markov PHD filter inspired by the well-known interacting multiple motion (IMM) model approach. The single-target IMM filter is known to be suboptimal, but has achieved considerable popularity because of its attractive balancing of the competing goals of computational tractability and tracking performance. Since the authors were avowedly pursuing a suboptimal approach, it is not surprising that their jump-Markov PHD filter is different from the one described in Section 11.4. The purpose of this section is to summarize their approach. Their time-update equation (Eq. (10) in [244]) is, when expressed in the notation of this book: ) ∫ ( pS (x′ ) · fk+1|k (x|o, x′ ) ˜ k|k (o, x′ )dx′ Dk+1|k (o, x) = bk+1|k (o, x) + ·D +bk+1|k (x|o, x′ ) (11.184)

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˜ k|k (o, x′ ) is a mode-mixed version of Dk|k (o, x): where (Eq. (9) in [244]) D ˜ k|k (o, x) = D

∑

χo,o′ · Dk|k (o′ , x).

(11.185)

o′

Also, the probability of target survival pS (o, x) = pS (x) is mode-independent. Their measurement-update equation (Eq. (11) in [244]) is as follows, once again expressed in the notation of this book, ∑ pD (x) · Lz (o, x) Dk+1|k+1 (o, x) = 1 − pD (x) + o (z) Dk+1|k (o, x) κk+1 (z) + τk+1

(11.186)

z∈Zk+1

where (Eq. (12) in [244]): ∫ o τk+1 (z) = pD (x) · Lz (o, x) · Dk+1|k (o, x)dx.

(11.187)

One question that arises is this: Is the authors’ approach consistent with the top-down statistical analysis advocated in Section 11.1? A different approach might begin by trying to specify an IMM version of the multitarget jump-Markov filter of Section 11.3.2. Then one would derive an IMM PHD filter from it. This approach would require that one first have a formulation of the IMM filter defined at the density-function level, not at the state-vector level. The authors have implemented their approach using sequential Monte Carlo (SMC, “particle”) techniques. 11.8.3

Best-Fitting-Gaussian PHD Filter of Wenling Li and Yingmin Jia

In this approach [143], an approximation is used to replace a jump-Markov linear system with an approximate single-model linear dynamical system. Specifically, the jump-Markov linear system k Xok+1|k = Fkok x + Gokk Wkok

(11.188)

is replaced by a linear system Xk+1|k = Φk x + Wk .

(11.189)

The best-fitting-Gaussian (BFG) approximation [106] permits this. The basic idea is, at each recursive cycle, the multimodel prior density is recursively approximated

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349

by its BFG approximation. This allows the multiple-model time-update to be replaced by a single-model time-update. The approach was implemented using unscented Kalman filter (UKF) Gaussian mixture techniques. It was tested with a filter in two-dimensional simulations involving a range-bearing sensor with medium, uniformly distributed Poisson clutter (λ = 24). The authors report improved performance in comparison to a filter based on the approach of Pasha et al. 11.8.4

JM-CPHD Filter of Georgescu et al.

This was the first attempt to propose a jump-Markov CPHD filter [91]. These authors employed the modeling approach identified as problematic in Section 11.3.1. That is, they use the multitarget jump-Markov state representation (o, {x1 , ..., xn }) rather than {(o1 , x1 ), ..., (on , xn )}.3 Their jump-Markov CPHD filter was derived using the “bin occupancy” approach of Erdinc et al. ([80], [179], pp. 599-609; Section 8.4.6.8). As a consequence, the authors do not address the conceptual and other issues discussed in Section 11.3.2 and more fully in Section IV-D of [170]. For example: • Shouldn’t performance suffer if a single mode o is imposed on all targets simultaneously? • What does the concept of a CPHD filter approximation even mean for distributions of the form fk+1|k (o, X|Z (k) )? That is, the CPHD filter is based on the assumption that the predicted multitarget distribution is i.i.d.c. But fk+1|k (o, X|Z (k) ) cannot be i.i.d.c. because of the discrete variable o (see [170] for more details). 11.8.5

Current Statistical Model (CSM) PHD Filter of Mengjun et al.

This approach is not, strictly speaking, a jump-Markov approach [197], but is included here for completeness. In the CSM approach, the state is assumed to include the acceleration variables in addition to position and velocity. Instead of a library of motion models, the CMS approach presumes that, at any given time-step, the distribution of the mean of the scalar acceleration is Rayleigh. The variance of this distribution can be expressed in terms of the current estimated scalar 3

This is easily seen from the fact that their cardinality distribution is mode dependent, see [91], Eq. (9). Thus a single value of o is being imposed on all targets simultaneously.

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acceleration. It is appended as an additional state variable, and it is time-updated and measurement-updated independently as a separate variable. The authors have implemented the filter using particle methods, and tested it in two-dimensional simulations assuming a range-bearing sensor, light, uniformly distributed Poisson clutter (λ = 10), and unity probability of detection. A PHD filter reported by Shaohua Hong et al. [270] employs a slight variation on this approach. It employs a constant-velocity (CV) model for nonmaneuvering targets and a CSM model for maneuvering targets. 11.8.6

The Variable State Space CPHD Filter of Chen et al.

Chen Xin, McDonald, and Kirubarajan have tested a Gaussian mixture implementation of the variable state space CPHD filter described in Section 11.6 [36]. Four targets in the plane are observed by a Cartesian, position-measuring sensor in light clutter (clutter rate λ = 3). Two targets appear at time t = 0, and another two at around midscenario. All four targets follow typical air traffic control trajectories: that is, straight-line segments occasionally punctuated by coordinated turns. The variable state space filter utilized the two models described at the beginning of Section 11.6: a constant velocity (CV) model and a coordinated turn (CT) model. At any given time tk , the number of targets in modes 1 or 2 were estimated using both the MAP estimator (k) nMAP ) k|k = arg sup pk|k (n|Z o

o

o

(11.190)

o

n

and the EAP estimator nEAP k|k = o

∑

o

o

o

n · pk|k (n|Z (k) ).

(11.191)

o

n≥0

The authors observed that the number of targets in the modes were accurately estimated using both methods, with the MAP approach being somewhat more accurate. However, the state estimates of the targets exhibited a downward bias. The authors observed that the same phenomenon occurs with single-target IMM filters.

Chapter 12 Joint Tracking and Sensor-Bias Estimation 12.1

INTRODUCTION

Current multitarget detection and tracking algorithms are typically based on the presumption that all sensors are correctly spatially and temporally registered. That is: • Spatial sensor registration—the position, velocity, and physical orientations of all sensors are known precisely with respect to some reference spatial coordinate system. • Temporal sensor registration—the exact time of the measurement collections for all sensors are known precisely, with respect to some reference clock. In real-world applications, neither of these assumptions are necessarily true, as the following examples illustrate: • The time stamps of measurements may be inaccurately aligned with each other, because different clocks have been used by different sensors. • In ground-target tracking, sensor position detections are typically overlaid on a geographical map. However, this map may be inaccurate because it has an unknown translational and/or rotational offset. • The presumed line-of-sight of a gimbaled optical or infrared sensor may actually be inaccurate, because the gimbal axes have become misaligned or because they were inaccurately calibrated to begin with.

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Such imperfections are called sensor biases, and can result in severely degraded target detection, tracking, and localization performance. For example, two sensors observing the same moving target can, if one is translationally misregistered, produce what appear to be two targets moving along parallel trajectories. Biases pose a particularly serious challenge if neither GPS nor accurate terrain maps are available, since the sensor platform’s inertial navigation system (INS) will drift as time passes. The purpose of this chapter is to address multitarget detection and tracking when sensor data is corrupted by unknown biases of unknown types. (Temporal bias will not be considered.) As we shall see, the problem has a solution in principle— Bayesian unified registration and tracking (BURT)—provided that (1) the sensors are all within localization range of each other; and (2) the number of unknown targets of opportunity is sufficiently large. The remainder of this Introduction is organized as follows: 1. Section 12.1.1: A simple example of joint target tracking and sensor registration: “gridlocking” of stationary sensor platforms. 2. Section 12.1.2: Gridlocking in general. 3. Section 12.1.3: A summary of the major lessons learned in this chapter. 4. Section 12.1.4: The organization of the chapter. 12.1.1

Example: “Gridlocking” of Sensor Platforms

Let us begin with the simplest registration problem. Two or more stationary sensor-bearing platforms are to cooperate in the detection and tracking of targets. Consequently, their positions must be known. Suppose, however, that the platforms lack accurate map references and, for whatever reason, do not have access to GPS. Despite this “GPS-denied” and “map-denied” environment, it is nevertheless possible for them to use their sensors and communications systems to determine their positions with respect to each other in a relative coordinate system, using a procedure sometimes called gridlocking. In what follows this process will be illustrated using three successively more detailed examples: 1. Section 12.1.1.1: Gridlocking without sensor biases. 2. Section 12.1.1.2: The impossibility of gridlocking when sensor biases exist. 3. Section 12.1.1.3: Simultaneous target-localization and gridlocking.

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353

Gridlocking Without Sensor Biases

Suppose that we have two motionless position-measuring sensors, with infinite fields of view, unity probability of detection, and no clutter. Suppose further that 1 2 they are located at positions x and x in some absolute coordinate frame. In their respective local coordinate systems, the measurement models of the two sensors are j

j

Zk+1 = x + Vk+1

(12.1)

(j = 1, 2)

j

where Vk+1 are zero-mean random vectors. Assume that the sensors measure the following positions of the locations of the other sensor: • Position of Sensor 2 as measured by Sensor 1: 1,2 2

1

1

1

Z k+1 = x − x + Vk+1 = ∆x + Vk+1 2

(12.2)

1

where ∆x = x − x is the separation vector between the first and second platforms. • Position of Sensor 1 as measured by Sensor 2: 2,1 1

2

2

2

Z k+1 = x − x + Vk+1 = −∆x + Vk+1 .

(12.3)

By averaging enough measurements over time, the first sensor can, in its own coordinate frame, estimate the location of the second sensor as 1,2

Z k+1 = ∆x,

(12.4)

and the second sensor can, in its coordinate frame, estimate the location of the first sensor as 2,1

Z k+1 = −∆x.

(12.5)

In other words, the two sensor platforms have deduced each other’s locations relative to each other. If we arbitrarily choose the position of one of the sensors as the origin—the first sensor, say—then we have determined the positions of both sensors in a relative coordinate system. The position of the first sensor is 1 x = x = 0 and the position of the second sensor is x = ∆x.

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12.1.1.2

Gridlocking with Biases

Now suppose that both sensors have unknown translational biases. In this case their measurement models become j

∗j

j

Zk+1 = x + t + Vk+1

(j = 1, 2)

(12.6)

∗j

where the vectors t are the translational biases. Given this, the sensors collect the following random measurements: • Position of Sensor 2 as measured by Sensor 1: 1,2 2

Z k+1

=

(12.7)

1

∗1

=

1

∗1

1

x − x + t + Vk+1 ∆x + t + Vk+1 .

(12.8)

• Position of Sensor 1 as measured by Sensor 2: 2,1 1

Z k+1

=

2

∗2

∗2

=

2

x − x + t + Vk+1

(12.9)

2

−∆x + t + Vk+1 .

(12.10)

The first sensor’s estimate of the second sensor’s position is 1,2

∗1

(12.11)

Z k+1 = ∆x + t and the second sensor’s estimate of the first sensor’s location is 2,1

∗2

(12.12)

Z k+1 = −∆x + t . ∗1 ∗2

∗1 ∗2

We now have two equations in the three unknowns ∆x, t , t . The biases t , t cannot be estimated unless ∆x is known. But such knowledge requires access to additional information of some kind. 12.1.1.3

Simultaneous Gridlocking and Target Localization

In this section, assume that we have the following additional information: • A motionless “target of opportunity” is located at an unknown position x0 .

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Given this we have, in addition to (12.8) and (12.10), access to the following two additional measurement collections: • Position of unknown target as measured by Sensor 1: 1,0

1

∗1

1

(12.13)

Z k+1 = x0 − x + t + Vk+1 . • Position of unknown target as measured by Sensor 2: 2,0

2

∗2

2

(12.14)

Z k+1 = x0 − x + t + Vk+1 . 1,2

1,0 2

Suppose that Z k+1 and Z k+1 are transmitted to the sensor located at x. 2 Then the fusion system at x can deduce the following: 1,2

Z k+1

∗1

=

2,1

Z k+1

(12.15)

∆x + t ∗2

(12.16)

=

−∆x + t

=

x0 + ∆x − x + t

=

x0 − x + t .

1,0

∗1

2

Z k+1 2,0

2

Z k+1

(12.17)

∗2

(12.18) ∗1 ∗2

2

We now have four linear equations in the following five unknowns: ∆x, t , t , x0 , x. 2 Choose x = 0 to be the origin of a relative coordinate system, which leaves four ∗1 ∗2 equations in four unknowns ∆x, t , t , x0 . We can then difference (12.15) through (12.18) to get: 1,0

x0

=

2,0

∆x

=

t

=

t

=

(12.19) 1,2

2,1

1,0

(12.20)

1,2

(12.21)

Z k+1 − Z k+1 + Z k+1 1,2

∗1

1,0

Z k+1 − Z k+1 + Z k+1 − Z k+1 2,0

∗2

1,2

Z k+1 − Z k+1

2,1

2,0

1,0

1,2

Z k+1 + Z k+1 − Z k+1 + Z k+1 − Z k+1 .

(12.22)

That is, we have simultaneously determined: 1. The positions of the platforms in a relative coordinate system (gridlocking).

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2. The sensor biases (sensor registration). 3. The location of the unknown target (target localization). This process can be described as joint tracking and sensor registration. 12.1.2

Gridlocking in General

Now consider the problem of joint tracking and bias estimation in general. The simple examples just presented rely on unrealistic assumptions: the sensors and the targets of opportunity are motionless and well separated; the sensors are positionmeasuring; the sensor biases are purely translational; and so on. Nevertheless they illustrate the following points: 1. Joint sensor registration and gridlocking are possible in principle, even when all sensors are biased, and even if we do not have access to GPS or other inertial information. 2. However, additional “ground truth” information must be available, in the form of an unknown number of unknown “targets of opportunity.” 3. The number of targets of opportunity must be sufficiently large. 4. Bias estimation (registration), gridlocking, and target tracking must be accomplished jointly and simultaneously using a single, fully unified algorithmic procedure. 5. In general, this procedure will be highly nonlinear. The challenge, then, is to devise a formal probabilistic framework that generalizes our simple example to scenarios of arbitrary complexity. This is the purpose of this chapter. 12.1.3

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter: • If there are a sufficient number of unknown “targets of opportunity” in a scenario, then—at least in principle—it is possible to devise an optimal procedure for estimating the sensor biases while simultaneously detecting and tracking those targets. See Section 12.3.

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• This procedure is a generalization of gridlocking—which, in turn, is a type of SLAM (simultaneous localization and mapping). Thus the “Bayesian unified registration and tracking” (BURT) approach in this chapter can be regarded as a generalization of SLAM. • The optimal BURT approach is computationally intractable in general. However, it can be approximated as a two-filter version of a PHD filter. See Section 12.4. (A CPHD filter approximation is also possible but will not be considered in this chapter.) • BURT-type PHD filters are apparently computationally tractable only for translational biases in the target-state variable. • In particular, a BURT-PHD filter for constant translational biases in the target-state variable, due to Ristic and Clark, appears to offer promising performance. See Section 12.5.1. • A heuristic BURT-PHD filter for constant translational biases in the targetstate variable appears to offer surprisingly effective performance. See Section 12.5.2. • These two BURT-PHD filters have been implemented and evaluated in simulation, for simple translational sensor biases. See Section 12.6. 12.1.4

Organization of the Chapter

The chapter is organized as follows: 1. Section 12.2: The modeling of general sensor biases. 2. Section 12.3: Optimal joint multitarget tracking and sensor registration: the single-filter and two-filter BURT filters. 3. Section 12.4: An approximation of the optimal BURT filter—the two-filter version the BURT-PHD filter. 4. Section 12.5: Single-filter versions of the BURT-PHD filter. 5. Section 12.6: Implemented PHD filters for joint sensor registration and tracking.

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MODELING SENSOR BIASES

The purpose of this section is to introduce a general model for sensor biases. It depends on the notion of the perceived sensor state versus the actual sensor state. ∗ As was noted in Section 10.2 of Chapter 10, a sensor has a state-vector x—for example, a state of the form ∗ ˙ η, x = (x, y, z, x, ˙ y, ˙ z, ˙ ℓ, θ, η, φ, θ, ˙ φ, ˙ µ, χ)

(12.23)

where x, y, z and x, ˙ y, ˙ z˙ are the position and velocity coordinates of the sensor˙ η, carrying platform, ℓ is its fuel level, θ, η, ϕ and θ, ˙ φ˙ are the sensor’s bodyframe coordinates and their rates, µ is the sensor mode, and χ is the current communications transmission path employed by the sensor. Thus the sensor’s measurement model actually has the form ∗

Zk+1 = ηk+1 (x, xk+1 ) + Vk+1

(12.24)

∗

where xk+1 is the actual sensor state at time tk+1 . ∗ Complicating matters still further, any of the variables xk|k , x or Zk+1 can be contaminated by a spatial bias—for example, translational biases ∗

Zk+1

=

ηk+1 (x + xb , xk+1 ) + Vk+1

Zk+1 Zk+1

= =

ηk+1 (x, xk+1 + xb ) + Vk+1 ∗ ηk+1 (x, xk+1 ) + zb + Vk+1

∗

∗

(12.25) (12.26) (12.27)

or, more generally, affine biases Zk+1

=

Zk+1

=

Zk+1

=

∗

ηk+1 (Tb x + xb , xk+1 ) + Vk+1 ∗

∗

∗

ηk+1 (x, xk+1 + T b xb ) + Vk+1 ∗ T˜b ηk+1 (x, xk+1 ) + zb + Vk+1

(12.28) (12.29) (12.30)

∗

where Tb , T b , T˜b are rotation matrices. In this case (12.24) actually has the form Zk+1

= =

∗

ηk+1 (b, x, xk+1 ) + Vk+1 ∗ ηk+1 (˚ x, xk+1 ) + Vk+1

(12.31) (12.32)

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where the vector b denotes the concatenation of all bias variables; and where the augmented state-vector ˚ x = (xT , bT )T (12.33) consolidates all unknown variables—the unknown state-vectors x and b—into a single unknown state-vector ˚ x. This model is general enough to represent the most common kinds of bias. ∗ Now, notationally suppress the sensor state xk+1 and assume that the sensor model is linear-Gaussian. Then consider the following examples: • Affine bias in the measurement: Zk+1

=

T Hk+1 x + b + Vk+1

(12.34)

ηk+1 (˚ x)

= =

ηk+1 (T, b, x) T Hk+1 x + b.

(12.35) (12.36)

• Affine bias in the target state:

12.3

Zk+1

=

Hk+1 (T x + b) + Vk+1

(12.37)

ηk+1 (˚ x)

= =

ηk+1 (T, b, x) Hk+1 T x + Hk+1 b.

(12.38) (12.39)

OPTIMAL JOINT TRACKING AND REGISTRATION

The multisensor-multitarget Bayes recursive filter was introduced in Section 10.2. The purpose of this section is to describe its extension to the joint tracking and registration problem. As in Section 10.2, suppose that there are s sensors and let: j

Z

:

measurement space for jth sensor

:

measurements for jth sensor

:

measurement sets for jth sensor

j

z j

Z

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j

j

j

Z (k)

:

Z 1 , ..., Z k : time sequence of measurement sets for jth sensor

Z

=

Z ⊎ ... ⊎ Z: multisensor measurement space

Z

=

Z ⊎ ... ⊎ Z: multisensor measurement set

Z (k)

:

Z (k) , ..., Z (k) : time sequence of multisensor measurement sets.

1

s

1

s

1

s

When the jth sensor has a bias, the multitarget likelihood function for this sensor has the form j

j

j

j

j

j

j

∗j

L j (b, X) = f k+1 (Z|b, X) abbr. = fk+1 (Z|b, X, x).

(12.40)

Z

Assume that the sensors are independent and let 1

s

b = (bT , ..., bT )T

(12.41)

be the joint bias vector. Then according to the discussion in Section 3.5.3, the multisensor-multitarget likelihood function has the form 1

fk+1 (Z|b, X)

=

s

1

=

s

(12.43)

fk+1 (Z, ..., Z|b, X) 1

=

(12.42)

fk+1 (Z ⊎ ... ⊎ Z|b, X) 1

1

s

s

s

f k+1 (Z|b, X) · · · f k+1 (Z|b, X).

(12.44)

Thus the total unknown state of the system is the pair (b, X). We are to estimate b and X simultaneously. Given this, two versions of the optimal BURT filter will be discussed: a single-filter version and a two-filter version. 12.3.1

Optimal BURT Filter: Single-Filter Version

The Bayes-optimal filter for the BURT problem—the optimal BURT filter—has the form ... → fk|k (b, X|Z (k) ) → fk+1|k (b, X|Z (k) ) → fk+1|k+1 (b, X|Z (k+1) ) → ...

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where fk+1|k (b, X|Z

(k)

)

∫

=

fk+1|k (b, X|b′ , X ′ )

(12.45)

·fk|k (b′ , X ′ |Z (k) )db′ δX ′ fk+1|k+1 (b, X|Z (k+1) )

(12.46) fk+1 (Zk+1 |b, X) · fk+1|k (b, X|Z fk+1 (Zk+1 |Z (k) ) ∫ fk+1 (Zk+1 |b, X)

= fk+1 (Zk+1 |Z (k) )

=

(k)

)

(12.47)

·fk+1|k (b, X|Z (k) )dbδX or, in more detailed notation, 1

∫

s

fk+1|k (b, ..., b, X|Z (k) )

=

1

s

1′

s′

fk+1|k (b, ..., b, X|b , ..., b , X ′ )(12.48) 1′

s′

·fk|k (b , ..., b , X ′ |Z (k) ) 1′

s′

·db · · · db δX ′ and 1

s

1

s

fk+1|k+1 (b, ..., b, X|Z (k+1) , ..., Z (k+1) ) ( 1 ) 1 s s 1 s f k+1 (Z k+1 |b, X) · · · f k+1 (Z k+1 |b, X) 1

s

1

(12.49)

s

·fk+1|k (b, ..., b, X|Z (k) , ..., Z (k) )

=

1

s

1

s

fk+1 (Z k+1 , ..., Z k+1 |Z (k) , ..., Z (k) ) and 1

=

s

1

s

fk+1 (Z, ..., Z|Z (k) , ..., Z (k) ) ∫ 1 s 1 1 s s f k+1 (Z|b, X) · · · fk+1 (Z|b, X) 1

s

1

s

(12.50)

1

s

·fk+1|k (b, ..., b, X|Z (k) , ..., Z (k) ) · db · · · dbδX.

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12.3.2

Optimal BURT Filter: Two-Filter Version

The single-filter optimal BURT filter is inconvenient for the purpose of computational approximation. Therefore, it is necessary to consider an alternative form of this filter. Since the system state has the hybrid form (b, X), the mixed-state “factored filter” analysis of Section 5.9 applies. We therefore know that the optimal BURT filter can be written in a two-filter form. From Bayes’ rule, fk|k (b, X|Z (k) ) = fk|k (b|Z (k) ) · fk|k (X|b, Z (k) )

(12.51)

where fk|k (b|Z (k) ) is a probability distribution on b; and where fk|k (X|b, Z (k) ) is a multitarget probability distribution on X, given that the multisensor bias is b. Also because of Bayes rule, the Markov transition density can be written as fk+1|k (b, X|b′ , X ′ ) = fk+1|k (b|b′ , X ′ ) · fk+1|k (X|b, b′ , X ′ ).

(12.52)

Assume that fk+1|k (b|b′ , X ′ ) fk+1|k (X|b, b′ , X ′ )

= =

fk+1|k (b|b′ ) fk+1|k (X|X ′ ).

(12.53) (12.54)

That is, the transition of the sensor biases does not depend on the past states of the targets; and the transition of the target states does not depend on the current or previous sensor biases. Then from Section 5.9 we know that the two-filter version of the optimal BURT filter has the form ... → fk|k (b) ... → fk|k (X|b) where:

→ ↑↓ →

fk+1|k (b) fk+1|k (X|b)

→ ↑↓ →

fk+1|k+1 (b) → ... fk+1|k+1 (X|b) → ...

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• Optimal BURT filter time-update: fk+1|k (b|Z (k) )

=

∫

fk+1|k (b|b′ ) · fk|k (b′ |Z (k) )db′ (12.55)

fk+1|k (X|b, Z (k) ) ( ∫ = f˜k+1|k (X|b′ , Z (k) )

=

∫

′

′

(k)

fk+1|k (b|b ) · fk|k (b |Z ·f˜k+1|k (X|b′ , Z (k) )db′ fk+1|k (b|Z (k) )

fk+1|k (X|X ′ )

)

(12.56) )

(12.57)

·fk|k (X ′ |b′ , Z (k) )δX ′ . • Optimal BURT filter measurement-update: fk+1|k+1 (b|Z (k+1) )

(12.58)

(k)

=

fk+1|k (b|Z ) · fk+1 (Zk+1 |b, Z fk+1 (Zk+1 |Z (k) )

(k)

)

and fk+1|k+1 (X|b, Z (k+1) ) =

(12.59)

fk+1 (Zk+1 |b, X) · fk+1|k (X|b, Z (k) ) fk+1 (Zk+1 |b, Z (k) )

where fk+1 (Zk+1 |Z (k) )

fk+1 (Zk+1 |b, Z (k) )

=

∫

fk+1|k (b|Z (k) )

(12.60)

=

·f (Z |b, Z (k) )db ∫ k+1 k+1 fk+1 (Zk+1 |b, X)

(12.61)

·fk+1|k (X|b, Z (k) )δX and where, for fixed b, the last equation is a conventional multitarget Bayes factor.

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The following special case is of interest, because it is often true that a sensor’s bias does not appreciably change over time. That is, at least approximately we can sometimes assume that fk+1|k (b|b′ ) = δb′ (b). (12.62) Given this assumption, (12.55) through (12.61) reduce to: fk+1|k (b|Z (k) )

=

fk+1|k (X|b, Z (k) )

=

fk|k (b|Z (k) ) (12.63) ∫ fk+1|k (X|X ′ ) · fk|k (X ′ |b, Z (k) )δX ′ (12.64)

and fk+1|k+1 (b|Z (k+1) )

=

fk|k (b|Z (k) ) · fk+1 (Zk+1 |b, Z (k) ) fk+1 (Zk+1 |Z (k) )

fk+1|k+1 (X|b, Z (k+1) )

(12.65) (12.66)

=

fk+1 (Zk+1 |b, X) · fk+1|k (X|b, Z fk+1 (Zk+1 |b, Z (k) )

(k)

)

where fk+1 (Zk+1 |Z (k) )

fk+1 (Zk+1 |b, Z (k) )

=

∫

fk+1|k (b|Z (k) )

(12.67)

=

·f (Z |b, Z (k) )db ∫ k+1 k+1 fk+1 (Zk+1 |b, X)

(12.68)

·fk+1|k (X|b, Z (k) )δX. 12.3.3

Optimal BURT Procedure

Assume that every sensor is within the field of view of every other sensor. Then the optimal BURT procedure consists of the following steps.1 1. Step 1: Arbitrarily select a sensor and adopt its coordinate system as the reference coordinate system for all sensors. 2. Step 2: Use the BURT filter to process a sequence Z (k) : Z1 , ..., Zk of multisensor measurement sets, collected from all targets in the scene (including the sensor-carrying platforms). 1

The procedure described here is slightly different than the one presented in [192].

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3. Step 3: At each time-step, integrate out X as a nuisance variable to get the marginal distribution ∫ fk+1|k+1 (b|Z (k+1) ) = fk+1|k+1 (X, b|Z (k+1) )δX. (12.69) 4. Step 4: Estimate the sensor biases using a MAP estimator: bk+1|k+1 = arg sup fk+1|k+1 (b|Z (k+1) ).

(12.70)

b

5. Step 5 (Optimal Registration): When the time sequence b1|1 , ..., bk+1|k+1 converges to a stable (that is, small-variance) value 1

s

bk+1|k+1 = (bTk+1|k+1 , ..., bTk+1|k+1 )T ,

(12.71)

we have achieved an optimal solution to the registration part of the problem. Note that his procedure will not necessarily converge. This will occur if, for example, there are too few targets of opportunity in the scenario. 6. Step 6: Employ this information to correctly register whatever multitarget tracking algorithms are being used on the platforms. If these algorithms are j

j

MHTs, for example, use (12.70) with b = bk+1|k+1 (j = 1, ..., s) in the extended Kalman filters (EKFs) that are employed in the MHTs. 7. Step 7 (Optimal Gridlocking): Use the JoM estimator, (5.9), or the MaM estimator, (5.10), to determine the number and states of the targets: JoM Xk+1|k+1 = arg sup X

c|X| · fk+1|k+1 (X|bk+1|k+1 , Z (k+1) ) . |X|!

(12.72)

Some of these targets will be those of the platforms that carry the sensors (localized relative to the coordinate system chosen in Step 1). This completes the joint gridlocking-registration-tracking process.

12.4

THE BURT-PHD FILTER

The optimal BURT filters of Sections 12.3.1 and 12.3.2 will be computationally intractable in general. Therefore, principled approximations are necessary. The purpose of this section is to introduce a PHD filter approximation of the optimal-BURT

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filter. This must be accomplished using the two-filter approach of Section 12.3.2. This is because it is not possible for the joint distribution fk+1|k (b, X|Z (k) ) to be Poisson (because of the variable b); whereas the conditional distribution fk+1|k (X|b, Z (k) ) can be assumed to be Poisson. Thus from (12.51), since fk|k (b, X|Z (k) ) = fk|k (b|Z (k) ) · fk|k (X|b, Z (k) )

(12.73)

it follows from (4.74) that the joint PHD will have the form Dk|k (b, X|Z (k) ) = fk|k (b|Z (k) ) · Dk|k (x|b, Z (k) ).

(12.74)

The section is organized as follows: 1. Section 12.4.1: The single-sensor BURT-PHD filter. 2. Section 12.4.2: The multisensor BURT-PHD filter, using the iteratedcorrector approach. 3. Section 12.4.3: The multisensor BURT-PHD filter, using the parallelcombination approach. 12.4.1

BURT-PHD Filter: Single-Sensor Case

For the purposes of this section, assume that only a single sensor is present. Then b is the bias vector for only that sensor. The PHD filter approximation of the optimal BURT filter has the two-filter form ... → fk|k (b|Z (k) ) ... → Dk|k (x|b, Z (k) )

12.4.1.1

→ ↑↓ →

fk+1|k (b|Z (k) ) Dk+1|k (x|b, Z (k) )

→ ↑↓ →

fk+1|k+1 (b|Z (k+1) ) → ... Dk+1|k+1 (x|b, Z (k+1) ) → ...

Single-Sensor BURT-PHD Filter: Time Update

For the sake of conceptual clarity, ignore the target-spawning model. Then the time-update equations are fk+1|k (b|Z

(k)

)

Dk+1|k (x|b, Z (k) )

=

∫

fk+1|k (b|b′ ) · fk|k (b′ |Z (k) )db′

=

∫

˜ k+1|k (x|b′ , Z (k) ) · fk|k+1 (b′ |b)db′ (12.76) D

(12.75)

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where ˜ k+1|k (x|b, Z (k) ) D

=

fk|k+1 (b′ |b)

=

bk+1|k (x) (12.77) ∫ + pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ |b, Z (k) )dx′ fk+1|k (b|b′ ) · fk|k (b′ |Z (k) ) fk+1|k (b|Z (k) )

(12.78)

and where, recall, fk|k+1 (b′ |b) is a retrodictive or “reverse” Markov transition density, as in (5.77). To see why these equations are true, note that from the integral formula for PHDs, (4.74), the PHDs of fk+1|k (X|b, Z (k) ) and f˜k+1|k (X|b′ , Z (k) ) in (12.57) and (12.57) are ∫ Dk+1|k (x|b, Z (k) ) = fk+1|k ({x} ∪ X|b, Z (k) )δX (12.79) ∫ ˜ k+1|k (x|b, Z (k) ) = D f˜k+1|k ({x} ∪ X|b′ , Z (k) )δX. (12.80) Therefore, from (12.55) and (12.57) we immediately get ∫ fk+1|k (b|Z (k) ) = fk+1|k (b|b′ ) · fk|k (b′ |Z (k) )db′ (12.81) ( ∫ ) fk+1|k (b|b′ ) · fk|k (b′ |Z (k) ) ˜ k+1|k (x|b′ , Z (k) )db′ ·D Dk+1|k (x|b, Z (k) ) = (12.82) fk+1|k (b|Z (k) ) ∫ ˜ k+1|k (x|b′ , Z (k) ) · fk|k+1 (b′ |b)db′ .(12.83) = D Since (12.57) is a conventional multitarget time-update for fixed b′ , it follows that the time-update for its PHD must be a conventional PHD filter time-update: ˜ k+1|k (x|b, Z (k) ) D

=

=

bk+1|k (x|b) + (12.84) ∫ pS (x′ |b) · fk+1|k (x|b, x′ ) · Dk|k (x′ |b, Z (k) )dx′ ∫ bk+1|k (x) + pS (x′ ) · fk+1|k (x|x′ ) (12.85) ·Dk|k (x′ |b, Z (k) )dx′

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where the last equation follows from the fact that target appearances and disappearances cannot depend on the sensor bias. Now consider a special case: the sensor bias is approximately constant over time. Then fk+1|k (b|b′ ) = δb′ (b) and these equations reduce to fk+1|k (b|Z (k) )

=

Dk+1|k (x|b, Z (k) )

=

fk|k (b|Z (k) ) (12.86) ∫ bk+1|k (x) + pS (x′ ) · fk+1|k (x|x′ ) (12.87) ·Dk|k (x′ |b, Z (k) )dx′ .

Remark 54 As an example of the retrodictive density in (12.78), let fk+1|k (b|b′ ) ′

fk|k (b |Z

(k)

)

= =

NQ (b − F b′ ) ′

NP (b − bk|k ).

(12.88) (12.89)

Then fk|k+1 (b′ |b) = NC (b′ − CP −1 bk|k − CF T Q−1 b)

(12.90)

C −1 = P −1 + F T Q−1 F.

(12.91)

where

12.4.1.2

Single-Sensor BURT-PHD Filter: Measurement Update

The measurement-update equations for this filter are fk+1|k+1 (b|Z (k+1) )

Dk+1|k+1 (x|b, Z (k+1) ) Dk+1|k (x|b, Z (k) )

(12.92) =

fk+1 (Zk+1 |b, Z (k) ) · fk+1|k (b|Z (k) ) fk+1 (Zk+1 |Z (k) )

=

1 − pD (b, x) +

(12.93)

∑ pD (b, x) · fk+1 (z|b, x) κk+1 (z) + τk+1 (z|b) z∈Zk+1

where the likelihood function fk+1 (Zk+1 |b, Z (k) ) for the bias is ∏ fk+1 (Zk+1 |b, Z (k) ) = e−λk+1 −Dk+1|k [pD |b] (κk+1 (z) + τk+1 (z|b)) z∈Zk+1

(12.94)

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and where )

=

∫

τk+1 (z|b)

=

∫

=

·Dk+1|k (x|b, Z (k) )dx ∫ pD (b, x) · Dk+1|k (x|b, Z (k) )dx.

fk+1 (Zk+1 |Z

(k)

Dk+1|k [pD |b]

fk+1|k (b|Z (k) ) · fk+1 (Zk+1 |b, Z (k) )db(12.95) pD (b, x) · fk+1 (z|b, x)

(12.96)

(12.97)

Remark 55 (Limitations) Notice that (12.93) is valid only for biases of the target state x, and not for biases of the sensor state or the measurements. The reason for this restriction is as follows. A translational bias on measurements z will cause clutter measurements to be translationally shifted, whereas this will not be the case for a translational bias on target states. In this case, κk+1 (z) would actually have the form κk+1 (z|b). Similar remarks apply to a translational bias of the ∗ ∗ sensor state x, since κk+1 (z) is actually an abbreviation for κk+1 (z|x). Thus the clutter is dependent on b in the joint state (b, X). Because of this and as was discussed in Section 8.7, the measurement-update for the PHD would involve a computationally problematic combinatorial sum over all partitions of the current measurement set. Remark 56 The bias likelihood function for the bias, (12.94), is highly nonlinear. Thus practical implementation of these equations will require sequential Monte Carlo (SMC) techniques. To see why (12.93) through (12.97) are true, note that, for fixed b, (12.59) is a conventional multitarget measurement-update. Therefore, the PHD of fk+1|k+1 (X|b, Z (k+1) ) is a conventional PHD filter measurement-update. Therefore, the equations corresponding to (12.58) through (12.59) are

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fk+1|k+1 (b|Z (k+1) )

(12.98) (k)

Dk+1|k+1 (x|b, Z (k+1) ) Dk+1|k (x|b, Z (k) )

=

fk+1 (Zk+1 |b, Z ) · fk+1|k (b|Z fk+1 (Zk+1 |Z (k) )

=

1 − pD (b, x) +

(k)

)

(12.99)

∑ pD (b, x) · fk+1 (z|b, x) κk+1 (z) + τk+1 (z|b) z∈Zk+1

where τk+1 (z|b)

=

∫

pD (b, x) · fk+1 (z|b, x)

(12.100)

·Dk+1|k (x|b, Z (k) )dx fk+1 (Zk+1 |Z (k) )

(12.101) =

∫

fk+1 (Zk+1 |b, Z (k) ) · fk+1|k (b|Z (k) )db.

For fixed b, (12.58) is a conventional multitarget Bayes normalization factor. Given our models and (8.56), it is fk+1 (Zk+1 |b, Z (k) ) = e−λk+1 −Dk+1|k [pD |b]

∏

(κk+1 (z) + τk+1 (z|b))

z∈Zk+1

(12.102) where Dk+1|k [pD |b] =

12.4.1.3

∫

pD (b, x) · Dk+1|k (x|b, Z (k) )dx.

(12.103)

Single-Sensor BURT-PHD Filter: State Estimation

State estimation is accomplished in the obvious fashion. First, estimate the current sensor bias: bk+1|k+1 = arg sup fk+1|k+1 (b|Z (k+1) ). (12.104) b

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Given this, apply the usual PHD filter state-estimation approach in Section 8.4.4 to Dk+1|k+1 (x|bk+1|k+1 , Z (k+1) ). That is, round off Nk+1|k+1 =

∫

Dk+1|k+1 (x|bk+1|k+1 , Z (k+1) )dx

(12.105)

to the nearest integer ν, and then find the states corresponding to the ν largest peaks of Dk+1|k+1 (x|bk+1|k+1 , Z (k+1) ). 12.4.2

BURT-PHD Filter: Multisensor Case Using Iterated Corrector 1

Suppose that there are 1

sensors with respective biases

s

s

b, ..., b

and let

s

T

T T

b = (b , ..., b ) be the joint multisensor bias-vector. Then apply the usual iterated-corrector approach of Section 10.5 by repeating (12.93) and (12.93) once for each sensor in turn. Thus for the first sensor we get 1

s

1

2

s

fk+1|k+1 (b, ..., b|Z (k+1) , Z (k) , ..., Z (k) ) 1

=

1

s

(12.106)

1

s

1

s

fk+1 (Z k+1 |b, ..., b, Z (k) ) · fk+1|k (b, ..., b|Z (k) , ..., Z (k) ) 1

1

s

fk+1 (Z k+1 |Z (k) , ..., Z (k) ) and 1

s

1

2

s

Dk+1|k+1 (x|b, ..., b, Z (k+1) , Z (k) , ..., Z (k) ) 1

s

(12.107) 1

s

Dk+1|k (x|b, ..., b, Z (k) , ..., Z (k) ) 1

=

1 − pD (x) + 1

1 1

∑

pD (x) · f k+1 (z|b, x)

.

1 1

1

κk+1 (z) + τk+1 (z|b)

1

z∈Z k+1

Similarly, for the second sensor we get 1

s

1

2

s

fk+1|k+1 (b, ..., b|Z (k+1) , Z (k+1) , ..., Z (k) ) ( ) 1 s 2 fk+1 (Z k+1 |b, ..., b, Z (k) ) 1

=

s

1

2

s

·fk+1|k (b, ..., b|Z (k+1) , Z (k) , ..., Z (k) ) 2

1

2

s

fk+1 (Z k+1 |Z (k+1) , Z (k) , ..., Z (k) )

(12.108)

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and 1

s

1

2

s

Dk+1|k+1 (x|b, ..., b, Z (k+1) , Z (k+1) , ..., Z (k) ) 1

s

(12.109) 1

2

s

Dk+1|k (x|b, ..., b, Z (k+1) , Z (k) , ..., Z (k) ) 2

=

∑

1 − pD (x) + 2

2

2 2

pD (x) · f k+1 (z|b, x) 2 2

2

κk+1 (z) + τk+1 (z|b)

z∈Z k+1

and so on. 12.4.3

BURT-PHD Filter: Multisensor Case Using Parallel Combination

In this case, the iterated-corrector procedure in the previous section is replaced by the procedure described in Section 10.6.

12.5

SINGLE-FILTER BURT-PHD FILTERS

The purpose of this section is to describe simpler versions of the BURT-PHD filter, in which—in effect—only the joint PHD Dk|k (x|b, Z (k) ) is propagated. The section is organized as follows: 1. Section 12.5.1: A single-filter BURT-PHD filter for the case of static sensor biases. 2. Section 12.5.2: Heuristic single-filter BURT-PHD filters. 12.5.1

Single-Filter BURT-PHD Filter for Static Biases

Ristic and Clark have shown that it is possible to derive a single-filter version of the BURT-PHD filter, given that the sensor biases are static. It has the following form [254], [253], [255]: ... → fk|k (b|Z (k) ) → ... → Dk|k (x|b, Z (k) ) → Dk+1|k (x|b, Z (k) )

fk+1|k+1 (b|Z (k+1) ) → ... ↗↑ → Dk+1|k+1 (x|b, Z (k+1) ) → ...

Note that the two filters are not actually coupled, since the bias distributions in the top filter are derived entirely in terms of the conditional PHDs in the bottom filter.

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Thus this filter has, in essence, a single-filter form. It is defined by the following equations: • Time update (target spawning neglected): Dk+1|k (x|b, Z (k) )

=

bk+1|k (x) (12.110) ∫ + pS (x′ ) · fk+1 (x|x′ ) · Dk|k (x′ |b, Z (k) )dx′ .

• Measurement update: Dk+1|k+1 (x|b, Z (k+1) ) Dk+1|k (x|b, Z (k) )

1 − pD (b, x)

=

(12.111)

∑ pD (b, x) · fk+1 (z|b, x) κk+1 (z) + τk+1 (z|b)

+

z∈Zk+1

and fk+1|k+1 (b|Z (k+1) ) =

fk+1 (Zk+1 |b, Z (k) ) · fk|k (b|Z (k) ) fk+1 (Zk+1 |Z (k) )

(12.112)

where τk+1 (z|b)

=

∫

pD (b, x) · fk+1 (z|b, x)

(12.113)

·Dk+1|k (x|b, Z (k) )dx fk+1 (Zk+1 |b, Z (k) )

=

e−λk+1 −Dk+1|k [pD |b] (12.114) ∏ · (κk+1 (z) + τk+1 (z|b)) z∈Zk+1

and where fk+1 (Zk+1 |Z

(k)

)

Dk+1|k [pD |b]

=

∫

=

·fk|k (b|Z (k) )db ∫ pD (b, x)

fk+1 (Zk+1 |b, Z (k) )

·Dk+1|k (x|b, Z (k) )dx.

(12.115)

(12.116)

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To see why these equations are true, recall that the two-filter version of the BURT-PHD filter is based on (12.51), fk+1|k+1 (b, X|Z (k+1) ) = fk+1|k+1 (b|Z (k+1) ) · fk+1|k+1 (X|b, Z (k+1) ), (12.117) from which follows, because of (4.74), Dk+1|k+1 (b, x|Z (k+1) ) = fk+1|k+1 (b|Z (k+1) ) · Dk+1|k+1 (x|b, Z (k+1) ) (12.118) and where, by (12.93), Dk+1|k+1 (x|b, Z (k+1) ) Dk+1|k (x|b, Z (k) )

=

1 − pD (b, x) +

(12.119)

∑ pD (b, x) · fk+1 (z|b, x) . κk+1 (z) + τk+1 (z|b) z∈Zk+1

Ristic and Clark noted that, if b is static, then fk+1|k+1 (b|Z (k+1) ) can be derived directly rather than recursively. An application of Bayes’ rule yields fk+1|k+1 (b|Z (k+1) ) = ∫

fk+1 (Z (k+1) |b) · f0|0 (b) fk+1 (Z (k+1) |b′ ) · f0|0 (b′ )db′

(12.120)

where f0|0 (b) is the prior distribution on b and fk+1 (Z (k+1) |b) is the likelihood function for the variable b. A second application of Bayes’ rule shows that fk+1 (Z (k+1) |b)

=

=

fk+1 (Zk+1 |b, Z (k) ) · fk (Zk |b, Z (k−1) ) (12.121) · · · f2 (Z2 |b, Z (1) ) · f1 (Z1 |b) fk+1 (Zk+1 |b, Z (k) ) · fk (Z (k) |b). (12.122)

However, from (12.94) we know that—given that the predicted multitarget distributions are Poisson for every k ≥ 1—that, for all l = 1, ..., k + 1, fl (Zl |b, Z (l−1) ) = e−λl −Dl|l−1 [pD |b]

∏

(κl (z) + τl (z|b)) .

(12.123)

z∈Zl

Consequently, fk+1 (Z (k+1) |b) can be computed directly from the PHDs (l) Dl|l−1 (x|b, Z ) for l ≥ 1. Thus the measurement-updated posterior on b

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can be computed recursively as fk+1|k+1 (b|Z (k+1) )

∝

fk+1 (Z (k+1) |b) · f0|0 (b) (k)

(12.124) (12.125)

=

fk+1 (Zk+1 |b, Z ) ·fk (Z (k) |b) · f0|0 (b)

∝

fk+1 (Zk+1 |b, Z (k) ) · fk|k (b|Z (k) ). (12.126)

As time progresses, the BURT-PHD filter will implicitly estimate the value of b. If it is desired that this value be explicitly estimated, then this can be accomplished using the MAP estimator: bk+1|k+1 = arg sup fk+1|k+1 (b|Z (k+1) ).

(12.127)

b

12.5.2

A Heuristic Single-Filter BURT-PHD Filter

It is possible to approach the problem of devising a BURT-PHD filter in a na¨ıve fashion. Consider the single-sensor case first. We na¨ıvely apply the usual PHD filtering equations to the joint PHD Dk|k (b, x|Z (k) ): ∫ Dk+1|k (b, x) = bk+1|k (x) + pS (x′ ) (12.128) ·fk+1|k (b, x|b′ , x′ ) ·Dk|k (b′ , x′ )db′ dx′ and fk+1|k (b, x|b′ , x′ )

=

fk+1|k (b|b′ ) · fk+1|k (x|x′ )

(12.129)

Dk+1|k+1 (b, x) Dk+1|k (b, x)

=

1 − pD (b, x)

(12.130)

+

∑ pD (b, x) · fk+1 (z|b, x) κk+1 (z) + τk+1 (z) z∈Zk+1

τk+1 (z)

=

∫

pD (b, x) · fk+1 (z|b, x)

(12.131)

·Dk+1|k+1 (b, x)dbdx. For the multisensor case, in place of (12.130) we apply one of the multisensor PHD measurement-update equations described in Chapter 10.

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Strictly speaking, such an approach is not theoretically justifiable. This is because, according to (12.74), the joint Dk+1|k+1 (b, x|Z (k+1) ) actually has the form Dk+1|k+1 (b, x|Z (k+1) ) = fk+1|k+1 (b|Z (k+1) ) · Dk+1|k+1 (x|b, Z (k+1) ). (12.132) Here, and as the analysis leading up to that equation demonstrates, the PHD measurement-update equation can be applied only to Dk+1|k+1 (x|b, Z (k+1) ), not to Dk+1|k+1 (b, x|Z (k+1) ). The na¨ıve BURT-PHD filter was proposed independently in 2011 by Lian, Han, Liu, and Chen [145] and by Mahler [192]. The two approaches differ in two minor respects. First, to address multisensor biases, Lian et al. employed the product-pseudolikelihood approach of Section 10.6.3; whereas Mahler employed the iterated-corrector approach of Section 10.5 of the same chapter. Second, to estimate the multisensor bias, Lian et al. proposed the EAP estimator bEAP k+1|k+1

∫ b · Dk+1|k+1 (b, x|Z (k+1) )dbdx , = ∫ Dk+1|k+1 (b′ , x′ |Z (k+1) )db′ dx′

(12.133)

whereas Mahler proposed the MAP estimator, AP bM k+1|k+1

= arg sup

∫

Dk+1|k+1 (b, x|Z (k+1) )dx.

(12.134)

b

Lian et al. also implemented and tested their approach in the multisensor case. Surprisingly—given the heuristic nature of the na¨ıve approach and given its substantial deviation from the rigorous, two-filter approach—it exhibited good performance (see Section 12.6.2). It is unclear why this should be the case.

12.6

IMPLEMENTED BURT-PHD FILTERS

The purpose of this section is to report two implemented BURT-PHD filters: the filter of Ristic and Clark as described in Section 12.5.1, and the filter of Lian, Han, Liu, and Chen, as described in Section 12.5.2.

Joint Tracking and Sensor-Bias Estimation

12.6.1

377

The BURT-PHD Filter of Ristic and Clark

This filter was discussed in Section 12.5.1. The authors implemented and tested it using sequential Monte Carlo (SMC) techniques. The discussion in this section is drawn from [255]. The following two asynchronous sensors were considered: • Range-bearing sensor, collects at odd-numbered time-steps, is located at (0, 0)T , with covariance diag((50km)2 , (0.5o )2 ), probability of detection pD = 0.95, uniformly distributed Poisson clutter with clutter rate λ = 10, and static measurement translational bias (6.8km, −3.50o )T . • Range-bearing sensor, collects at even-numbered time-steps, is located at (120km, 35km)T , with variance diag((50km)2 , (0.5o )2 ), probability of detection pD = 0.95, uniformly distributed Poisson clutter with clutter rate λ = 10, and static measurement translational bias (−5km, 2o )T . Because the sensors collect measurements asynchronously, the single-sensor PHD filter can be applied (that is, no multisensor PHD filter is necessary). These sensors were used to observe three to five moving, appearing and disappearing targets. Because the primary purpose of the experiments was to determine the accuracy of the bias estimates, the track trajectories were very short. The authors report that their filter’s estimates of the biases were quite close to the actual values: (6.863km, −3.578o )T and (−5.104km, 1.806o )T , respectively. 12.6.2

The BURT-PHD Filter of Lian et al.

This filter was discussed in Section 12.5.2. In [145], the authors described the performance of a sequential Monte Carlo (SMC) implementation of it, in simple two-dimensional scenarios. The following three sensors were considered: • Range-bearing sensor, located at (600, 400)T , with covariance diag(2.5m)2 , (2.5mrad)2 ), probability of detection pD = 0.8, uniformly distributed Poisson clutter with clutter rate λ = 60, and static measurement translational bias (50m, −50mrad)T . • Range-only sensor, located at (0, 0)T , with variance (2.5m)2 , probability of detection pD = 0.9, uniformly distributed Poisson clutter with clutter rate λ = 50, and static measurement translational bias 30m.

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• Bearing-only sensor, located at (−600, −400)T , with variance (2.5mrad)2 , probability of detection pD = 0.7, uniformly distributed Poisson clutter with clutter rate λ = 40, and static measurement translational bias −40mrad. These sensors were used to observe up to four moving, appearing and disappearing targets moving along curvilinear trajectories. The authors reported that, after a settling-in period of about 15 time steps, their SMC-BURT-PHD filter converged to the correct estimates of the biases of the three sensors. It also essentially correctly estimated the number of targets at any given time—whereas a conventional SMC-PHD filter exhibited a significant downward bias. Finally, their filter accurately localized the target locations, with a significantly smaller error than the standard SMC-PHD filter, as measured using the OSPA metric (Section 6.2.2). Lian et al. also compared their filter with the multisensor JPDA filter, suitably generalized to address joint tracking and registration. The authors reported that the multisensor JPDA filter outperformed the SMC-BURT-PHD filter when the clutter rate was relatively low, but the reverse was true when the clutter rate was large.

Chapter 13 Multi-Bernoulli Filters 13.1

INTRODUCTION

Beyond the underlying modeling assumptions, the measurement-update equations for the classical PHD filter require the following simplifying assumption: • The predicted multitarget distribution fk+1|k (X|Z (k) ) is approximately Poisson, for every k ≥ 0. Likewise, the classical CPHD filter requires the following two simplifying assumptions: • The multitarget distributions fk|k (X|Z (k) ) approximately i.i.d.c. for every k ≥ 0.

and fk+1|k (X|Z (k) ) are

The main subject of this chapter, the cardinality-balanced multitarget multiBernoulli (CBMeMBer) filter, follows this pattern. It requires the following simplifying assumption: • The multitarget distributions fk|k (X|Z (k) ) and fk+1|k (X|Z (k) ) are approximately multi-Bernoulli for every k ≥ 0, where multi-Bernoulli RFSs were defined in Section 4.3.4. The CBMeMBer filter is conceptually different than the PHD and CPHD filters, however, in that the information in fk|k (X|Z (k) ) or fk+1|k (X|Z (k) ) is not compressed into summary statistical moments. Rather, an attempt is being made to:

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• Directly approximate fk|k (X|Z (k) ) and fk+1|k (X|Z (k) ) as full multitarget probability distributions. In certain circumstances, this approximation will not be a good one. Let pk|k (n|Z (k) ) be the cardinality distribution of fk|k (X|Z (k) ). If fk|k (X|Z (k) ) is multi-Bernoulli, then the variance of pk|k (n|Z (k) ) can never exceed the mean of pk|k (n|Z (k) )—see (4.133). Thus: • The CBMeMBer filter cannot be expected to perform well when the variance of pk|k (n|Z (k) ) exceeds its mean—that is, when target number is being poorly estimated. As was noted in Section 5.10.6, the derivation of the measurement-update equations for the original MeMBer filter made use of an ill-considered first-order Taylor’s linearization ([179], p. 681, Eq. (17.176)). This linearization caused a serious upward bias in the target number estimate [310]. To correct for this bias, Vo, Vo, and Cantoni [310] devised the CBMeMBer filter, which is the primary subject of this chapter. Vo, Vo, and their associates have extended the CBMeMBer filter to address unknown clutter and detection-profile backgrounds [312], [313]. This additional work will be described in Section 18.7.1 A secondary purpose of this chapter is to describe the Bernoulli filter, which was briefly introduced in Section 5.10.7. The CBMeMBer filter is in certain respects more general than the Bernoulli filter, in that the Bernoulli filter can track at most a single target. In other respects, however, the Bernoulli filter is more general than the CBMeMBer filter. Specifically, its clutter model can be arbitrary, whereas the clutter model for the CBMeMBer filter is presumed to be Poisson. The discussions of both the CBMeMBer filter and the Bernoulli filter in the following sections will be at a fairly high level. For details—especially in regard to implementation issues—see the book Particle Filters for Random Set Models by Ristic [250]. 13.1.1

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter: 1

Ouyang, Ji, and Li have pointed out a limitation of the CBMeMBer filter. They proposed a heuristic remedy for this problem, but it does not seem to be generally applicable [228].

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• The Bernoulli filter is—given its modeling assumptions—the Bayes-optimal approach for detecting and tracking at most a single target in an arbitrary clutter and detection background. See Section 13.2. • The CBMeMBer filter is conceptually different than the PHD and CPHD filters. Whereas the latter assume that the multitarget probability distribution fk|k (X|Z (k) ) can be approximated by statistical moments, the CBMeMBer filter is based on a direct approximation of fk|k (X|Z (k) ) itself. • This multi-Bernoulli approximation is not accurate when the CBMeMBer filter is not accurately estimating target number—specifically, when the variance of the cardinality distribution exceeds the mean of the cardinality distribution. • The instantaneous computational complexity of the CBMeMBer filter is roughly the same as that of the PHD filter: O(mn), where m is the current number of measurements and n is the current number of target tracks. However, n tends to increase with time, so that pruning and merging of tracks is required. • The performance of the Gaussian mixture (GM) implementation of the CBMeMBer filter is not appreciably better than that of the GM-CPHD filter. However, it is significantly more computationally efficient. • Both the computational efficiency and tracking performance of the sequential Monte Carlo (SMC) implementation of the CBMeMBer filter are appreciably better than that of the SMC-CPHD filter. • Consequently, the SMC-CBMeMBer filter is probably most appropriate for problems with larger pD , but with significant motion and/or measurement nonlinearities. • The CBMeMBer filter can be extended to incorporate multiple motion models, using jump-Markov techniques. See Section 13.5. 13.1.2

Organization of the Chapter

The chapter is organized as follows: 1. Section 13.2: The Bernoulli filter. 2. Section 13.3: The multisensor Bernoulli filter.

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3. Section 13.4: The cardinality-balanced multitarget multi-Bernoulli (CBMeMBer) filter. 4. Section 13.5: A jump-Markov version of the CBMeMBer filter.

13.2

THE BERNOULLI FILTER

Given its modeling assumptions, the Bernoulli filter is: • The Bayes-optimal approach for detection and tracking in a single-sensor scenario that is known to contain at most a single target, in an arbitrary clutter background and detection profile. Under certain modeling assumptions, the Bernoulli filter is just the multitarget Bayes filter, given that target number is known a priori to be 0 or 1—that is, when the initial multitarget distribution has the form f0|0 (X) = 0 if |X| ≥ 2. When clutter is i.i.d.c., the Bernoulli filter is identical to the single-target CPHD filter (see Section 8.5.6.5). The Bernoulli filter was independently proposed by B.-T. Vo in his doctoral dissertation [298], and by Mahler in [179], pp. 514-528. Vo’s terminology, “Bernoulli filter,” is more technically accurate and descriptive, and has also become the accepted usage. It is therefore adopted in place of Mahler’s usage in [179], “joint target-detection and tracking” (JoTT) filter. As was noted in [179], the Bernoulli filter is a generalization of the integrated probabilistic data association (IPDA) filter, derived by Musicki, Evans and Stankovic using a bottom-up methodology [215].2 Challa, Vo, and Wang subsequently demonstrated that, given the same modeling assumptions, the IPDA filter could be derived using the FISST methodology [32]. The Bernoulli filter consists of two coupled filters of the form ... →

pk|k (Z (k) )

... →

sk|k (x|Z (k) )

→ ↑↓ →

pk+1|k (Z (k) ) sk+1|k (x|Z (k) )

→ ↑↓ →

pk+1|k+1 (Z (k+1) )

→ ...

sk+1|k+1 (x|Z (k+1) )

→ ...

where pk|k is the probability that the target exists at time tk ; and where, if it does exist, sk|k (x) is its track distribution—that is, the probability (density) that it has 2

The Bernoulli filter generalizes the IPDA filter in that (1) target appearance is modeled; (2) probability of detection is state-dependent; and (3) the state-independent clutter process is arbitrary rather than Poisson.

Multi-Bernoulli Filters

state x. At any time-step, the Bernoulli filter is filter as follows:  1 − pk|k  pk|k · sk|k (x) fk|k (X|Z (k) ) =  0

383

related to the multitarget Bayes

if if if

X=∅ X = {x} . |X| ≥ 2

(13.1)

The purpose of this section is to summarize this filter. A detailed tutorial, by Ristic, Vo, Vo, and Farina, can be found in [262]. See also Ristic’s book, Particle Filters for Random Set Models [250]. The section is organized as follows: 1. Section 13.2.1: Modeling assumptions for the Bernoulli filter. 2. Section 13.2.2: Bernoulli filter time-update equations. 3. Section 13.2.3: Bernoulli filter measurement-update equations. 4. Section 13.2.4: State estimation for the Bernoulli filter. 5. Section 13.2.5: Error estimation for the Bernoulli filter. 6. Section 13.2.6: The Bernoulli filter is equivalent to an exact PHD filter. 7. Section 13.2.7: Implementing the Bernoulli filter. 8. Section 13.2.8: Algorithmic implementations of the Bernoulli filter. 13.2.1

Bernoulli Filter: Modeling

The following models are required for the Bernoulli filter: • Probability that, if it is in the scene, the target will not disappear if it has state x—pS (x) abbr. = pS,k+1|k (x). • Probability that, if it is not in the scene, the target will appear or reappear— pB abbr. = pB,k+1|k . • Spatial distribution of the target if it appears—ˆbk+1|k (x). Thus the PHD of the target-birth RFS is bk+1|k (x) = pB · ˆbk+1|k (x). • Single-target Markov density—Mx (x′ ) abbr. = fk+1|k (x|x′ ).

(13.2)

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• Single-sensor, single-target likelihood function—Lz (x) abbr. = fk+1 (z|x). • Multiobject probability distribution for an arbitrary clutter RFS—κk+1 (Z). 13.2.2

Bernoulli Filter: Time-Update

Suppose that we are given the prior probability of existence pk|k and the prior track distribution sk|k (x) at time tk . Then the time-update equations for the Bernoulli filter are ([179], p. 519): pk+1|k

=

sk+1|k (x)

=

pB · (1 − pk|k ) + pk|k · sk|k [pS ] pB · (1 − pk|k ) · ˆbk+1|k (x) + sk|k [pS Mx ]

(13.3) (13.4)

pk+1|k where sk|k [pS ] sk|k [pS Mx ] 13.2.3

=

∫

pS (x′ ) · sk|k (x′ )dx′

(13.5)

=

∫

pS (x′ ) · Mx (x′ ) · sk|k (x′ )dx′ .

(13.6)

Bernoulli Filter: Measurement Update

Suppose that we are given the predicted probability of existence pk+1|k and the predicted track distribution sk+1|k (x) at time tk+1 . Let Zk+1 be the newly collected measurement set. Then the measurement-update equations for the Bernoulli filter are ([179], p. 520): (13.7)

pk+1|k+1 ∑

(Zk+1 −{z}) 1 − sk+1|k [pD ] + z∈Zk+1 sk+1|k [pD Lz ] · κk+1 κk+1 (Zk+1 ) ∑ κk+1 (Zk+1 −{z}) p−1 z∈Zk+1 sk+1|k [pD Lz ] · k+1|k − sk+1|k [pD ] + κk+1 (Zk+1 )

=

and sk+1|k+1 (x) sk+1|k (x) =

∑ (Zk+1 −{z}) 1 − pD (x) + pD (x) z∈Zk+1 Lz (x) · κk+1 κk+1 (Zk+1 ) ∑ (Zk+1 −{z}) 1 − sk+1|k [pD ] + z∈Zk+1 sk+1|k [pD Lz ] · κk+1 κk+1 (Zk+1 )

(13.8)

Multi-Bernoulli Filters

385

where sk+1|k [pD ] sk+1|k [pD Lz ]

=

∫

pD (x) · sk+1|k (x)dx

=

∫

pD (x) · Lz (x) · sk+1|k (x)dx.

(13.9) (13.10)

When Zk+1 = ∅, the summations vanish by convention. That is, pk+1|k+1 sk+1|k+1 (x)

=

=

1 − sk+1|k [pD ] p−1 k+1|k

(13.11)

− sk+1|k [pD ]

1 − pD (x) · sk+1|k (x). 1 − sk+1|k [pD ]

(13.12)

Remark 57 In 2007, Vo, Vo, and Cantoni proposed a version of the Bernoulli filter that addresses state-dependent Poisson clutter [309]. The clutter process in this case has the specific form κk+1 (Z|x) = e−λk+1 (x)

∏

κk+1 (z|x)

(13.13)

zseZ

where κk+1∫(z|x) is the state-dependent clutter intensity function and where λk+1 (x) = κk+1 (z|x)dz is the state-dependent clutter rate. This filter was shown to significantly outperform conventional approaches such as the probabilistic data association (PDA) filter. 13.2.4

Bernoulli Filter: State Estimation

State estimation for the Bernoulli filter requires that the following two questions be answered: Is a target present? If so, what is its state? The answers to these questions involve the JoM or MaM multitarget state estimators of (5.9) and (5.10). In the case of the MaM estimator, a target can be declared to exist at time tk+1 if pk+1|k+1 > 1/2—or, equivalently, if ([179], Eq. (14.212)) 1 pk+1|k > 2 − sk+1|k [pD ] +

∑

. (13.14) z∈Zk+1

sk+1|k [pD Lz ] ·

κk+1 (Zk+1 −{z}) κk+1 (Zk+1 )

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Given this, the estimated state is3 x ˆMaM k+1|k+1 = arg sup sk+1|k+1 (x).

(13.15)

x

In the case of the JoM estimator, let c > 0 be a constant with the same units as x, which is equal in magnitude to the desired target-localization accuracy. Then a target is declared to exist if4 pk+1|k+1 < c · sup sk+1|k+1 (x).

(13.16)

x

The target state estimate is the same as for the MaM estimator. 13.2.5

Bernoulli Filter: Error Estimation

Error estimation requires two things: (1) an estimate of the error in the target number estimate, and (2) an estimate of the error in the target state estimate (if such exists). The former is given by the variance ([179], Eq. (14.229)): 2 σk+1|k+1 = pk+1|k+1 · (1 − pk+1|k+1 ).

(13.17)

The latter is given by the covariance of sk+1|k+1 (x) ([179], Eq. (14.232)): ∫ Pk+1|k+1 = (x − x ¯k+1|k+1 )(x − x ¯k+1|k+1 )T · sk+1|k+1 (x)dx (13.18) where x ¯k+1|k+1 is the JoM or MaM estimate of the target state. 13.2.6

The Bernoulli Filter as an Exact PHD Filter

From (4.75) we know that the PHD of the multitarget distribution fk|k (X|Z (k) ) in (13.1) is Dk|k (x) = pk|k · sk|k (x) (13.19) with expected number of targets Nk|k = 3 4

∫

Dk|k (x)dx = pk|k .

(13.20)

Erratum: There is a typo in the corresponding equation in [179]. Specifically, the factor fk+1|k (x) in Eq. (14.213) should be included within the arg supx in Eq. (14.214). Note that Eq. (14.215) in [179] is valid only if pD is constant.

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Thus when there is at most a single target, Dk|k (x) contains exactly the same information as the two Bernoulli-filter items pk|k and sk|k (x). Consequently, the Bernoulli filter is equivalent to a PHD filter ... →

Dk|k (x)

→

Dk+1|k (x)

→

Dk+1|k+1 (x)

→ ...

where, now, the measurement-update step is exact in the sense that it no longer requires the assumption that fk+1|k (X|Z (k) ) is Poisson. The time-update equation for this PHD filter is, because of the particular motion model assumed for the Bernoulli filter: ∫ Dk+1|k (x) = bk+1|k (x) + pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′ (13.21) where bk+1|k (x) = pB · (1 − pk|k ) · ˆbk+1|k (x).

(13.22)

The measurement-update equation is (see [171] and [179], pp. 631-632): Dk+1|k+1 (x) Dk+1|k (x) =

(13.23)

∑ (Zk+1 −{z}) 1 − pD (x) + pD (x) z∈Zk+1 Lz (x) · κk+1 κk+1 (Zk+1 ) ∑ (Zk+1 −{z}) 1 − Dk+1|k [pD ] + z∈Zk+1 Dk+1|k [pD Lz ] · κk+1 κk+1 (Zk+1 )

where

13.2.7

Dk+1|k [pD ]

=

∫

pD (x) · Dk+1|k (x)dx

(13.24)

Dk+1|k [pD Lz ]

=

∫

pD (x) · Lz (x) · Dk+1|k (x)dx.

(13.25)

Bernoulli Filter: Practical Implementation

The Bernoulli filter can be implemented in exact closed form using Gaussian mixture techniques. Alternatively, it can be implemented using sequential Monte Carlo techniques. For GM implementation, the track distribution is approximated as a Gaussian mixture: νk|k

sk|k (x) ∼ =

∑ i=1

k|k

wk|k · NP k|k (x − xi ) i

(13.26)

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∑νk|k where i=1 wk|k = 1. For SMC implementation, it is approximated as a Dirac mixture: νk|k ∑ ∼ sk|k (x) = wk|k · δxk|k (x). (13.27) i

i=1

For greater detail, see [303] or the book Particle Filters for Random Set Models by Ristic [250]. 13.2.8

Bernoulli Filter: Implementations

The tutorial [262] by Ristic, Vo, Vo, and Farina discusses several applications of the Bernoulli filter. The reader is referred there for details.

13.3

THE MULTISENSOR BERNOULLI FILTER

Since the Bernoulli filter is just a special case of the general multitarget Bayes filter, multiple independent sensors can be addressed using the iterated-corrector approach. For conceptual clarity, consider the two-sensor case. Suppose that measure1

2

ment sets Z k+1 and Z k+1 are collected by the two sensors at the same time tk+1 . Then (13.7) and (13.8) are applied to the first sensor: (13.28)

p˜k+1|k+1 1

1 − sk+1|k [pD ] +

∑

1 1 1

1

z∈Z k+1

1

1

sk+1|k [pD Lz1 ] ·

1

κk+1 (Z k+1 −{z}) 1

1

κk+1 (Z k+1 )

= 1

p−1 k+1|k − sk+1|k [pD ] +

∑

1

z∈Z k+1

1

1

1 1 1

sk+1|k [pD Lz1 ] ·

1

κk+1 (Z k+1 −{z}) 1

1

κk+1 (Z k+1 )

and s˜k+1|k+1 (x) sk+1|k (x) 1

(13.29) 1

1 − pD (x) + pD (x)

∑

1

z∈Z k+1

1

1

1 1

Lz1 (x) ·

1

κk+1 (Z k+1 −{z}) 1

1

κk+1 (Z k+1 )

=

. 1

1 − sk+1|k [pD ] +

∑

1 1 1

1

z∈Z k+1

sk+1|k [pD Lz1 ] ·

1

1

1

κk+1 (Z k+1 −{z}) 1

1

κk+1 (Z k+1 )

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Then (13.7) and (13.8) are applied for the second sensor, resulting in the two-sensor measurement-update: (13.30)

pk+1|k+1 2

1 − s˜k+1|k+1 [pD ] +

∑

2

s˜k+1|k+1 [pD Lz2 ] ·

2

2

z∈Z k+1

2

2

2

2

κk+1 (Z k+1 −{z}) 2

2

κk+1 (Z k+1 )

= 2

p˜−1 ˜k+1|k+1 [pD ] + k+1|k+1 − s

∑

2

z∈Z k+1

2

2

2 2 2

s˜k+1|k+1 [pD Lz2 ] ·

2

κk+1 (Z k+1 −{z}) 2

2

κk+1 (Z k+1 )

and sk+1|k+1 (x) s˜k+1|k+1 (x)

(13.31)

2

2

1 − pD (x) + pD (x)

∑

2

z∈Z k+1

2

2

2 2

Lz2 (x) ·

2

κk+1 (Z k+1 −{z}) 2

2

κk+1 (Z k+1 )

=

. 2

1 − s˜k+1|k+1 [pD ] +

∑

2 2 2

2

z∈Z k+1

s˜k+1|k+1 [pD Lz2 ] ·

2

2

2

κk+1 (Z k+1 −{z}) 2

2

κk+1 (Z k+1 )

An application of the multisensor Bernoulli filter, to detection and tracking of road-constrained targets using TDOA/FDOA (time difference of arrival/frequency difference of arrival) measurements, has been reported by B.-T. Vo, Chong Meng See, and Wee Teck Ng [303]. The authors implemented the multisensor Bernoulli filter in exact closed form, using the UKF variant of the Gaussian mixture (GM) approach. They also used the approach described in Section 9.5.6 to account for state-dependent probability of detection. Road segments were modeled as ellipses. The primary challenge of filtering using TDOA/FDOA measurements is the fact that the actual target is heavily obscured by a large number of “ghost targets.” These are due to the large number of bearing-only triangulations created by clutter. Especially when the clutter rate is large or the probability of detection is small, it can be essentially impossible to initialize the target. The authors employed two range-dependent TDOA/FDOA sensors, with the measurements of both corrupted by uniformly distributed Poisson clutter. The characteristics of these sensors were as follows: • First sensor: maximum probability of detection pD,max = 0.95 and clutter rate λ = 100. • Second sensor: maximum probability of detection pD,max = 0.75 and clutter rate λ = 10.

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The multisensor Bernoulli filter was tested against an appearing and disappearing target under two conditions: without road-constraint information, and with this information. As expected, the filter’s performance in the first case was very poor, largely because of the large clutter rate for the first sensor and the small probability of detection for the second sensor. With road information, however, performance was very good. The target’s appearances and disappearances were detected (with a small delay), and it was tracked accurately when it was present. The authors also found that, as expected, performance was better when both sensors were used rather than one.

13.4

THE CBMEMBER FILTER

As previously noted in Section 5.10.6, the CBMeMBer filter is based on the approximation of multitarget posterior distributions as multi-Bernoulli distributions in the sense of Section 4.3.4: νk|k

Gk|k [h|Z (k) ] ∼ =

∏

i i (1 − qk|k + qk|k · sik|k [h]).

(13.32)

i=1

Also as previously noted, this approximation is not fully general: if pk|k (n) is the cardinality distribution of a multi-Bernoulli RFS, then its variance is always smaller than its mean. The CBMeMBer filter time-update step is exact in the following sense. Suppose that both the target appearance process and the multitarget Markov density are both multi-Bernoulli. Then if fk|k (X|Z (k) ) is multi-Bernoulli, so is the predicted distribution fk+1|k (X|Z (k) ). The same is not true for the measurement-update step, however. If fk+1|k (X|Z (k) ) is multi-Bernoulli then fk+1|k+1 (X|Z (k+1) ) is usually not multi-Bernoulli. Thus one must determine a multi-Bernoulli distribution that approximates it. In this case one gets a multitarget filter that has the form ... →

fk|k (X|Z (k) )

→

fk+1|k (X|Z (k) )

→

fk+1|k+1 (X|Z (k+1) ) → ...

where fk+1|k (X|Z (k) ) is multi-Bernoulli if fk|k (X|Z (k) ) is multi-Bernoulli, and fk+1|k+1 (X|Z (k+1) ) is approximately multi-Bernoulli if fk+1|k (X|Z (k) ) is multi-Bernoulli. The CBMeMBer filter results from a particular choice of approximation for fk+1|k+1 (X|Z (k+1) ). The CBMeMBer filter propagates the multi-Bernoulli parameters rather than the multi-Bernoulli distributions. That is, it propagates a track table

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T0|0 → T1|0 → T1|1 → · · · → Tk|k → Tk+1|k → Tk+1|k+1 → · · · where, for a given value of k, Tk|k consists of a list of νk|k target tracks: ν

ν

ν

k|k k|k k|k 1 Tk|k = {(ℓ1k|k , qk|k , s1k|k (x)) , ..., (ℓk|k , qk|k , sk|k (x))}.

(13.33)

Here, • ℓik|k is the identifying label of the ith track at time tk . i • 0 < qk|k < 1 is the probability that the ith track is an actual target (that is, its probability of existence) at time tk .

• sik|k (x) is the probability distribution (track distribution) of the ith track at time tk . Thus the CBMeMBer filter has the following form: ν

...

→

k|k i {(ℓik|k , qk|k , sik|k (x))}i=1

→

k+1|k i {(ℓik+1|k , qk+1|k , sik+1|k (x))}i=1

→

k+1|k+1 i {(ℓik+1|k+1 , qk+1|k+1 , sik+1|k+1 (x))}i=1 → ...

ν

ν

Its computational complexity is O(mn), where m is the current number of measurements and n is the current number of tracks ([310], p. 414). However, n increases without bound over time, thus requiring the use of track-pruning and track-merging techniques. Remark 58 (“Spooky action at a distance”) The “spookiness” phenomenon in PHD and CPHD filters was noted in Section 9.2. Vo and Ma have noted that the CBMeMBer filter also exhibits this phenomenon, though to a significantly lesser extent [304]. Spookiness is probably a consequence of the approximations used to derive the measurement-update equations for the CBMeMBer filter. The purpose of this section is to describe the CBMeMBer filter. It is organized as follows: 1. Section 13.4.1: Modeling assumptions for the CBMeMBer filter. 2. Section 13.4.2: Time update equations for the CBMeMBer filter.

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3. Section 13.4.3: Measurement update equations for the CBMeMBer filter. 4. Section 13.4.4: Merging and pruning for the CBMeMBer filter. 5. Section 13.4.5: Multitarget state and error estimation for the CBMeMBer filter. 6. Section 13.4.6: More efficient track management for the CBMeMBer filter. 7. Section 13.4.7: Gaussian-mixture and particle implementation of the CBMeMBer filter. 8. Section 13.4.8: Practical implementations of the CBMeMBer filter. 13.4.1

CBMeMBer Filter: Modeling

The CBMeMBer filter requires the following models: • Target probability of survival: pS (x′ ) abbr. = pS,k+1|k (x′ ). • Single-target Markov density: Mx (x′ ) abbr. = fk+1|k (x|x′ ). • Target probability of detection: pD (x) abbr. = pD,k+1|k (x), assumed to be large. • Single-sensor, single-target likelihood function, Lz (x) abbr. = fk+1 (z|x). • Poisson clutter with clutter rate λk+1 and spatial distribution ck+1 (z), with λk+1 assumed to be not too large, where the clutter intensity function is (13.34)

κk+1 (z) = λk+1 · ck+1 (z). 13.4.2

CBMeMBer Filter: Predictor

The time-update for the CBMeMBer filter is the same as that for the original MeMBer filter ([179], pp. 661-662). Suppose that we are given the prior track table νk|k i Tk|k = {(ℓik|k , qk|k , sik|k (x))}i=1 . (13.35) We are to determine the time-updated track table ν

k+1|k i Tk+1|k = {(ℓik+1|k , qk+1|k , sik+1|k (x))}i=1 .

(13.36)

It consists of persisting tracks and birth (appearing) tracks persist birth Tk+1|k = Tk+1|k ∪ Tk+1|k

(13.37)

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where persist Tk+1|k birth Tk+1|k

ν

=

k|k {(ℓi , qi , si (x))}i=1

(13.38)

=

bk B B {(ℓB i , qi , si (x))}i=1

(13.39)

and where there are νk|k persisting tracks and bk appearing tracks. The appearing tracks are specified on the basis of whatever a priori knowledge we might have about the appearances of targets. The persisting tracks have the form, for i = 1, ..., νk|k ,

ℓi

=

ℓik|k

(13.40)

qi

=

i qk|k · sik|k [pS ]

(13.41)

=

sik|k [pS Mx ] sik|k [pS ]

(13.42)

=

∫

pS (x′ ) · sik|k (x′ )dx′

(13.43)

=

∫

pS (x′ ) · Mx (x′ ) · sik|k (x′ )dx′ .

(13.44)

si (x)

where sik|k [pS ] sik|k [pS Mx ]

Because of the birth tracks, the number of tracks will tend to increase with time: (13.45)

νk+1|k = νk|k + bk . 13.4.3

CBMeMBer Filter: Corrector

Suppose that we are given the predicted track table ν

k+1|k i Tk+1|k = {(ℓik+1|k , qk+1|k , sik+1|k (x))}i=1 .

(13.46)

and suppose that a new measurement set Zk+1 = {z1 , ..., zmk+1 } is collected with |Zk+1 | = mk+1 . We are to determine the form of the time-updated track table ν

k+1|k+1 i Tk+1|k+1 = {(ℓik+1|k+1 , qk+1|k+1 , sik+1|k+1 (x)) }i=1 .

(13.47)

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It consists of legacy tracks and measurement-updated tracks legacy meas Tk+1|k+1 = Tk+1|k+1 ∪ Tk+1|k+1

(13.48)

where ν

legacy Tk+1|k+1 meas Tk+1|k+1

=

k+1|k L L {(ℓL i , qi , si (x))}i=1

(13.49)

=

mk+1 U U {(ℓU j , qj , sj (x))}j=1

(13.50)

and where there are νk+1|k legacy tracks and mk+1 measurement-updated tracks. Thus the total number of tracks is νk+1|k+1 = νk+1|k + mk+1 . Because of the measurement-updated tracks, the number of tracks will tend to increase with time. Given this, the measurement-update equations for the CBMeMBer filter are as follows: • Corrector equations for legacy tracks for i = 1, ..., νk+1|k ([310], Eqs. (14,15)): ℓL i

=

ℓik+1|k

qiL

=

i qk+1|k ·

sL i (x)

=

sik+1|k (x) ·

(13.51) 1 − sik+1|k [pD ] i 1 − qk+1|k · sik+1|k [pD ]

1 − pD (x) . 1 − sik+1|k [pD ]

(13.52) (13.53)

• Corrector equations for measurement-updated tracks for i = 1, ..., νk+1|k and j = 1, ..., mk+1 ([310], Eqs. (27,38)): ℓU j

=

ℓ∗k+1|k ∑νk+1|k i=1

qjU

(13.54) i i qk+1|k (1−qk+1|k )·sik+1|k [pD Lzj ] i (1−qk+1|k ·sik+1|k [pD ])2

= κk+1 (zj ) +

sU j (x)

∑ =

∑νk+1|k i=1

i νk+1|k qk+1|k i i=1 1−qk+1|k

∑νk+1|k i=1

i qk+1|k ·sik+1|k [pD Lzj ]

(13.55)

i 1−qk+1|k ·sik+1|k [pD ]

· sik+1|k (x) · pD (x) · Lzj (x)

i qk+1|k i 1−qk+1|k

(13.56) · sik+1|k [pD Lzj ]

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where sik+1|k [pD ] sik+1|k [pD Lzj ]

=

∫

pD (x) · sik+1|k (x)dx

(13.57)

=

∫

pD (x) · Lzj (x) · sik+1|k (x)dx

(13.58)

and where ℓ∗k+1|k is the label of the predicted track that has the largest contribution to the current measurement-updated probability of existence qjU in (13.55) ([310], p. 414). 13.4.4

CBMeMBer Filter: Merging and Pruning

As time progresses, the number of tracks will increase without bound, and so merging and pruning will be necessary. Techniques similar to those for Gaussian mixture implementation can be employed to reduce the number of tracks ([179], pp. 665-666). Suppose that two tracks ℓi , qi , si (x) and ℓj , qj , sj (x) are such that qi +qj < 1. Then they are eligible for merging if the association density ∫ pi,j = si (x) · sj (x)dx (13.59) exceeds some threshold. In this case the merged track is ℓ, q, s(x) where ℓ

=

ℓ∗

(13.60)

q

=

(13.61)

s(x)

=

qi + qj si (x) · sj (x) pi,j

(13.62)

where ℓ∗ is the label of the track that has largest probability of existence: ℓ∗ = ℓi if qi > qj . Once tracks with small probabilities of existence have been discarded, the remaining tracks are merged in order to keep within memory and computational limits. 13.4.5

CBMeMBer Filter: State and Error Estimation

Following merging and pruning, Vo, Vo, and Cantoni proposed two general approaches for estimating the number and states of the targets ([310], p.414).

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• Method 1: Choose a target-detection threshold τ , and then select those tracks i that have the largest existence probabilities: qk+1|k+1 > τ . Then select the means or modes of the corresponding distributions sk+1|k+1 (x). • Method 2: The cardinality distribution is given by (4.130):

=

(13.63)

pk+1|k+1 (n) (νk+1|k+1 ) ∏ i (1 − qk+1|k+1 ) i=1

·σνk+1|k+1 ,n

(

ν

k+1|k+1 qk+1|k+1

1 qk+1|k+1 1 qk+1|k+1

, ...,

ν

)

.

k+1|k+1 1 − qk+1|k+1

Estimate the number of targets by determining the MAP estimate n ˆ k+1|k+1 = arg sup pk+1|k+1 (n).

(13.64)

n≥0

Then select those n ˆ k+1|k+1 tracks that have the largest probabilities of existence. Finally, select the means or modes of the corresponding track distributions sk+1|k+1 (x). 13.4.6

CBMeMBer Filter: Track Management

The track management scheme described in (13.40), (13.51), (13.54), and (13.60) has the advantage of being simple to implement. However, it has been noted by Wong, Vo, and Vo that this approach is not effective when targets intersect or are otherwise closely-spaced [325]. This is because tracks in close proximity will inevitably fall within the track-merging threshold. To address this problem, these authors proposed a more sophisticated track management approach based on the method of Shafique and Shah [269].5 In brief, the approach has three stages: 1. Stage 1: Search for possible associations between the current estimate and the estimates in previous time-steps. 5

Strictly speaking, in [325] Wong et al. did not propose the method of Shafique and Shah for use with the CBMeMBer filter. Rather, this method was applied to the IO-MeMBer filter of Section 20.5. However, it applies equally well to the CBMeMBer filter.

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2. Stage 2: Previously missed estimates are considered for association with those current estimates which have not yet been associated. 3. Stage 3: Current estimates that have not yet been associated are considered to be possible newly appearing targets, and are assigned new labels. 4. Stage 4: If an old estimate is not associated after a sufficiently large number of times, then it is considered to be a disappearing target and is eliminated. 13.4.7

CBMeMBer Filter: Gaussian-Mixture and Particle Implementation

Gaussian mixture and sequential Monte Carlo implementation of the CBMeMBer filter are described in [310], pp. 414-417. For more details, the reader is directed there. As usual, for GM implementation one must assume that the probability of detection and probability of target survival are constant: pD (x) = pD and pS (x′ ) = pS . 13.4.8

CBMeMBer Filter: Performance

Vo, Vo, and their associates have implemented and tested the CBMeMBer filter in a number of applications. In this section, discussion is limited to two such implementations: the baseline implementations described by Vo, Vo, and Cantoni in their original paper; and an application for tracking using audio and visual data. The section also includes a summary of a series of papers by Zhang et al., in which the CBMeMBer filter is applied to multitarget detection and tracking using managed sensor networks. 13.4.8.1

CBMeMBer Filter: Baseline Simulations

Vo, Vo, and Cantoni implemented the CBMeMBer filter using both Gaussian mixture and particle techniques, with the following results. Gaussian mixture implementation ([310], pp. 420-421). The GM-CBMeMBer filter was tested in a scenario involving a single linear-Gaussian sensor, observing 10 targets that appear and disappear along linear trajectories. In a uniform clutter background with clutter rate 10, the filter was able to correctly detect and track the targets, including a simultaneous crossing of three targets at roughly mid-scenario. The CBMeMBer filter’s average localization accuracy was the same as that of the GM-PHD filter (23m), but worse than that of the GM-CPHD filter (17m). When

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the clutter rate was increased to 50, the GM-CBMeMBer filter evidenced a slight upward bias in the target-number estimate (whereas the GM-PHD and GM-CPHD filters remained unbiased). The authors noted that the GM-CBMeMBer filter performed well up to a clutter rate of 20, and down to a probability of detection of 0.90. Particle implementation ([310], pp. 417-419): In this case, the SMCCBMeMBer filter was tested in a scenario involving a single range-bearing sensor, with 10 targets that appear and disappear along curvilinear trajectories. A coordinated-turn motion model was assumed. In a high-SNR scenario with clutter rate 10, the SMC-CBMeMBer filter was able to successfully detect and track the targets, including at simultaneous target crossings. The filter had a better average localization performance (50m) than either the SMC-CPHD filter (60m) or SMCPHD filter (70m). This is probably due to the difficulty of multitarget state estimation for SMCPHD and SMC-CPHD filters, compared to the simplicity of estimation for the SMC-CBMeMBer filter. This relationship held true for larger clutter rates—though, once again, the SMC-CBMeMBer filter exhibited a slight upward bias in the targetnumber estimate for clutter rates exceeding 20. Overall Evaluation: On the basis of their experiments, Vo et al. concluded that the CBMeMBer filter is best suited for problems with the following characteristics: • State-dependent probability of detection. • Nonlinearities extreme enough to require particle implementation. 13.4.8.2

CBMeMBer Filter: Audio-Visual Tracking

In [114], Hoseinnezhad, Vo, Vo, and Suter applied the CBMeMBer filter to the problem of tracking people using a video camera equipped with two side-mounted microphones. In this problem, the targets are not always audible, and—because of occlusions or crossings—not always visible. Targets were modeled as moving rectangular templates with unknown widths and heights. Time difference of arrival (TDOA) techniques were used to process the microphone measurements. Kernelbased background-subtraction techniques and morphological techniques were used to process the video data. Also, an “active speaker” model was employed, in which probability of detection is set high for video data (0.95, because targets are almost always visible) but low for audio (0.40, because targets are usually nonspeaking).

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The approach was successfully tested against real audio-video data involving two people. In part, good performance was due to the complementary nature of the two data sources. Nonspeaking targets could be tracked visually and obscured targets could be tracked using audio. 13.4.8.3

CBMeMBer Filter: Tracking Using Managed Sensor Networks

In a series of papers, Zhang and his associates have applied the CBMeMBer filter to the problem of detecting and tracking multiple targets using a network of managed sensors [327], [328], [122], [120], [123], [121], [124]. The authors assume that the sensors are organized into sensor-clusters, each managed by its own “cluster head” (CH). A CH is activated if its sensors are capable of efficiently sensing at least some targets. within each cluster, those sensors with better information about the targets—as determined using an RFS-based sensor management objective function—transmit their measurements to the CH. The CH then sequentially processes its local information using a CBMeMBer filter. Since this approach employs the sensor management approach described in Part V, further discussion is deferred until Section 26.6.3.1.

13.5

JUMP-MARKOV CBMEMBER FILTER

Dunne and Kirubarajan have extended the CBMeMBer filter to more effectively track rapidly maneuvering targets [65], using the jump-Markov techniques described in Chapter 11 [66].6 The filtering equations for the JM-CBMeMBer filter are essentially identical to (13.40) through (13.58), with a few minor alterations. The main differences are that the track distributions sik|k (x) now have the form sik|k (o, x), and that the integral on the augmented target state (o, x) has the form ∑ ∫ f (o, x)dx. o 13.5.1

Jump-Markov CBMeMBer Filter: Modeling

The jump-Markov CBMeMBer filter requires the following models: • Target probability of survival: pS (o′ , x′ ) abbr. = pS,k+1|k (o′ , x′ ). 6

Jin-Long Yang, Hong-Bing Ji, and Hong-Wei Ge proposed a jump-Markov CBMeMBer filter in 2012 [119], based on generalization of the interacting multiple model (IMM) technique. This work came to my attention too late for consideration in this book.

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• Single-target Markov density: Mo,x (o′ , x′ )

= =

fk+1|k (o, x|o′ , x′ ) χo,o′ · fk+1|k (x|o′ , x′ ),

(13.65) (13.66)

where χo,o′ is the mode transition matrix. • Target probability of detection: pD (o, x) abbr. = pD,k+1|k (o, x), assumed to be large. • Single-sensor, single-target likelihood function, Lz (o, x) abbr. = fk+1 (z|o, x). • Poisson clutter with clutter rate λk+1 and spatial distribution ck+1 (z), with λk+1 assumed to be not too large, where the clutter intensity function is (13.67)

κk+1 (z) = λk+1 · ck+1 (z).

Remark 59 (Gaussian mixture implementation) Gaussian mixture (GM) implementation of the jump-Markov CBMeMBer filter is similar to GM implementation of the jump-Markov PHD and CPHD filters, as described in Section 11.7.1. The probability of detection and probability of target survival must be assumed to have functional dependence only on the jump variable: pD (o, x) = poD and ′ pS (o′ , x′ ) = poS . 13.5.2

Jump-Markov CBMeMBer Filter: Predictor

We are given the prior track table ν

k|k i Tk|k = {(ℓik|k , qk|k , sik|k (o, x))}i=1 .

(13.68)

We are to determine the form of the time-updated track table ν

k+1|k i Tk+1|k = {(ℓik+1|k , qk+1|k , sik+1|k (o, x))}i=1

(13.69)

consisting of persisting tracks and appearing tracks persist birth Tk+1|k = Tk+1|k ∪ Tk+1|k

(13.70)

where ν

persist Tk+1|k

=

k|k {(ℓi , qi , si (o, x))}i=1

(13.71)

birth Tk+1|k

=

bk B B {(ℓB i , qi , si (o, x))}i=1

(13.72)

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and where there are νk|k persisting tracks and bk appearing tracks. The persisting tracks have the form, for i = 1, ..., νk|k ,

=

ℓik|k

(13.73)

qi

=

(13.74)

si (o, x)

=

i qk|k · sik|k [pS ] sik|k [pS Mo,x ] sik|k [pS ]

ℓi

(13.75)

where sik|k [pS ]

=

∑∫

pS (o′ , x′ ) · sik|k (o′ , x′ )dx′

(13.76)

o′

sik|k [pS Mo,x ]

=

∑∫

pS (o′ , x′ ) · Mo,x (o′ , x′ ) · sik|k (o′ , x′ )dx′ .(13.77)

o′

13.5.3

Jump-Markov CBMeMBer Filter: Corrector

Suppose that we are given the predicted track table ν

k+1|k i Tk+1|k = {(ℓik+1|k , qk+1|k , sik+1|k (o, x))}i=1 .

(13.78)

Suppose that a new measurement set Zk+1 = {z1 , ..., zmk+1 } is collected with |Zk+1 | = mk+1 . We are to determine the form of the time-updated track table ν

k+1|k+1 i Tk+1|k+1 = {(ℓik+1|k+1 , qk+1|k+1 , sik+1|k+1 (o, x)) }i=1 .

(13.79)

It consists of legacy tracks and measurement-updated tracks legacy meas Tk+1|k+1 = Tk+1|k+1 ∪ Tk+1|k+1

(13.80)

where legacy Tk+1|k+1 meas Tk+1|k+1

ν

=

k+1|k L L {(ℓL i , qi , si (o, x))}i=1

(13.81)

=

mk+1 U U {(ℓU j , qj , sj (o, x))}j=1

(13.82)

and where there are νk+1|k legacy tracks and mk+1 measurement-updated tracks. Thus the total number of tracks is νk+1|k+1 = νk+1|k + mk+1 . Given this, the measurement-update equations for the CBMeMBer filter are as follows ([310], Eqs. (14,15,27,38)).

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• Corrector equations for legacy tracks for i = 1, ..., νk+1|k ([310], Eqs. (14,15)): ℓL i

=

qiL

=

sL i (o, x)

=

ℓik+1|k

(13.83)

i qk+1|k ·

1

1 − sik+1|k [pD ] i − qk+1|k · sik+1|k [pD ]

sik+1|k (o, x) ·

1 − pD (o, x) . 1 − sik+1|k [pD ]

(13.84) (13.85)

• Corrector equations for measurement-updated tracks for i = 1, ..., νk+1|k and j = 1, ..., mk+1 ([310], Eqs. (27,38)): ℓU j

=

ℓ∗k+1|k ∑νk+1|k

(13.86) i i qk+1|k (1−qk+1|k )·sik+1|k [pD Lzj ] i (1−qk+1|k ·sik+1|k [pD ])2

i=1

qjU

= κk+1 (zj ) +

∑νk+1|k i=1

i qk+1|k ·sik+1|k [pD Lzj ]

(13.87)

i 1−qk+1|k ·sik+1|k [pD ]

and sU j (o, x) ∑νk+1|k i=1

=

(13.88) i qk+1|k i 1−qk+1|k

· sik+1|k (o, x) · pD (o, x) · Lzj (o, x)

∑νk+1|k i=1

i qk+1|k i 1−qk+1|k

· sik+1|k [pD Lzj ]

where sik+1|k [pD ]

=

∑∫

pD (o, x) · sik+1|k (o, x)dx

(13.89)

o

sik+1|k [pD Lzj ]

=

∑∫

pD (o, x) · o ·sik+1|k (o, x)dx

Lzj (o, x)

(13.90)

and where ℓ∗k+1|k is the label of the predicted track that has the largest contribution to the current measurement-updated probability of existence in (13.55) ([310], p. 414).

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Jump-Markov CBMeMBer Filter: Performance

In [65], [66], Dunne et al. reported performance results for both particle (SMC) and Gaussian mixture (GM) implementations of the JM-CBMeMBer filter. In a two-dimensional scenario, three motion models were assumed: a single constantvelocity (CV) model and two constant turn rate (CT) models, one for right-hand turns and one for left-hand turns. The scenario contained four targets with different trajectories: straight-line, sinusoidal, elliptical, and “∝”-shaped. They were observed by a linear-Gaussian sensor with uniform Poisson clutter (clutter rate λ = 10) and probability of detection pD = 0.95. The performances of four filters—SMC-CBMeMBer, GM-CBMeMBer, SMC-JM-CBMeMBer, and GM-JMCBMeMBer—were compared, using the OSPA distance (see Section 6.2.2) and the estimates of target number. The authors report that all four filters estimated target number reasonably well with the GM-JM-CBMemBer filter performing best, followed by the GMCBMemBer filter, the SMC-JM-CBMeMBer filter, and the SMC-CBMeMBer filter. Similar results were reported for overall performance, as measured using the OSPA metric (Section 6.2.2). The authors also assessed the ability of the two jump-Markov CBMeMBer filters to estimate the currently correct target motion model. Both filters proved to be quite capable in this respect.

Chapter 14 RFS Multitarget Smoothers 14.1

INTRODUCTION

Let us be given a single sensor that observes a single target with state x, with no missed detections or clutter. Given a time sequence Z k : z1 , ..., zk of measurements, recall that the single-target recursive Bayes filter propagates the measurement-updated probability density fk+1|k+1 (x|Z k+1 ) =

fk+1 (zk+1 |x, Z k ) · fk+1|k (x|Z k ) fk+1 (zk+1 |Z k )

(14.1)

where fk+1|k (x|Z k ) fk+1 (zk+1 |Z k )

=

∫

fk+1|k (x|x′ , Z k ) · fk|k (x′ |Z k )dx′

(14.2)

=

∫

fk+1 (zk+1 |x, Z k ) · fk+1|k (x|Z k )dx

(14.3)

and where it is usually assumed that fk+1|k (x|x′ , Z k ) k

fk+1 (zk+1 |x, Z )

=

fk+1|k (x|x′ )

(14.4)

=

fk+1 (zk+1 |x).

(14.5)

At each step k of the recursion, a Bayes-optimal state estimator is used to construct an estimate of x from fk|k (x|Z k ). However, the Bayes filter is not the only way to exploit the information in Z k to arrive at an estimate of the target’s trajectory. It is also possible to exploit

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the entire time history Z k to arrive at more accurate track estimates at all of the preceding time-steps ℓ = 0, 1, ..., k. A Bayes smoother is an algorithm that computes the probability distributions fℓ|k (x|Z k )

(14.6)

for ℓ = 0, ..., k, and then uses a Bayes-optimal state estimator to compute a smoothed estimate of x at time tℓ from fℓ|k (x|Z k ). That is, it employs the entire measurement-stream Z k to determine the best estimate of x at each of the intermediate times tℓ . While this requires off-line batch processing rather than real-time processing, it is useful for track reconstruction. Various Bayes smoothers have been proposed. The two most common are the forward-backward smoother and the two-filter smoother. The obvious multitarget generalization of a Bayes smoother would be an algorithm that computes the multitarget distributions fℓ|k (X|Z (k) )

(14.7)

for ℓ = 0, ..., k, and which then uses a Bayes-optimal multitarget state estimator to compute a smoothed estimate of X at time tℓ from the distribution fℓ|k (X|Z (k) ). The purpose of this chapter is to consider the following multitarget generalizations of the forward-backward smoother: • The general multitarget forward-backward smoother. • A special case of the general smoother, in which target number is assumed a priori to be at most 1: the Bernoulli forward-backward smoother. • A Poisson approximation of the general multitarget smoother: the PHD forward-backward smoother. • An i.i.d.c. approximation of the general multitarget smoother, assuming that there are no target appearances: the zero target appearances (ZTA) CPHD smoother. (This is the smoother analog of the ZFA-CPHD filter of Section 8.6.) 14.1.1

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter:

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• The single-target forward-backward smoother can be solved in exact closed form using Gaussian mixture methods (Section 14.2.3). • Finite-set statistics methods lead to principled multitarget smoothers. • In particular, the general multitarget forward-backward smoother provides a Bayes-optimal solution to the multitarget smoothing problem (Section 14.3). This smoother is given by (see (14.58))

fℓ|k (X|Z

(k)

) = fℓ|ℓ (X|Z

(ℓ)

)

∫

fℓ+1|k (Y |Z (k) ) · fℓ+1|ℓ (Y |X) δY. (14.8) fℓ+1|ℓ (Y |Z (ℓ) )

• A special case of (14.8)—the Bernoulli forward-backward smoother—optimally and tractably addresses the single-target detection and smoothing problem in clutter with missed detections, and performs successfully when implemented using either particle or Gaussian mixture methods (Section 14.4). • The forward-backward smoothed PHDs Dℓ|k (x|Z (k) ) are the PHDs of the smoothed multitarget distributions fℓ|k (X|Z (k) ). Consequently, the rigorous approach for determining the Dℓ|k (x|Z (k) ) is to determine the formulas for the PHDs of the right side of (14.8). These are (see (14.96)): Dℓ|k (x|Z (k) ) Dℓ|ℓ (x|Z (ℓ) )

ℓ+1|ℓ

=

1 − pS

(x)

ℓ+1|ℓ

+pS

(x)

∫

(14.9) fℓ+1|ℓ (y|x) · Dℓ+1|k (y|Z (k) ) dy. Dℓ+1|ℓ (y|Z (ℓ) )

• PHD forward-backward smoothers have somewhat (about 30%) better target localization accuracy than PHD filters, but tend to be adversely affected by missed detections or target disappearances (Section 14.5). • A fast particle implementation of the PHD forward-backward smoother, based on target labeling, can address significantly large numbers of targets in significantly dense clutter (Section 14.5.3). • A generalization of the PHD forward-backward smoother to the CPHD filter case does not appear to be possible for computational reasons. However, such a smoother is possible under the simplifying assumption that target appearances are negligible (Section 14.6).

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14.1.2

Organization of the Chapter

The chapter is organized as follows: 1. Section 14.2: The single-sensor, single-target forward-backward smoother, including closed-form Gaussian mixture implementation. 2. Section 14.3: smoother.

The general single-sensor, multitarget forward-backward

3. Section 14.4: The Bernoulli forward-backward smoother, for the case when target number is known a priori to be no larger than 1. 4. Section 14.5: The PHD forward-backward smoother. 5. Section 14.6: The zero target appearance (ZTA) CPHD forward-backward smoother—a CPHD smoother, assuming that target appearances are negligible.

14.2

SINGLE-TARGET FORWARD-BACKWARD SMOOTHER

This smoother is defined by the equation [2]: fℓ|k (x|Z k ) = fℓ|ℓ (x|Z ℓ )

∫

fℓ+1|ℓ (y|x) · fℓ+1|k (y|Z k ) dy. fℓ+1|ℓ (y|Z ℓ )

(14.10)

It is employed using the following three steps: 1. Forward recursion: Use the recursive Bayes filter and the initial distribution f0|0 (x) to compute the distributions fℓ+1|ℓ (y|Z l ) for ℓ = 0, ..., k − 1 and fℓ|ℓ (y|Z ℓ ) for ℓ = 1, ..., k. 2. Backward recursion: Starting with ℓ = k − 1, computed as k

fk−1|k (x|Z ) = fk−1|k−1 (x|Z

k−1

)

∫

fk−1|k (x|Z k ) can be

fk|k (y|Z k ) · fk|k−1 (y|x) dy. fk|k−1 (y|Z k−1 ) (14.11)

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Given fk−1|k (x|Z k ), fk−2|k (x|Z k ) can be computed as ∫

fk−1|k (y|Z k ) · fk−1|k−2 (y|x) dy. fk−1|k−2 (y|Z k−2 ) (14.12) Continuing in this fashion we eventually get to a formula for f1|k (y|Z k ), from which we compute f0|k (y|Z k ) as fk−2|k (x|Z k ) = fk−2|k−2 (x|Z k−2 )

k

f0|k (x|Z ) = f0|0 (x)

∫

f1|k (y|Z k ) · f1|0 (y|x) dy. f1|0 (y)

(14.13)

3. State estimation: At each stage of the backward recursion, apply a Bayesoptimal state estimator to fℓ|k (x|Z k ) to get smoothed estimates of x at times tℓ = tk , tk−1 , ..., t0 . The section is organized as follows: • Section 14.2.1: Derivation of the single-target forward-backward smoother. • Section 14.2.2: The Vo-Vo alternative formulation of the forward-backward smoother. • Section 14.2.3: The Vo-Vo exact closed-form Gaussian mixture solution of the forward-backward filter. 14.2.1

Derivation of Forward-Backward Smoother

Let y be the target state at time tℓ+1 , in which case the total probability theorem and Bayes’ rule gives us ∫ k fℓ|k (x|Z ) = fℓ,ℓ+1|k (x, y|Z k )dy (14.14) ∫ = fℓ+1|k (y|Z k ) · fℓ|ℓ+1,k (x|y, Z k )dy. (14.15) The density fℓ|ℓ+1,k (x|y, Z k ) defines the backward state transition from y at time tℓ+1 to x at time tℓ .1 Assume that in this transition, x is independent of 1

The notation fℓ,ℓ+1|k (x, y|Z k ) indicates that x is the target state at time tℓ , y is the target state at time tℓ+1 , and that x, y are conditioned on the measurements through time tk ; and similarly for fℓ|ℓ+1,k (x|y, Z k ).

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those measurements zℓ+1 , ..., zk that are in its future: fℓ|ℓ+1,k (x|y, Z k ) = fℓ|ℓ+1,ℓ (x|y, Z ℓ ).

(14.16)

Then: k

fℓ|k (x|Z )

= =

∫

fℓ+1|k (y|Z k ) · fℓ|ℓ+1,ℓ (x|y, Z ℓ )dy (14.17) ∫ fℓ|ℓ+1,ℓ (x|y, Z ℓ ) fℓ|ℓ (x|Z ℓ ) fℓ+1|k (y|Z k ) · dy.(14.18) fℓ|ℓ (x|Z ℓ )

Bayes’ rule then yields fℓ|k (x|Z k ) = fℓ|ℓ (x|Z ℓ )

∫

fℓ+1|k (y|Z k ) · fℓ+1|ℓ,ℓ (y|x, Z ℓ ) dy fℓ+1|ℓ (y|Z ℓ )

(14.19)

where fℓ+1|ℓ,ℓ (y|x, Z ℓ ) defines the usual forward state transition from x to y. Further assume that y does not depend on Z ℓ —that is, fℓ+1|ℓ,ℓ (y|x, Z ℓ ) = fℓ+1|ℓ (y|x). Then, as claimed, fℓ|k (x|Z k ) = fℓ|ℓ (x|Z ℓ ) 14.2.2

∫

fℓ+1|k (y|Z k ) · fℓ+1|ℓ (y|x) dy. fℓ+1|ℓ (y|Z ℓ )

(14.20)

Vo-Vo Alternative Form of the Forward-Backward Smoother

Because of the denominator of the quotient in the integral on the right side of (14.20), it would appear to be impossible to implement (14.10) exactly using Gaussian mixture techniques. However, Vo and Vo have shown [302] that the forward-backward smoother can be reformulated in such a manner as to permit exact closed-form GM implementation. In fact, their approach appears to be: • The first general exact closed-form Gaussian mixture solution of the singletarget forward-backward smoother. Specifically, for ℓ = 0, ..., k − 1, define the unitless function (the backward corrector) ∫ fℓ+1|k (y|Z k ) Bℓ|k (x) = · fℓ+1|ℓ (y|x)dy (14.21) fℓ+1|ℓ (y|Z ℓ )

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and note that Bℓ|ℓ (x) = 1. Also, for ℓ = 1, ..., k define the unitless function Lℓ (zℓ |x) =

fℓ (zℓ |x) . fℓ (zℓ |Z ℓ−1 )

(14.22)

Then (14.10) can be equivalently replaced by the following equations, for ℓ = 0, ..., k − 1 ([302], Eqs. (15-18)): fℓ|k (x|Z k )

=

Bℓ|k (x)

=

fℓ|ℓ (x|Z ℓ ) · Bℓ|k (x) (14.23) ∫ Bℓ+1|k (y) · Lℓ+1 (zℓ+1 |y) · fℓ+1|ℓ (y|x)dy. (14.24)

To see why, note that (14.10) becomes fℓ|k (x|Z k ) = fℓ|ℓ (x|Z ℓ ) · Bℓ|k (x).

(14.25)

From this follows fℓ|k (x|Z k ) fℓ|ℓ−1 (x|Z ℓ−1 )

= = =

fℓ|ℓ (x|Z ℓ ) · Bℓ|k (x) fℓ|ℓ−1 (x|Z ℓ−1 ) fℓ (zℓ |x) · Bℓ|k (x) fℓ (zℓ |Z ℓ−1 ) Lℓ (zℓ |x) · Bℓ|k (x)

(14.26) (14.27) (14.28)

and thus, as claimed,

Bℓ−1|k (x)

=

∫

=

∫

fℓ|k (y|Z k ) · fℓ|ℓ−1 (y|x)dy fℓ|ℓ−1 (y|Z ℓ−1 )

(14.29)

Lℓ (zℓ |y) · Bℓ|k (y) · fℓ|ℓ−1 (y|x)dy.

(14.30)

The Vo-Vo alternative forward-backward smoother is employed using the following three steps: 1. Forward recursion: Use the recursive Bayes filter and the initial distribution f0|0 (x) to compute the distributions fℓ+1|l (y|Z ℓ ) for ℓ = 0, ..., k − 1 and fℓ|ℓ (y|Z ℓ ) for ℓ = 1, ..., k.

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2. Backward recursion: Starting with ℓ = k−1, fk−1|k (x|Z k ) and Bk−1|k (x) can be computed as Bk−1|k (x)

=

∫

Lk (zk |y) · fk|k−1 (y|x)dy

(14.31)

fk−1|k (x|Z k )

=

fk−1|k−1 (x|Z k−1 ) · Bk−1|k (x).

(14.32)

=

∫

(14.33)

Then for ℓ = k − 2, Bk−2|k (x)

Bk−1|k (y) · Lk−1 (zk−1 |y)

·fk−1|k−2 (y|x)dy fk−2|k (x|Z k )

fk−2|k−2 (x|Z k−2 ) · Bk−2|k (x).

=

(14.34)

Continuing in this fashion we eventually get, with ℓ = 0, B0|k (x)

=

∫

f0|k (x|Z k )

=

f0|0 (x) · B0|k (x).

B1|k (y) · L1 (z1 |y) · f1|0 (y|x)dy

(14.35) (14.36)

3. State estimation: At each stage of the backward recursion, apply a Bayesoptimal state estimator to fℓ|k (x|Z k ). 14.2.3

Vo-Vo Exact Closed-Form GM Forward-Backward Smoother

Because (14.23) and (14.24) no longer involve a quotient, Gaussian mixture implementation becomes possible. Suppose that for all ℓ = 1, ..., k, fℓ (z|x) fℓ|ℓ−1 (y|x)

= =

fℓ|ℓ−1 (y|Z ℓ−1 )

=

(14.37) (14.38)

NRℓ (z − Hℓ x) NQℓ−1 (y − Fℓ−1 x) νℓ|ℓ−1

∑

ℓ|ℓ−1

wiℓ|ℓ−1 · NP ℓ|ℓ−1 (y − xi

)

(14.39)

i

i=1 nℓ|k

Bℓ|k (y)

=

∑ i=1

ℓ|k

ci

ℓ|k

· NC ℓ|k (y − ci ). i

(14.40)

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From (14.23) and (14.24) it follows that the smoothed distribution fℓ|k (x|Z k ) is also a Gaussian mixture. Specifically, in Section K.24 it is shown that nℓ|k

Bℓ−1|k (x) =

∑

ℓ−1|k

ℓ|k

· NQ

ci

ℓ|k T ℓ−1 +Fℓ−1 Di Fℓ−1

(di

(14.41)

− Fℓ−1 x)

i=1

where ℓ|k

ci ℓ|k · NR +H C ℓ|k H T (zℓ − Hℓ ci ) (14.42) ℓ ℓ i ℓ fℓ (zℓ |Z ℓ−1 )

ℓ−1|k

ci

=

νℓ|ℓ−1

fℓ (zℓ |Z

ℓ−1

)

=

(Di )−1

=

ℓ|k ℓ|k (Di )−1 di

=

∑

wiℓ|ℓ−1 · NR

ℓ|ℓ−1

HℓT ℓ +Hℓ Pi i=1 ℓ|k (Ci )−1 + HℓT Rℓ−1 Hℓ ℓ|k ℓ|k (Ci )−1 ci + HℓT Rℓ−1 zℓ

ℓ|k

(z − Hℓ x)

(14.43) (14.44) (14.45)

or, equivalently, ℓ|k

ℓ|k

di

=

ℓ|k Di

=

Kℓ,k

=

ℓ|k

(14.46)

+ Kℓ,k (zℓ − Hℓ ci )

ci

ℓ|k (I − Kℓ,k Hℓ )Ci ℓ|k ℓ|k Ci HℓT (Hℓ Ci HℓT

(14.47) + Rℓ )

−1

(14.48)

.

Then, the smoothed distribution is k

fℓ−1|k (x|Z )

=

νℓ|ℓ nℓ−1|k ∑ ∑ l=1

wlℓ|ℓ ciℓ−1|k

(14.49)

l=1 ℓ|k

·NQ

ℓ|k ℓ|ℓ T ℓ−1 +Di +Fℓ−1 Pl Fℓ−1

ℓ|ℓ

− Fℓ−1 xl )

(di

ℓ|ℓ

·NE ℓ|ℓ (y − ei,l ) i,l

where ℓ|ℓ

(Ei,l )−1 ℓ|ℓ ℓ|ℓ (Ei,l )−1 ei,l

ℓ|ℓ

ℓ|k

=

T (Pl )−1 + Fℓ−1 (Qℓ−1 + Di )−1 Fℓ−1

(14.50)

=

ℓ|ℓ ℓ|ℓ (Pl )−1 xl

(14.51)

T + Fℓ−1 (Qℓ−1 +

ℓ|k ℓ|k Di )−1 di

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or, equivalently, ℓ|ℓ

ei,l

ℓ|ℓ

ℓ Ki,l

ℓ|ℓ

ℓ xl + Ki,l (di

=

ℓ (I − Ki,l Fℓ−1 )Pl

=

ℓ|ℓ T ℓ|ℓ Pl Fℓ−1 (Pl

ℓ|ℓ

Ei,l

ℓ|k

=

− Fℓ−1 xl ) ℓ|ℓ

+ Qℓ−1 +

(14.52) (14.53)

ℓ|k Di )−1 .

(14.54)

For a somewhat different formulation and more complete implementation details, see [302]. Remark 60 (Two-filter smoother) Next to the forward-backward smoother, this smoother is probably the next most familiar one. It has the form [136] fℓ|k (x|Z k ) = ∫

fℓ|ℓ (x|Z ℓ ) · fℓ+1 (Z ℓ+1 |x) fℓ|ℓ (y|Z ℓ ) · fℓ+1 (Z ℓ+1 |y)dy

(14.55)

where fℓ+1 (Z ℓ+1 |x) can be determined recursively using the following “backwardforward information filter”: fℓ (Z ℓ |x) = fℓ (zℓ |x)

∫

fℓ+1 (Z ℓ+1 |y) · fℓ+1|ℓ (y|x)dy.

(14.56)

The two-filter smoother is not well-suited for particle implementation, which has led Klass, Briers, de Freitas, Doucet, Maskell, and Lang to propose a more appropriate alternative form [136], [28].

14.3

GENERAL MULTITARGET FORWARD-BACKWARD SMOOTHER

This is the obvious multitarget analog of the forward-backward smoother. Only the single-sensor case will be considered here, although the discussion applies equally well to the multisensor-multitarget case. Suppose that a single sensor observes multiple targets with state set X. Given a time sequence Z (k) : Z1 , ..., Zk of measurements, assume—in addition to the usual assumptions underlying the multitarget Bayes filter—that the backward multitarget state transition obeys the following conditions, for ℓ = 0, ..., k − 1: fℓ|ℓ+1,k (X|Y, Z (k) ) = fℓ|ℓ+1,ℓ (X|Y, Z (ℓ) ).

(14.57)

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Then the multitarget forward-backward Bayes smoother is defined by the following direct generalization of (14.10): ∫ fℓ+1|k (Y |Z (k) ) · fℓ+1|ℓ (Y |X) fℓ|k (X|Z (k) ) = fℓ|ℓ (X|Z (ℓ) ) δY (14.58) fℓ+1|ℓ (Y |Z (ℓ) ) where fℓ+1|ℓ (Y |X) is the multitarget Markov density for the standard multitarget motion model. Equation (14.58) is applied using the same three-step procedure described at the end of Section 14.2. The p.g.fl. form of the forward-backward smoother is easily shown to be: Gℓ|k [h] =

∫

where F˜ℓ+1|ℓ [r, h] =

δ F˜ℓ+1|ℓ fℓ+1|k (X ′ |Z (k) ) [0, h] · δX ′ δX ′ fℓ+1|ℓ (X ′ |Z (ℓ) ) ∫

hX · Gℓ+1|ℓ [r|X] · fℓ|ℓ (X|Z (ℓ) )δX

and where Gℓ+1|ℓ [r|X] =

∫

(14.59)

(14.60)

′

r X · fℓ+1|ℓ (X ′ |X)δX ′

(14.61)

is the p.g.fl. of the multitarget Markov density fℓ+1|ℓ (X ′ |X). Equation (14.59) is the smoother analog of the p.g.fl. form of Bayes’ rule, (5.58). Equation (14.58) can be equivalently replaced by the multitarget analogs of (14.23) and (14.24): fℓ|k (X|Z (k) )

=

Bℓ|k (X)

=

fℓ|ℓ (X|Z (ℓ) ) · Bℓ|k (X) (14.62) ∫ Bℓ+1|k (Y ) · Lℓ+1 (Zℓ+1 |Y ) · fℓ+1|ℓ (Y |X)δY(14.63)

where Bℓ|k (X)

=

Lℓ (Zℓ |X)

=

∫

fℓ+1|k (Y |Z (k) ) · fℓ+1|ℓ (Y |X)δY fℓ+1|ℓ (Y |Z (ℓ) )

fℓ (Zℓ |X) fℓ (Zℓ |Z (ℓ−1) )

(14.64) (14.65)

and where, as usual, the integrals are set integrals. In general, (14.58) or (14.62) and (14.63) will usually not be computationally tractable. Principled approximations are required, and these are the subjects of the following subsections.

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BERNOULLI FORWARD-BACKWARD SMOOTHER

The Bernoulli filter, which was described in Section 13.2, is a general and Bayesoptimal approach to single-target joint detection and tracking. It is a special case of the multitarget Bayes filter, in which target number is known, a priori, to be no greater than 1. Thus the multitarget distribution, (13.1), has the form

fℓ|ℓ (X|Z (ℓ) ) =

 

1 − pℓ|ℓ pℓ|ℓ · sℓ|ℓ (x)  0

if if if

X=∅ X = {x} otherwise

(14.66)

where sℓ|ℓ (x) abbr. = sℓ|ℓ (x|Z (ℓ) ) is the distribution of the target track at time tℓ and abbr. pℓ|ℓ = pℓ|ℓ (Z (ℓ) ) is its probability of existence. The Bernoulli forward-backward smoother is a similarly general and Bayesoptimal approach for single-target detection and smoothing. It was first proposed by Clark in 2009 [43]; first implemented using particle methods by Clark, Vo, and Vo [50]; and subsequently implemented by Vo, Clark,Vo, and Ristic [301]. Nagappa and Clark described a fast particle implementation in [220]. Subsequently, Vo and Vo discovered an exact closed-form Gaussian mixture implementation, using the alternative forward-backward formulation described earlier in Section 14.3 (see [302]). Clark has proposed a Bernoulli two-filter smoother [50], although this will not be described here. The purpose of this section is to describe the alternative formulation of the Bernoulli forward-backward smoother due to Vo and Vo. The development here is more direct than theirs, which employed a very general formalism. The section is organized as follows: 1. Section 14.4.1: Modeling assumptions for the Bernoulli forward-backward smoother. 2. Section 14.4.2: The Bernoulli forward-backward smoother equations. 3. Section 14.4.3: Exact Gaussian mixture implementation of the Bernoulli forward-backward smoother. 4. Section 14.4.4: Implementations of the Bernoulli forward-backward smoother.

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14.4.1

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Bernoulli Forward-Backward Smoother: Modeling

As in Section 13.2.1, assume the following models (using notation that is slightly different than in that section): • Probability that, if the target has state x′ at time tℓ , then it will not ℓ+1|ℓ disappear: pS (x′ ). ℓ+1|ℓ

• Probability that, if the target is not present at time tℓ , it will appear: pB

.

• Spatial distribution of the target if it appears at time tℓ+1 : ˆbℓ+1|ℓ (x). • Single-target Markov density: fℓ+1|ℓ (x|x′ ). • Single-sensor, single-target likelihood function at time tℓ+1 : fℓ+1 (z|x). • Multiobject probability distribution for an arbitrary clutter RFS: κℓ+1 (Z). 14.4.2

Bernoulli Forward-Backward Smoother: Equations

Define the backward correctors 0 θℓ|k

=

1 θℓ|k (x)

=

1 − pℓ+1|k k+1|k · (1 − pB ) (14.67) 1 − pℓ+1|ℓ ∫ k+1|k p · pℓ+1|k sℓ+1|k (y) ˆ + B · bℓ+1|ℓ (y)dy pℓ+1|ℓ sℓ+1|ℓ (y) 1 − pℓ+1|k ℓ+1|ℓ · (1 − pS (x)) (14.68) 1 − pℓ+1|ℓ ∫ ℓ+1|ℓ p (x) · pℓ+1|k sℓ+1|k (y) + S · fℓ+1|ℓ (y|x)dy pℓ+1|ℓ sℓ+1|ℓ (y)

and the forward correctors L0ℓ+1 (Zℓ+1 ) =

1 Lℓ

(14.69)

and

=

L1ℓ+1 (Zℓ+1 |x) ( ) 1 − pℓ+1 (x) 1 D ∑ fℓ+1 (z|x)·κℓ+1 (Zℓ+1 −{z}) +pℓ+1 Lℓ D (x) z∈Zℓ+1 κℓ+1 (Zℓ+1 )

(14.70)

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where

Lℓ = 1 − pℓ+1|ℓ + pℓ+1|ℓ

(

sℓ+1|ℓ [1 − pℓ+1 D ] +

∑

z∈Zℓ+1

ℓ+1 sℓ+1|ℓ [pℓ+1 ]·κℓ+1 (Zℓ+1 −{z}) D Lz κℓ+1 (Zℓ+1 )

)

(14.71) and where sℓ+1|ℓ [1 − pℓ+1 D ]

=

∫

(1 − pℓ+1 D (x)) · sℓ+1|ℓ (x)dx

(14.72)

ℓ+1 sℓ+1|ℓ [pℓ+1 D Lz ]

=

∫

pℓ+1 D (x) · fℓ+1 (z|x) · sℓ+1|ℓ (x)dx.

(14.73)

0 1 Note that θℓ|ℓ = 1 and θℓ|ℓ (x) = 1. Then as will be shown in Section K.25, the equations for the Bernoulli forward-backward smoother are as follows:

pℓ|k

=

sℓ|k (x)

=

0 1 − (1 − pℓ|ℓ ) · θℓ|k

(14.74)

1 pℓ|ℓ · sℓ|ℓ (x) · θℓ|k (x)

(14.75)

pℓ|k 0 θℓ|k

=

0 θℓ+1|k · L0ℓ+1 (Zℓ+1 ) ∫ ℓ+1|ℓ 1 +pB θℓ+1|k (x) · L1ℓ+1 (Zℓ+1 |x) · ˆbℓ+1|ℓ (x)dx

1 θℓ|k (x)

=

0 θℓ+1|k · L0ℓ+1 (Zℓ+1 ) · (1 − pS (x)) (14.77) ∫ ℓ+1|ℓ 1 +pS (x) θℓ+1|k (y) · L1ℓ+1 (Zℓ+1 |y) · fℓ+1|ℓ (y|x)dy.

(14.76)

ℓ+1|ℓ

These equations are employed as follows: 1. Forward recursion: Use the Bernoulli filter and the initial items p0|0 , s0|0 (x) to compute the items pℓ+1|ℓ , sℓ+1|ℓ (x) for ℓ = 0, ..., k−1 and pℓ|ℓ , sℓ|ℓ (x) for l = 1, ..., k.

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0 1 2. Backward recursion: Starting with ℓ = k − 1, the items θk−1|k , θk−1|k (x), pk−1|k (x), and sk−1|k (x) can be computed as 0 θk−1|k

=

L0k (Zk ) k|k−1

+pB

(14.78) ∫

L1k (Zk |x) · ˆbk|k−1 (yx)dx k|k−1

1 θk−1|k (x)

=

L0k (Zk ) · (1 − pS (x)) (14.79) ∫ k|k−1 +pS (x) L1k (Zk |y) · fk|k−1 (y|x)dy

pk−1|k

=

0 1 − (1 − pk−1|k−1 ) · θk−1|k

pk−1|ℓ · sk−1|k−1 (x) · sk−1|k (x)

(14.80)

1 θk−1|k (x)

=

(14.81)

. pk−1|k

Given this, if ℓ = k − 2 then we get 0 θk−2|k

=

0 θk−1|k · L0k−1 (Zk−1 ) ∫ k−1|k−2 1 +pB θk−1|k (y) · L1k−1 (Zk−1 |y)

(14.82)

·ˆbk−1|k−2 (y)dy 1 θk−2|k (x)

k−1|k−2

=

0 θk−1|k · L0k−1 (Zk−1 ) · (1 − pS (x)) (14.83) ∫ k−1|k−2 1 +pS (x) θk−1|k (y) · L1k−1 (Zk−1 |y)

·fk−1|k−2 (y|x)dy pk−2|k

=

0 1 − (1 − pk−2|k−2 ) · θk−2|k

pk−2|k−2 · sk−2|k−2 (x) · sk−2|k (x)

(14.84)

1 θk−2|k (x)

.

= pk−2|k

(14.85)

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Continuing in this fashion we eventually get, for ℓ = 0, 0 θ0|k

=

0 θ1|k · L01 (Z1 ) ∫ 1|0 1 +pB θ1|k (y) · L11 (Z1 |y) · ˆb1|0 (y)dy

1 θ0|k (x)

=

0 θ1|k · L01 (Z1 ) · (1 − pS (x)) (14.87) ∫ 1|0 1 +pS (x) θ1|k (y) · L11 (Z1 |y) · f1|0 (y|x)dy

p0|k

=

0 1 − (1 − p0|0 ) · θ0|k

1|0

p0|0 · s0|0 (x) · s0|k (x)

(14.86)

(14.88)

1 θ0|k (x)

=

(14.89)

. p0|k

3. State estimation: At each stage of the backward recursion, apply a Bayesoptimal state estimator as in Section 13.2.4 to pℓ|k , sℓ|k (x) to get smoothed estimates of x at times tℓ = tk , tk−1 , ..., t0 . 14.4.3

Bernoulli Forward-Backward Smoother: Exact GM Implementation

This smoother is based on the following assumptions. Suppose that for all ℓ = 1, ..., k, ℓ|ℓ−1

ℓ|ℓ−1

=

pS

(14.90)

fℓ (z|x)

= =

pℓD

(14.91) (14.92)

fℓ|ℓ−1 (y|x)

=

NQℓ−1 (y νℓ|ℓ−1

sℓ|ℓ−1 (x)

=

pS

(x) pℓD (x)

NRℓ (z − Hℓ x) ∑

(14.93)

− Fℓ−1 x) ℓ|ℓ−1

wiℓ|ℓ−1 · NP ℓ|ℓ−1 (x − xi

)

(14.94)

i

i=1 nℓ|k 1 θℓ|k (x)

=

∑

ℓ|k

ci

ℓ|k

· NC ℓ|k (x − ci ).

(14.95)

i

i=1

Then the Bernoulli forward-backward smoother equations, (14.74) through (14.77), can be solved in exact closed form. The specific formulas for this implementation will not be described here. For more complete implementation details, see [302].

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421

Bernoulli Forward-Backward Smoother: Results

Two implementations are described in this section: an SMC implementation and a GM implementation. SMC implementation by Clark, Vo, Vo, and Ristic [50], [301]. The target appears at k = 11 and disappears at k = 94, while following a curvilinear trajectory between these two times. It is observed by a range-bearing sensor with probability of detection pD = 0.88 and in uniformly distributed Poisson clutter with clutter rate λ = 30. The target is declared to be present if pℓ|k > 0.5, in which case its state estimate is the expected value of sℓ|k (x). The smoother was not run in full batch mode (that is, backwards smoothing from ℓ = k, ..., 0) but, rather, recursively with a two-step lag: ℓ = k−1, k−2. Performance was measured using the OSPA metric (Section 6.2.2). The authors report that the smoother performed better than a corresponding SMC-PHD filter. It initialized and terminated the track two time-steps earlier than the PHD filter, and the state estimates were slightly improved. Exact GM implementation by Vo, Vo, and Mahler [302]. The target appears at k = 10 and disappears at k = 80, while following a slightly curvilinear trajectory. It is observed by a linear-Gaussian sensor with probability of detection pD = 0.98 and in uniformly distributed Poisson clutter with clutter rate λ = 7. The target is declared to be present if pℓ|k > 0.5, in which case its state estimate is the expected value of sℓ|k (x). The smoother was run using one-step, two-step, and three-step lags. Performance was measured using the OSPA metric. The authors reported the expected results: with all three lags, the smoother initialized and terminated the tracks correctly. State estimation was successively improved as the lag increased from 1 to 2 to 3.

14.5

PHD FORWARD-BACKWARD SMOOTHER

This smoother was independently and simultaneously proposed by: • Nadarajah and Kirubarajan [217], [218], using the “physical-space” representation of PHD filters (mentioned in Section 8.4.6.8; see also [179], pp. 599-609). • Mahler, Vo, and Vo [199], [196], using finite-set statistics p.g.fl. techniques. The purpose of this section is to describe the PHD forward-backward smoother and its implementations. The section is organized as follows:

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1. Section 14.5.1: The defining equation for the initial form of the PHD forward-backward smoother. 2. Section 14.5.2: A sketch of the p.g.fl. derivation of the forward-backward smoother. 3. Section 14.5.3: A fast sequential Monte Carlo implementation of the PHD forward-backward smoother, due to Nagappa and Clark. 4. Section 14.5.4: An alternative formulation of the PHD forward-backward smoother, due to Vo and Vo. 5. Section 14.5.5: The exact closed-form Gaussian mixture solution of the PHD forward-backward smoother, due to Vo and Vo. 6. Section 14.5.6: Implementations of the PHD forward-backward smoother. 14.5.1

PHD Forward-Backward Smoother Equation

For ℓ = 0, ..., k − 1 let us be given: • bℓ+1|ℓ (x), the PHD of the target appearance process at time tℓ+1 . • fℓ+1|ℓ (y|x), the Markov transition density from time tℓ to time tℓ+1 . • pS,ℓ+1|ℓ (x), the probability of target survival at time tℓ+1 . Then the PHD forward-backward smoother equation is, for ℓ = 0, ..., k − 1, given by ([196], p. 5, Proposition 1): Dℓ|k (x|Z (k) ) Dℓ|ℓ (x|Z (ℓ) )

ℓ+1|ℓ

=

1 − pS

(x)

ℓ+1|ℓ

+pS

(x)

∫

(14.96) fℓ+1|ℓ (y|x) · Dℓ+1|k (y|Z (k) ) dy Dℓ+1|ℓ (y|Z (ℓ) )

ℓ+1|ℓ

=

(x) (14.97) ∫ (k) fℓ+1|ℓ (y|x) · Dℓ+1|k (y|Z ) ℓ+1|ℓ +pS (x) dy bℓ+1|ℓ (y) + ρℓ+1|ℓ (y)

1 − pS

where the second equation follows from the PHD filter time-update equation, (8.15), where ∫ ℓ+1|ℓ ρℓ+1|ℓ (y) = pS (x) · fℓ+1|ℓ (y|x) · Dℓ|ℓ (x|Z (ℓ) )dx. (14.98)

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If there are no target appearances or disappearances and if exactly one target is known to exist, then (14.96) reduces to the single-target forward-backward smoother equation, (14.10). The derivation of (14.96) requires the following assumptions: • The multitarget distributions fℓ|ℓ (X|Z (ℓ) ) are Poisson for ℓ = 0, 1, ..., k. • The multitarget distributions fℓ+1|ℓ (X|Z (ℓ) ) are Poisson for ℓ = 0, ..., k−1. If in addition the smoothed multitarget distributions fℓ+1|ℓ (X|Z (ℓ) ) are assumed to be Poisson, then it is additionally possible to determine the formula for the smoothed cardinality distribution ([196], p. 7, Proposition 2):

pℓ|k (n) = e

∫

E(y)dy

ℓ|ℓ−1 n ∑ Dℓ|ℓ [1 − pS ]n−i i=0

(n − i)!

[ ] bℓ+1|ℓ i · Dℓ+1|k 1 − Dℓ+1|ℓ

(14.99)

where E(y)

=

bℓ+1|ℓ (y) · Dℓ+1|k (y|Z (k) ) Dℓ+1|ℓ (y|Z (ℓ) )

(14.100)

−Dℓ+1|k (y|Z (k) ) + Dℓ+1|ℓ (y|Z (ℓ) )

ℓ|ℓ−1

Dℓ|ℓ [1 − pS ] [ ] bℓ+1|ℓ Dℓ+1|k 1 − Dℓ+1|ℓ

= =

−bℓ+1|ℓ (y) − Dℓ|ℓ (y|Z (ℓ) ) ∫ ℓ|ℓ−1 (1 − pS (x)) · Dℓ|ℓ (x|Z (ℓ) )dx (14.101) ) ∫ ( bℓ+1|ℓ (y) 1− (14.102) Dℓ+1|ℓ (y|Z (ℓ) ) ·Dℓ+1|k (y|Z (k) )dy.

It is also possible to derive formulas for the mean and variance of pℓ|k (n) ([196], p. 7, Proposition 3), though these will not be given here. The PHD forward-backward smoother is employed using a three-step process similar to that for the single-target forward-backward smoother: 1. Forward recursion: Use the conventional PHD filter and the initial PHD D0|0 (x) to compute the PHDs Dℓ+1|ℓ (y|Z (ℓ) ) for ℓ = 0, ..., k − 1 and Dℓ|ℓ (y|Z (ℓ) ) for ℓ = 1, ..., k.

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2. Backward recursion: Starting with ℓ = k − 1, the PHD Dk−1|k (x|Z k ) can be computed using Dk−1|k (x|Z (k) ) Dk−1|k−1 (x|Z (ℓ) )

(14.103)

k|k−1

=

1 − pS

(x) ∫ fk|k−1 (y|x) · Dk|k (y|Z (k) ) k|k−1 + pS (x) dy. Dk|k−1 (y|Z (k−1) )

Given Dk−1|k (x|Z (k) ), Dk−2|k (x|Z (k) ) can be computed as Dk−2|k (x|Z (k) ) Dk−2|k−2 (x|Z (k−2) )

(14.104)

k−1|k−2

=

1 − pS

(x) ∫ fk−1|k−2 (y|x) · Dk−1|k (y|Z (k) ) k−1|k−2 +pS (x) dy. Dk−1|k−2 (y|Z (k−2) )

Continuing in this fashion, we eventually get to a formula for D1|k (y|Z (k) ), from which we compute D0|k (y|Z (k) ) as D0|k (x|Z (k) ) D0|0 (x)

1|0

=

(14.105)

1 − pS (x) 1|0

+pS (x)

∫

f1|0 (y|x) · D1|k (y|Z (k) ) dy. D1|0 (y)

3. State estimation: At each stage of the backward recursion, apply the usual PHD filter state-estimation approach to Dℓ|k (x|Z (k) ) to get smoothed estimates of the multitarget state set X at times tℓ = tk , tk−1 , ..., t0 . 14.5.2

Derivation of the PHD Forward-Backward Smoother

The basic idea behind the derivation is as follows. By definition, • The forward-backward smoothed PHD is the PHD of the forward-backward smoothed multitarget distribution, which was defined in (14.58).

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It is most easily derived via direct substitution into the p.g.fl. forwardbackward smoother equation, (14.59). The derivation consists of the following steps: 1. Step 1: Derive the function F˜ℓ+1|ℓ [r, h] for the standard multitarget motion model: ℓ+1|ℓ ℓ+1|ℓ F˜ℓ+1|ℓ [r, h] = GB + pS Mrℓ+1|ℓ )] ℓ+1|ℓ [r] · Gℓ|ℓ [h(1 − pS

(14.106)

where GB ℓ+1|ℓ [r] is the p.g.fl. of the target appearance process and where Mrℓ+1|ℓ (x) =

∫

r(x′ ) · fℓ+1|ℓ (x′ |x)dx′ .

(14.107)

2. Step 2: Into (14.106), substitute the equations bℓ+1|ℓ [r−1] GB , ℓ+1|ℓ [r] = e

Gℓ|ℓ [h] = eDℓ|ℓ [h−1]

(14.108)

to get F˜ℓ+1|ℓ [r, h] = exp

(

bℓ+1|ℓ [r − 1] − Nℓ|ℓ ℓ+1|ℓ ℓ+1|ℓ ℓ+1|ℓ +Dℓ|ℓ [h(1 − pS + pS M r )]

)

. (14.109)

3. Step 3: Construct the functional derivatives of F˜ℓ+1|ℓ [r, h] with respect to r: δ F˜ℓ+1|ℓ X′ [0, h] = F˜ℓ+1|ℓ [0, h] · γℓ+1|ℓ (14.110) ′ δX where ℓ+1|ℓ

γℓ+1|ℓ (x′ ) = bℓ+1|ℓ (x′ ) + Dℓ|ℓ [pS

ℓ+1|ℓ

M x′

].

(14.111)

4. Step 4: Construct the first functional derivative of (δ F˜ℓ+1|ℓ /δX ′ )[0, h] with respect to h and then set h = 1: δ F˜ℓ+1|ℓ [0, 1] δX ′ δx 

=

X′ θℓ+1|ℓ

·

+F˜ℓ+1|ℓ [0, 1] ·

(14.112) 

δ F˜ℓ+1|ℓ δx [0, 1] ℓ+1|ℓ ℓ+1|ℓ ∑ Dℓ|ℓ (x)·pS (x)·Mx′ (x) x′ ∈X ′ θℓ+1|ℓ (x′ )

.

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5. Step 5: Into (14.59), substitute (14.112) and the equations fℓ+1|ℓ (X|Z (k) ) ′

fℓ+1|k (X |Z

(k)

)

′

= =

X e−Nℓ+1|ℓ · Dℓ+1|ℓ

e

−Nℓ+1|k

·

X′ Dℓ+1|k .

(14.113) (14.114)

6. Step 6: Derive the PHD smoother equation using algebra, based on Campbell’s theorem (4.96), and the formula ∫ ( =

θℓ+1|ℓ · Dℓ+1|k Dℓ+1|ℓ

){x′ }∪X ′

δX ′

(14.115)

θℓ+1|ℓ (x′ ) · Dℓ+1|k (x′ ) Dℓ+1|ℓ (x′ ) (∫ ) θℓ+1|ℓ (x′ ) · Dℓ+1|k (x′ ) ′ · exp dx Dℓ+1|ℓ (x′ )

for the PHD of the function f˜(X ′ ) =

(

θℓ+1|ℓ · Dℓ+1|k Dℓ+1|ℓ

)X ′

.

(14.116)

Remark 61 The proof given in [196] is more general than the one just sketched, in that one can relax the assumption that the intermediary smoothed distributions fℓ+1|k (X ′ |Z (k) ) are Poisson. 14.5.3

Fast Particle-PHD Forward-Backward Smoother

Mahler, Vo, and Vo proposed a particle implementation of (14.97) in [196]. Subsequently, Nagappa and Clark noted that, because of the backward-smoothing step, this implementation is computationally expensive: O(n2 ν 2 ), where n is the current number of tracks and ν is the current number of particles assigned per target. To remedy this problem, they devised a significantly faster implementation, one with the following advantages [220]: • The computational complexity is O(nν 2 ) rather than O(n2 ν 2 ). • Is independent of the clutter rate. The Nagappa-Clark SMC-PHD forward-backward smoother is based on two ideas:

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1. Append a label state variable τ to the target state, ˚ x = (τ, x). 2. During Markov transitions, never allow states to change labels: fℓ+1|ℓ (τ, x|τ ′ , x′ ) = fℓ+1|ℓ (x|x′ ) · δτ,τ ′ .

(14.117)

The addition of the label variable τ necessitates the following redefinitions: pℓD (τ, x) fℓ (z|τ, x)

= =

pℓD (x) fℓ (z|x).

(14.118) (14.119)

Given this, (14.97) becomes: Dℓ|k (τ, x|Z (k) ) Dℓ|ℓ (τ, x|Z (ℓ) )

ℓ+1|ℓ

=

1 − pS +

∫

(14.120)

(x) ℓ+1|ℓ

pS

(x) · fℓ+1|ℓ (y|x) · Dℓ+1|k (τ, y|Z (ℓ) ) dy bℓ+1|ℓ (τ, y) + ρℓ+1|ℓ (τ, y)

where ρℓ+1|ℓ (τ, y)

=

∑∫

ℓ+1|ℓ

pS

(x) · δτ,τ ′ · fℓ+1|ℓ (y|x)

(14.121)

τ′

=

·Dℓ|ℓ (τ ′ , x|Z (ℓ) )dx ∫ ℓ+1|ℓ pS (x) · fℓ+1|ℓ (y|x) · Dℓ|ℓ (τ, x|Z (ℓ) )dx.(14.122)

For the purpose of the forward recursion, the target-birth PHD bℓ+1|ℓ (τ, y) is defined as follows. For each new measurement zj a Gaussian component is created, along with a unique label, resulting in the Gaussian mixture: mℓ+1

bℓ+1|ℓ (τ, x) =

∑

bℓ+1 · δτ,τ ℓ+1 · NP ℓ+1 (x − xℓ+1 ). j j j

(14.123)

j

j=1

Again for the purpose of the forward recursion, this representation can be approximated as a Dirac mixture as follows. Particles are drawn from each NP ℓ+1 (x − j

xℓ+1 ), assigning them the corresponding label. Then j m ˜ ℓ+1

bℓ+1|ℓ (τ, y) ∼ =

∑ l=1

bℓ+1 · δτ,˜τ ℓ+1 · δx˜ℓ+1 (x) l l

l

(14.124)

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where for each l, τ˜lℓ+1 is equal to τjℓ+1 for some j. The forward recursion of the smoother is then implemented as a conventional SMC-PHD filter (Section 9.6.2) in order to obtain particle representations of the PHDs Dℓ|ℓ (τ, x|Z (ℓ) ) and Dℓ+1|ℓ (τ, x|Z (ℓ) ). Two points should be made: • Once a particle has been created, it and its surviving resampled copies retain the same label thereafter. • Thus when the forward recursion terminates with the construction of the particle representation of Dk|k (τ, x|Z (k) ), the full complement T of assigned labels has been created. • Since the particle representation of Dk−1|k (τ, x|Z (k) ) derives from the particle representation of Dk−1|k−1 (τ, x|Z (k−1) ), the labels of the former are the same as the labels of the latter. More generally, the labels of Dℓ|k (τ, x|Z (k) ) are also labels drawn from T . For the backward recursion, assume that

Dℓ|ℓ (τ, y|Z

(k)

)

νℓ|ℓ ∑

=

ℓ|ℓ

(14.125)

wi · δτ ℓ|ℓ ,τ · δxℓ|ℓ (y) i

i

i=1 νℓ+1|k

Dℓ+1|k (τ, y|Z (k) )

∑

=

ℓ+1|k

wl

· δτ ℓ+1|k ,τ · δxℓ+1|k (y). (14.126) l

l

l=1

Then (14.120) becomes

=

Dℓ|k (τ, x|Z (k) )  νℓ|ℓ ∑ ℓ|ℓ wi ·  ∫ + i=1

ℓ+1|ℓ

1 − pS

(14.127) 

ℓ|ℓ

(xi )

ℓ+1|ℓ ℓ|ℓ ℓ|ℓ ℓ|ℓ pS (xi )·fℓ+1|ℓ (y|xi )·Dℓ+1|k (τi ,y|Z (ℓ) ) ℓ|ℓ ℓ|ℓ bℓ+1|ℓ (τi ,y)+ρℓ+1|ℓ (τi ,y)

dy



·δτ ℓ|ℓ ,τ · δxℓ|ℓ (y) i

i

where

ρℓ+1|ℓ (τ, y) =

νℓ|ℓ ∑ i=1

ℓ+1|ℓ

ℓ|ℓ

wi · δτ ℓ|ℓ ,τ · pS i

ℓ|ℓ

ℓ|ℓ

(xi ) · fℓ+1|ℓ (y|xi )

(14.128)

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and where ∫

ℓ+1|ℓ

pS

ℓ|ℓ

ℓ|ℓ

ℓ|ℓ

(xi ) · fℓ+1|ℓ (y|xi ) · Dℓ+1|k (τi , y|Z (ℓ) ) ℓ|ℓ

dy (14.129)

ℓ|ℓ

bℓ+1|ℓ (τi , y) + ρℓ+1|ℓ (τi , y) ℓ+1|k

νℓ+1|k

=

∑ wi

ℓ+1|ℓ

· δτ ℓ+1|k ,τ ℓ|ℓ · pS ℓ|ℓ

ℓ+1|k

ℓ+1|ℓ

· pS

wi =

ℓ|ℓ

ℓ|ℓ

|xi ) (14.130)

ℓ|ℓ

ℓ+1|k

) + ρℓ+1|ℓ (τi , xl

ℓ|ℓ

ℓ+1|k

(xi ) · fℓ+1|ℓ (xi ℓ+1|k

bℓ+1|ℓ (τi , xi

ℓ+1|k

ℓ+1|k

bℓ+1|ℓ (τi , xl

l=1

ℓ|ℓ

(xi ) · fℓ+1|ℓ (xl

i

l

ℓ|ℓ

|xi )

ℓ|ℓ

) (14.131)

.

ℓ+1|k

) + ρℓ+1|ℓ (τi , xi

) ℓ+1|k

The last equation follows from the fact that the label τl must always be equal ℓ|ℓ to one of the labels τi . Thus the particle representation of the backward recursion equation, (14.120), is ℓ|k

wi

(14.132)

ℓ|ℓ

wi ℓ+1|k ℓ+1|ℓ

=

1 − pS

ℓ|ℓ

ℓ+1|ℓ

· pS

wi

(xi ) +

ℓ|ℓ

ℓ|ℓ

ℓ+1|k

(xi ) · fℓ+1|ℓ (xi ℓ+1|k

bℓ+1|ℓ (τi , xi

ℓ|ℓ

ℓ|ℓ

|xi )

.

ℓ+1|k

) + ρℓ+1|ℓ (τi , xi

)

The elimination of the summation in (14.130) is what accounts for the improved computational characteristics of this implementation of the SMC-PHD forwardbackward smoother. 14.5.4

Alternative PHD Forward-Backward Smoother

This is constructed in direct analogy with (14.23) through (14.49), by defining a recursive backward corrector. For ℓ = 0, ..., k − 1, define the backward corrector Bℓ|k (x) =

ℓ+1|ℓ ℓ+1|ℓ 1−pS (x)+pS (x)

∫

Dℓ+1|k (y|Z (k) ) ·fℓ+1|ℓ (y|x)dy (14.133) Dℓ+1|ℓ (y|Z (ℓ) )

and for ℓ = 1, ..., k the forward corrector Lℓ (Zℓ |x) = 1 − pℓD (x) +

∑ pℓ (x) · fℓ (z|x) D κℓ (z) + τℓ (z) z∈Zℓ

(14.134)

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where τℓ (z) =

∫

pℓD (x) · fℓ (z|x) · Dℓ|ℓ−1 (x|Z (ℓ−1) )dx.

(14.135)

Then (14.96) can be equivalently replaced by [302], Eqs. (79-83) Dℓ|k (x|Z (k) ) Bℓ|k (x)

= =

Dℓ|ℓ (x|Z (ℓ) ) · Bℓ|k (x) 1−

ℓ|ℓ−1 pS (x)

ℓ|ℓ−1

+pS

(x)

∫

(14.136) (14.137)

Bℓ|k (x) · Lℓ (Zℓ |x) · fℓ|ℓ−1 (y|x)dy

for ℓ = 0, ..., k − 1. This is easily demonstrated as follows. From (14.136) we get Dℓ|k (x|Z k ) Dℓ|ℓ−1 (x|Z (ℓ−1) )

=

Dℓ|ℓ (x|Z ℓ ) · Bℓ|k (x) Dℓ|ℓ−1 (x|Z (ℓ−1) )

(14.138)

=

Lℓ (Zℓ |x) · Bℓ|k (x).

(14.139)

Thus, as claimed, ℓ|ℓ−1

Bℓ−1|k (x)

(x) (14.140) ∫ (k) Dℓ|k (y|Z ) ℓ|ℓ−1 +pS (x) · fℓ|ℓ−1 (y|x)dy Dℓ|ℓ−1 (y|Z (ℓ) )

=

1 − pS

=

1 − pS

ℓ|ℓ−1

14.5.5

(x) (14.141) ∫ ℓ|ℓ−1 +pS (x) Bℓ|k (x) · Lℓ (Zℓ |x) · fℓ|ℓ−1 (y|x)dy.

Gaussian-Mixture PHD Smoother

Vo and Vo have demonstrated that, as with the single-target forward-backward smoother, the PHD forward-backward smoother, (14.96), can be implemented in exact closed form using GM methods [314], [302]. The forward recursion is implemented using the conventional GM-PHD filter formulas (Section 9.5.4). The key is the alternative form of the PHD backward recursion described in the previous section.

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This smoother is based on the following assumptions. Suppose that for all ℓ = 1, ..., k, ℓ|ℓ−1

pS

ℓ|ℓ−1

(x) pℓD (x)

fℓ (z|x) fℓ|ℓ−1 (y|x)

=

pS

(14.142)

= = =

pℓD

(14.143) (14.144) (14.145)

NRℓ (z − Hℓ x) NQℓ−1 (y − Fℓ−1 x)

and νℓ|ℓ−1

Dℓ|ℓ−1 (y|Z ℓ−1 )

=

∑

ℓ|ℓ−1

wiℓ|ℓ−1 · NP ℓ|ℓ−1 (y − xi

)

(14.146)

i

i=1 νℓ|ℓ

Dℓ|ℓ (y|Z ℓ )

=

∑

ℓ|ℓ

(14.147)

ℓ|k

(14.148)

wiℓ|ℓ · NP ℓ|ℓ (y − xi ) i

i=1 nℓ|k

Bℓ|k (y)

=

∑

ℓ|k

ci

· NC ℓ|k (y − ci ). i

i=1

Then the PHD forward-backward smoother equations, (14.136) and (14.137), can be solved in closed form. The equations for this implementation will not be further discussed here. For more complete implementation details, see [302]. As already noted, a major consequence of the Vo-Vo approach is that the exact closed-form solution is preserved in the single-sensor special case (see Section 14.2.3). 14.5.6

Implementations of the PHD Forward-Backward Smoother

The following experimental implementations will be described in this section: • Conventional SMC implementation by Mahler, Vo, and Vo [196]. • Fast SMC implementation by Nagappa and Clark [220]. • Exact GM implementation by Vo, Vo, and Mahler [302]. • SMC implementation by Nadarajah and Kirubarajan, including multiple motion-model implementation [217], [218].

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Conventional SMC implementation by Mahler, Vo, and Vo [196]. Five appearing and disappearing targets, following curvilinear trajectories, were observed by a single range-bearing sensor with probability of detection pD = 0.98 and clutter rate λ = 7. The SMC-PHD smoother was applied with a smoother-lag time of 5, and compared to an SMC-PHD filter. The authors reported that the smoother achieved about 33% better localization accuracy than the SMC-PHD filter, but exhibited approximately the same accuracy in target-number estimation. Further investigation revealed a curious anomaly: the smoother successfully removed the effect of false alarms, but did not respond well to missed detections or to target disappearances. The authors conjectured that this behavior is attributable to the Poisson approximation made in both the filter and smoother. Because this approximation attempts to model target number using a single parameter, the smoother may lack enough degrees of freedom to account for sudden decreases in target number (whether due to missed detections or target disappearances). Fast SMC implementation by Nagappa and Clark [220]. Twelve appearing and disappearing targets, following curvilinear trajectories, were observed by a single range-bearing sensor with probability of detection pD = 0.98 and a clutter rate λ = 30. The authors observed the same behavior as reported in [196]. While exhibiting significantly better localization accuracy than the filter, the smoother experienced some difficulty with missed detections and disappearing targets. Because of its greater computational efficiency, however, the smoother was able to accommodate larger numbers of targets, and in denser clutter. The authors verified that the fast smoother is linear in the number of targets, and essentially flat with respect to the clutter rate. Exact closed-form GM implementation by Vo, Vo, and Mahler [302]. Four persisting targets appear simultaneously at the origin and follow linear trajectories along the coordinate axes thereafter. They are observed by a linear-Gaussian sensor with probability of detection pD = 0.98 and uniformly distributed Poisson clutter with clutter rate λ = 7. The smoother was run using one-step, two-step, and three-step lags. Performance, as measured using the OSPA metric (Section 6.2.2), conformed to expectations: the smoother initialized and terminated the tracks correctly with all three lags; and state estimation was improved as the lag increased from 1 to 2 to 3. SMC implementation by Nadarajah and Kirubarajan [217], [218]. These authors used SMC techniques to implement two versions of the PHD smoother: with and without multiple motion models. The simulation results for both versions will be reported in turn.

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For the simulations involving an a priori motion model, the authors tested their algorithm in a two-dimensional scenario involving a single range-bearing sensor and three appearing and disappearing targets, with the first target existing for only a short period of time. Probability of detection was pD = 0.9, clutter rate was either λ = 20 or λ = 50, and the smoother lag ranged from 1 to 5. The PHD smoother, with smoother lags 1, 2, 3, 4, and 5, was compared to a conventional SMC-PHD filter using the Wasserstein multitarget miss-distance (see Section 6.2.1.2). For λ = 20, the authors reported that the PHD smoother significantly outperformed the PHD filter for lags 1, 2, and 3, but with no discernible improvement for lags 4 and 5. Similar results were observed for λ = 50, although—as expected—the performance of both the smoother and the filter were degraded compared to the λ = 20 case. For the simulations involving multiple motion models, two models were used: a constant-velocity model and a coordinated-turn model. In this case, two maneuvering targets were present throughout the entire scenario, and the clutter rate was λ = 20. The multiple-model PHD smoother was compared to a multiplemodel PHD filter. Once again, the smoother significantly outperformed the filter in regard to tracking performance. The smoother also exhibited somewhat better behavior in regard to estimating which motion model was in effect at any given time.

14.6

ZTA-CPHD SMOOTHER

Is there a CPHD forward-backward smoother? It is possible to derive exact formulas for such a smoother, but they are computationally intractable. Suppose, however, that target appearances in each filtering cycle are negligible—in which case we can also neglect the CPHD filter’s target appearance model. Then the resulting smoother—the “zero target appearances” (ZTA) CPHD forward-backward smoother—becomes tractable. The purpose of this section is to briefly describe this smoother (but without proof, since the result appears to be a relatively minor one). In the multitarget forward-backward smoother equation, (14.58), assume that the distributions fℓ|ℓ (X|Z (ℓ) ), fℓ+1|ℓ (Y |Z (ℓ) ), fℓ+1|k (Y |Z (k) ) are i.i.d.c. in the sense of Section 4.3.2. The corresponding cardinality and spatial distributions are

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pℓ|ℓ (n), pℓ+1|ℓ (n), pℓ+1|k (n) and sℓ|ℓ (x), sℓ+1|ℓ (x), sℓ+1|k (x). Define: ℓ+1|ℓ

ψℓ|ℓ

=

θℓ,k

=

ℓ+1|ℓ

sℓ|ℓ [1 − pS ] = 1 − sℓ|ℓ [pS ] ∑ (1 − ψℓ|ℓ )n · pℓ+1|k (n) (n+1) · Gℓ|ℓ (ψℓ|ℓ ) n! · pℓ+1|ℓ (n)

(14.149) (14.150)

n≥0

+ θℓ,k

∑ (1 − ψℓ|ℓ )n · pℓ+1|k (n + 1) (n+1) · Gℓ|ℓ (ψℓ|ℓ ). n! · pℓ+1|ℓ (n + 1)

=

(14.151)

n≥0

Then using the p.g.fl. forward-backward smoother equation, (14.59), the ZTACPHD smoother equations can be shown to be:

Gℓ|k (x)

=

n ∑ pℓ+1|k (n) · θℓ,k (n) · Gℓ|ℓ (x · ψℓ|ℓ ) · xn n! · pℓ+1|ℓ (n)

(14.152)

n≥0

=

∑ (1 − ψℓ|ℓ )n · pℓ+1|k (n) n ·p ψℓ|ℓ ℓ+1|ℓ (n) n≥0 ∑ · pℓ|ℓ (i) · Ci,n · (ψℓ|ℓ x)i

(14.153)

i≥n

and Dℓ|k (x) sℓ|ℓ (x)

ℓ+1|ℓ

=

(1 − pS ℓ+1|ℓ

+pS

(x)) · θℓ,k + (x) · θℓ,k

∫

(14.154)

sℓ+1|k (y) · fℓ+1|ℓ (y|x) dy sℓ+1|ℓ (y)

where Ci,n is the binomial coefficient as defined in (2.1). It is easily shown that if pℓ+1|ℓ (n), pℓ+1|k (n), and pℓ|ℓ (n) are Poisson distributions, this formula reduces to the PHD smoother equation, (14.97), but with no target appearances, that is, bℓ+1|ℓ (y) = 0. The ZTA-CPHD smoother can be implemented in exact closed form using Gaussian mixture techniques, in the same manner as described in Section 14.5.5.

Chapter 15 Exact Closed-Form Multitarget Filter 15.1

INTRODUCTION

The single-sensor, multitarget Bayes filter is the optimal approach for singlesensor, multitarget detection, tracking, and identification. However, it is also computationally intractable except for very simple tracking problems. Thus far, a number of filters have been described that tractably approximate the multitarget Bayes filter. These include PHD filters, CPHD filters, and multi-Bernoulli filters. Let us briefly review the basic approximation philosophy motivating these filters. • PHD and CPHD filters: In the theory of PHD and CPHD filters (Chapter 8), no attempt is made to try to accurately approximate the multitarget posterior distribution fk|k (X|Z (k) ) itself. Rather, the information in fk|k (X|Z (k) ) is lossily compressed into multitarget moments of various kinds. Consequently, this information loss is expected to result in a significant degradation of performance in comparison to the optimal filter. • Multi-Bernoulli filters: The theory of multi-Bernoulli filters (Chapter 13) goes one step further, in that an attempt is being made to approximate fk|k (X|Z (k) ) with some degree of accuracy. Specifically, fk|k (X|Z (k) ) is approximated by a multi-Bernoulli distribution. However, this approximation is not general, since the target-number variance must always be smaller than the target-number mean. It is also not exact, since several additional approximations must be assumed in order to achieve closed-form formulas. One can therefore ask if there is a solution that is both general and exact:

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• Is there an exact closed-form (and computationally tractable) solution of the multitarget Bayes filter, in the same sense that the Kalman filter is an exact closed-form solution of the single-target Bayes filter? As was discovered by B.-T. Vo and B.-N. Vo, the answer is—somewhat surprisingly—Yes [295], [296]. Their exact closed-form solution is based on the fact that target tracks must be distinctly labeled if temporal track-continuity is to be achieved. This, in turn, means that the concept of a random finite set (RFS) must be generalized to that of a labeled random finite set (labeled RFS, for short). The discovery of the Vo-Vo exact-closed form multitarget filter has the following significant consequences: • The Vo-Vo exact closed-form multitarget filter is apparently the very first tractable, provably Bayes-optimal multitarget detection and tracking algorithm. • Consequently, it is apparently also the first multitarget detection and tracking algorithm to have a provably Bayes-optimal track-management scheme. • The approach is easily extended to permit, in a theoretically rigorous fashion, the inclusion of target-type tags—and thus to permit provably Bayes-optimal joint multitarget detection, tracking, and target identification. • Using suitable approximations, the filter can be made efficient in regard to both tracking performance and computational throughput. • The “spookiness” phenomenon, discussed in Section 9.2, affects PHD and CPHD filters and, to a lesser extent, the CBMeMBer filter of Section 13.4. Because the Vo-Vo filter is an exact closed-form solution of the general multitarget Bayes filter, one would expect that it would not exhibit this phenomenon.1 Vo and Vo have verified that this is indeed the case, that is, the exact closed-form filter does not exhibit spooky behavior [304]. The purpose of this chapter is to describe in detail the theory and practice of the Vo-Vo exact closed-form multitarget Bayes filter. Because of the importance of the result, a significant amount of space will be devoted to the demonstration that the Vo-Vo filter is indeed exact closed-form and thus Bayes-optimal. The remainder of this introduction is organized as follows: 1

For, the contrary result would mean that—despite its optimality—the multitarget Bayes filter itself exhibits spookiness.

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1. Section 15.1.1: The concept of an exact closed-form solution of the singlesensor, single-target Bayes filter. 2. Section 15.1.2: The concept of an exact closed-form solution of the singlesensor, multitarget Bayes filter. 3. Section 15.1.3: Overview of the Vo-Vo filter approach. 4. Section 15.1.4: A summary of the major lessons learned in this chapter. 5. Section 15.1.5: The organization of the chapter. 15.1.1

Exact Closed-Form Solution of the Single-Target Bayes Filter

The Kalman filter is an exact, algebraically closed-form solution of the singlesensor, single-target Bayes filter, in the following three-stage sense. First, suppose that both target motion and the prior track distribution are linear-Gaussian: fk+1|k (x|x′ ) k

fk|k (x|Z )

=

NQk (x − Fk x′ )

(15.1)

=

NPk|k (x − xk|k ).

(15.2)

Then the Bayes filter time-update formula (prediction integral) can be evaluated exactly, and results in an exact closed-form linear-Gaussian predicted track density function ([179], pp. 35-36): ∫ k fk+1|k (x|Z ) = fk+1|k (x|x′ ) · fk|k (x′ |Z k )dx′ (15.3) =

NPk+1|k (x − xk+1|k )

(15.4)

where Pk+1|k

=

Qk + Fk Pk|k FkT

(15.5)

xk+1|k

=

Fk xk|k .

(15.6)

Second, suppose that both the sensor likelihood function and the predicted track distribution are linear-Gaussian: fk+1 (z|x) k

fk+1|k (x|Z )

=

NRk+1 (z − Hk+1 x)

(15.7)

=

NPk+1|k (x − xk+1|k ).

(15.8)

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Then the Bayes filter measurement-update formula (Bayes’ rule) can be evaluated in exact closed form, resulting in the linear-Gaussian measurement-updated density function ([179], pp. 37-39): fk+1k+1 (x|Z k+1 )

fk+1 (zk+1 |x) · fk+1|k (x|Z k ) fk+1 (zk+1 |y) · fk+1|k (y|Z k )dy

=

∫

=

NPk+1|k+1 (x − xk+1|k+1 )

(15.10)

(15.9)

where −1 Pk+1|k+1

=

−1 −1 Pk+1|k + HkT Rk+1 Hk

(15.11)

−1 Pk+1|k+1 xk+1|k+1

=

−1 −1 Pk+1|k xk+1|k + HkT Rk+1 zk+1 .

(15.12)

The family of linear-Gaussian distributions is thereby said to be a family of conjugate priors for the family of likelihood functions Lz (x) = NRk+1 (z − Hk+1 x). Third, it follows that the entire Bayes filter can be solved in exact closed form if: • The Markov transition densities fk+1|k (x|x′ ) = NQk (x − Fk x′ )

(15.13)

are linear-Gaussian. • The sensor likelihood functions fk+1 (z|x) = NRk+1 (z − Hk+1 x)

(15.14)

are linear-Gaussian. • The initial distribution f0|0 (x) = NP0|0 (x − x0|0 )

(15.15)

is linear-Gaussian. Stated more formally: Let D be the family of linear-Gaussian probability density functions fp (x) parametrized by the space P of parameters p = (x, P ). Then this family has the following three properties:

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439

1. Exact closed-form closure with respect to the prediction integral: if fpk|k ∈ D with pk|k = (xk|k , Pk|k ); and if f+ (x) =

∫

fk+1|k (x|x′ ) · fpk|k (x′ )dx′

(15.16)

is the Markov prediction of fpk|k using the specified Markov transition ′ density fk+1|k (x|x ) = NQk (x − Fk x′ ); then there exists a fpk+1|k ∈ D with pk+1|k = (xk+1|k , Pk+1|k ) such that f+ = fpk+1|k .

(15.17)

2. Exact closed-form closure with respect to Bayes’ rule: if fpk+1|k ∈ D with pk+1|k = (xk+1|k , Pk+1|k ) and if f z (x) = ∫

fk+1 (z|x) · fpk+1|k (x) fk+1 (z|y) · fpk+1|k (y)dy

(15.18)

is the Bayes-rule update of fpk+1|k using the specified likelihood function fk+1 (z|x) = NRk+1 (z − Hk+1 x) and any specific measurement z, then there exists a fpk+1|k+1 ∈ D with pk+1|k+1 = (xk+1|k+1 , Pk+1|k+1 ) such that f z = fpk+1|k+1 . (15.19) (The family D is therefore a class of conjugate priors for the family of likelihood functions Lz (x) = fk+1 (z|x).) 3. Exact closed-form closure with respect to Bayes-optimal state estimation. Suppose that some Bayes-optimal state estimator is applied to fpk+1|k+1 — two examples being the expected a posteriori (EAP) estimator and the maximum a posteriori (MAP) estimator: EAP [fpk+1|k+1 ]

=

∫

M AP [fpk+1|k+1 ]

=

sup fpk+1|k+1 (x).

x · fpk+1|k+1 (x)dx

(15.20) (15.21)

x

Suppose that, among all of these Bayes-optimal estimators, there is at least one such that its state estimate can be exactly constructed from the parameter pk+1|k+1 without knowledge of fpk+1|k+1 . In our case, both the EAP and

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MAP estimators satisfy this property: EAP [fpk+1|k+1 ] = M AP [fpk+1|k+1 ] = π(pk+1|k+1 ) = xk+1|k+1 (15.22) where π : (x, P ) ?→ x is the projection operator. If the first two properties are satisfied, then the family D of distributions is said to be: • An exact closed-form solution of the single-target Bayes filter. In this case it follows that the Bayes filter can be replaced, in exact closed form, by a filter that propagates parameters in the smaller (and hopefully more computationally advantageous) space P: ... →

pk|k

→

pk+1|k

→

pk+1|k+1

→ ...

If the third property is also satisfied, then we are justified in saying that the family D of distributions is also: • Exact closed-form with respect to Bayes-optimal state estimation. In this case, the filter has the form

... → 15.1.2

pk|k

→

pk+1|k

→

π(pk+1|k+1 ) ↑ pk+1|k+1

→ ...

Exact Closed-Form Solution of the Multitarget Bayes Filter

At the outset, assume that the following specifications for the multitarget measurement and motion models: • The multitarget likelihood function fk+1 (Z|X) is the one corresponding to the standard multitarget measurement model—that is, the superposition of a multi-Bernoulli target-detection RFS and a Poisson clutter RFS (7.21). • The multitarget Markov density fk+1|k (X|X ′ ) is the one corresponding to the modified standard multitarget motion model. By this is meant the superposition of a multi-Bernoulli target-survival RFS and a multi-Bernoulli target appearance RFS. That is, the p.g.fl. of this model is appearing targets

surviving targets

?? ? ? ?? ? ? X′ Gk+1|k [h|X ′ ] = GB k+1|k [h] · (1 − pS + pS Mh )

(15.23)

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where the p.g.fl. GB k+1|k [h] of the target appearance RFS is multi-Bernoulli rather than Poisson, as in the standard multitarget motion model of (7.66): B νk+1|k

GB k+1|k [h]

=

∏

l l (1 − qk+1|k + qk+1|k · ˆblk+1|k [h]).

(15.24)

l=1

Given this, an exact closed-form solution of the multitarget Bayes filter is a family D of parametrized multitarget probability distributions fp (X) with p ∈ P for some parameter space P, which satisfies the following conditions: 1. Exact closed-form closure with respect to the multitarget prediction integral: if fpk|k ∈ D and if f+ (X) =

∫

fk+1|k (X|X ′ ) · fpk|k (X ′ )δX ′

(15.25)

is the Markov prediction of fpk|k using the multitarget Markov transition density fk+1|k (X|X ′ ) for the modified standard multitarget motion model, then there is an fpk+1|k ∈ D such that f+ = fpk+1|k .

(15.26)

2. Exact closed-form closure with respect to multitarget Bayes’ rule: if fpk+1|k ∈ D and fk+1 (Z|X) · fpk+1|k (X) f Z (X) = ∫ (15.27) fk+1 (Z|Y ) · fpk+1|k (Y )δY is the Bayes-rule update of fpk+1|k using the multitarget likelihood function fk+1 (Z|X) for the standard multitarget measurement model, and any specified measurement set Z, then there is an fpk+1|k+1 ∈ D such that f Z = fpk+1|k+1 .

(15.28)

In this case the multitarget Bayes filter can be replaced by an equivalent filter on the parameter space P. Mahler showed that Condition 1 is satisfied if D is the class of multiBernoulli distributions, as defined in Section 4.3.4 ([179], pp. 675-677). Thus, suppose that the initial multitarget distribution f0|0 (X|Z (0) ) is multi-Bernoulli. Then so is the first time-updated distribution f1|0 (X|Z (0) ).

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However, this progression fails with the next step. The first measurementupdated distribution f1|1 (X|Z (1) ) can be only approximately multi-Bernoulli—as is the case, for example, with the CBMeMBer filter. One might then ask: Can the class of multi-Bernoulli distributions be generalized to a larger class D of multitarget distributions that is exact closed-form under time-update and measurement-update? Better yet, is it potentially computationally tractable? The answer is, Yes. The crucial insight, due to B.-T. Vo, is the following. Let D be the class of generalized labeled multi-Bernoulli distributions, to be defined in Section 15.3.4.1. Then D turns out to be the desired solution class. 15.1.3

Overview of the Vo-Vo Filter Approach

The core idea underlying the Vo-Vo approach is this: • If target-tracks are distinctly labeled, then computationally tractable exact closed-form closure becomes possible. That is, kinematic states x must be replaced by labeled states ˚ x = (x, ℓ) where ℓ is a discrete label variable that is uniquely associated with this track as time progresses. Thus conventional multitarget states X = {x1 , ..., xn } must ˚ = {˚ be replaced by labeled multitarget states X x1 , ...,˚ xn } with ˚ xi = (xi , ℓi ) 2 where ℓ1 , ..., ℓn are distinct. Given this, an RFS multitarget state Ξ becomes a labeled RFS multitarget state ˚ Ξ—that is, an RFS whose instantiations are labeled multitarget states. Recall (Section 4.3.2) that an (unlabeled) i.i.d.c. RFS Ξ has a distribution of the form ∏ fΞ (X) = |X|! · p(|X|) s(x). (15.29) x∈X

As we shall see in (15.85), the multitarget probability density function of a labeled multi-Bernoulli RFS ˚ Ξ generalizes the concept of an i.i.d.c. RFS: ˚ ˚ f˚ ˚ X ˚L | · ω(XL ) Ξ (X) = δ|X|,|

∏

s(x, ℓ)

(15.30)

˚ (x,ℓ)∈X

where: 2

It is unknown at this time if it there exists a practically useful exact closed-form solution of the multitarget Bayes filter with the standard multitarget measurement model, but with unlabeled target states.

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˚L denotes the set of labels of the targets in X. ˚ 3 • X • sℓ (x) = s(x, ℓ) is the spatial probability density function of the track that has label ℓ. ∑ • ω(L) ≥ 0 with L ω(L) = 1 is the weight (probability) of the following hypothesis: there are |L| targets whose labels are the elements of L, and such that the sℓ (x) with ℓ ∈ L are their respective track distributions. Equation (15.30) defines a labeled multi-Bernoulli distribution, hereafter ˚ is the labeled version of abbreviated as an “LMB distribution.” If fk+1 (Z|X) the standard multitarget likelihood function, then the Bayes’ rule update of a LMB distribution has the form ∫

˚ · f (X) ˚ fk+1 (Zk+1 |X) = δ|X|,| ˚ X ˚L | ˚ ˚ ˚ fk+1 (Zk+1 |Y ) · f (Y )δ Y

∑ ˚ θ∈TZk+1

∏

˚ ˚ ω θ (X L)

˚

sθ (x, ℓ)

˚ (x,ℓ)∈X

(15.31) ˚ where sθ (x, ℓ) are probability distributions in x for each fixed ℓ; and where ∑ ∑ ˚ ω θ (L) = 1; (15.32) ˚ θ∈TZk+1

L

and where the first summation is taken over all measurement-to-track associations ˚ θ ∈ TZk+1 between the measurements in Zk+1 and the set of labels in L, where TZk+1 denotes the set of all such associations (see Section 15.3.4 for more details). If we replace TZk+1 by an arbitrary index set O, we get a generalized LMB distribution—or “GLMB distribution” or “Vo-Vo prior” for short. Given this, it can further be shown that: • The Bayes’ rule update of a GLMB distribution, using the multitarget likelihood function for the standard multitarget measurement model of (7.21), is also a GLMB distribution. • The time-update of a GLMB distribution, using the labeled version of the multitarget Markov density for the modified standard multitarget motion model of (15.23), is a GLMB distribution. In other words: the family D of GLMB distributions forms an exact closedform and potentially tractable solution of the multitarget Bayes filter, hereafter called the “Vo-Vo filter.” 3

˚ = {(x1 , ℓ1 ), ..., (xn , ℓn )} then X ˚L = {ℓ1 , ..., ℓn }. For example, if X

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Furthermore, GLMB distributions are easily extended to include target features and target identification (see Section 15.4.4). Remark 62 (Another exact closed-form solution) As will be seen in Section 20.4, there exists another closed-form solution of the multitarget Bayes filter—but not for the standard multitarget measurement model. Rather, this one applies to a measurement model, due to B.-N. Vo, in which measurements are pixelized images with pixel-to-pixel independence—see Section 20.2. In this case, the solution-class D also turns out to be a family of multi-Bernoulli distributions. 15.1.4

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter: • The Vo-Vo filter is the first provably Bayes-optimal, but computationally tractable, multitarget filter for the standard multitarget motion and measurement models. • In particular, its track-management scheme is provably Bayes-optimal— apparently the first tractable multitarget tracking filter for which such a claim can be made. • As a consequence, the track management approaches in the multihypothesis tracker (MHT)—as well as in its many offshoots and “target-existence probability” generalizations—appear to be heuristic. Unless, that is, it can someday be demonstrated that MHT, or any of its generalizations, also constitutes an exact closed-form solution of the multitarget Bayes filter. • Stated differently: The Vo-Vo filter can be thought of as the first theoretically rigorous, Bayes-optimal formulation of an MHT-type (that is, data association-based) tracking algorithm. • It is also the first theoretically rigorous, Bayes-optimal formulation of an association-based tracking algorithm that includes target-identification capability (see Section 15.4.4). • Practical implementation of the Vo-Vo solution is based on δ-GLMB distributions of the form ∑ ˚ = ˚ f˚O (X) ωo · f˚o (X), (15.33) o∈O

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˚ are component functions indexed by o ∈ O. The number where f˚o (X) of components grows combinatorially, so components with small weights ωo must be pruned. The loss due to pruning can be characterized exactly. That is, if components with index set O′ are to be discarded, then: ∑ ∥f˚O − f˚O′ ∥1 = ωo . (15.34) o∈O−O′

where ∥f˚∥1 denotes the L1 norm. See Section 15.6.3. 15.1.5

Organization of the Chapter

The chapter is organized as follows: • Section 15.2: Introduction to the theory of labeled random finite sets (labeled RFSs). • Section 15.3: Examples of labeled RFSs: labeled i.i.d.c. RFSs, labeled Poisson RFSs, labeled multi-Bernoulli (LMB) RFSs, and generalized LMB (GLMB) RFSs. • Section 15.4: Modeling assumptions for the Vo-Vo filter, including GLMB forms of the modified standard multitarget Markov density and the standard multitarget likelihood function. This section includes an overview of the VoVo filter (Section 15.4.2). • Section 15.5: Demonstration that the family of GLMB distributions solves the multitarget Bayes filter, with respect to the standard multitarget measurement model and the modified standard multitarget motion model. • Section 15.6: A sketch of the practical implementation of the Vo-Vo filter. • Section 15.7: Performance results.

15.2

LABELED RFSS

In the typical theory and practice of multitarget tracking, the state x of a single target is purely kinematic. Its state variables consist of position x, y, z and velocity vx , vy , νz and perhaps also acceleration variables and body-frame orientation variables. In the real world, however, every target inherently possesses an additional state variable: its identity. This identity-variable can be:

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• Explicit: a class identifier (for example, jet fighter versus transport plane) or a specific identifier (for example, an aircraft tail number). • Implicit: a track label ℓ that usually does not tell us anything about identity, but which serves the purpose of distinguishing the time-evolving tracks in the scene from each other. This is the common practice in purely kinematic multitarget tracking. The purpose of this section is to examine the concept and the statistics of target labeling in greater depth. The following topics are considered: 1. Section 15.2.1: Labeling of single targets. 2. Section 15.2.2: Distinct labeling of multiple targets. 3. Section 15.2.3: Set integrals for labeled targets. 15.2.1

Target Labels

It is sometimes asserted that the RFS approach inherently cannot deal with track continuity. This is supposedly because the elements of finite sets are orderindependent, and thus are indistinguishable except for their dynamic behavior. Thus—or again so it is often asserted—one cannot determine which track at time tk+1 evolved from which track at time tk . These assertions are untrue. As was noted in [179], pp. 506-507, track management becomes possible—indeed, Bayes-optimally possible—if individual target states are distinctly labeled. In this case any complete state specification of a single target must have the form ˚ x = (x, ℓ) ∈ X × L

(15.35)

where x is in the kinematic state space X and ℓ is an element of a discrete space L of labels. Since the number of distinct targets is unbounded, the label space L must have the general form L = {ϖ1 , ..., ϖi , ...}

(15.36)

where ϖ1 , ..., ϖi , ... is a countable number of distinct labels drawn, once and for all, from some convenient alphabet. Remark 63 (The Vo-Vo labeling convention) For purposes of implementation of the Vo-Vo multitarget filter, in (15.146) the label space will be given the specific

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form L = {0, 1, ...} × {1, 2, ...}. Specifically, a label will have the form ℓ = (k, i) where k is the time that a given track was created, and i is an index that uniquely distinguishes this track from all other tracks created at that time. For all i = 1, 2, ... define L(i) = {ϖ1 , ..., ϖi }.

(15.37)

Fn (L) = {L ⊆ L| |L| = n}

(15.38)

Also, let the symbol denote the class of finite subsets of L of cardinality n. 15.2.2

Labeled Multitarget State Sets

Suppose that there are multiple targets to be considered. Then the state set of labeled targets will have the form ˚ = {˚ X x1 , ...,˚ xn } = {(x1 , ℓ1 ), ..., (xn , ℓn )} ⊆ X × L.

(15.39)

Pairs of the form (x1 , ℓ), (x2 , ℓ) with x1 ̸= x2 are physically unrealizable. This is because it is not possible for the same target to simultaneously have two different kinematic states (for example, two different positions) at the same time. Thus {(x1 , ℓ1 ), ..., (xn , ℓn )} is not a physically well defined multitarget state representation unless the ℓ1 , ..., ℓn are distinct.4 Define the projection operators ˚ x ?→ ˚ xX and ˚ x ?→ ˚ xL on points ˚ x = (x, ℓ) by ˚ xX ˚ xL

= =

x ℓ.

(15.40) (15.41)

˚ ?→ X ˚X and X ˚ ?→ X ˚L on finite sets Likewise, define the projection operators X ˚ = {(x1 , ℓ1 ), ..., (xn , ℓn )} by X ˚X X ˚L X 4

= =

{x1 , ..., xn } {ℓ1 , ..., ℓn }.

(15.42) (15.43)

In this case, one must verify that the resulting space of distinctly labeled target states has appropriate topological properties—see [94], pp. 196-198.

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˚ ⊆ X × L that has distinct Then a labeled multitarget state set is a finite subset X labels, i.e.: ˚L | = |X|. ˚ |X (15.44) Consequently, a labeled RFS state set is a random labeled multitarget state set ˚ Ξ ⊆ X × L, in which case |˚ ΞL | = |˚ Ξ|. (15.45) Thus: • It is possible for two different targets to have the same kinematic state—that is, (x, ℓ1 ), (x, ℓ2 ) with ℓ1 ̸= ℓ2 is allowed. • A target cannot have different kinematic states—that is, (x1 , ℓ), (x2 , ℓ) with x1 ̸= x2 is not allowed. Remark 64 (Bayes-optimal track management) It follows that the multitarget Bayes filter will—inherently—optimally maintain track continuity. For, suppose ˚′ is the track set at time tk and (xk|k , ℓ) ∈ X ˚′ is the state of one of that X ˚ its tracks. Further suppose that X is the track set at time tk+1 . Then after ˚ is the state a single recursion of the multitarget Bayes filter, (xk+1|k+1 , ℓ) ∈ X of the same track at the next time tk+1 ([94], pp. 196-198). This results in a Bayes-optimal approach to track management. The difficulty, of course, is that this optimal approach to target labeling will be computationally intractable for all but the simplest practical problems. 15.2.3

Set Integrals for Labeled Multitarget States

˚ be a function of a labeled state set X ˚ that has the following properties: Let f˚(X) ˚ ˚ are u−|X| 1. The units of measurement of f˚(X) where u are the units of measurement of x ∈ X. ∫ 2. f˚({(x1 , ℓ1 ), ..., (xn , ℓn )})dx1 · · · dxn exists for all ℓ1 , ..., ℓn ∈ L and n ≥ 1.

3. For each n ≥ 1, ∫

f˚({(x1 , ℓ1 ), ..., (xn , ℓn )})dx1 · · · dxn = 0

for all but a finite number of n-tuples (ℓ1 , ..., ℓn ) ∈ Ln .

(15.46)

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˚ exists and is defined to be Then the set integral of f˚(X) ∫

˚ X ˚= f˚(X)δ

∑ 1 n! n≥0

∫

∑

f˚({(x1 , ℓ1 ), ..., (xn , ℓn )})dx1 · · · dxn .

(ℓ1 ,...,ℓn )∈Ln

(15.47) If ˚ h(x, ℓ) is any unitless test function taking values in [0, 1], the probability ˚ is generating function (p.g.fl.) of f˚(X) ˚˚[˚ G f h] =

∫

˚ ˚ ˚ X ˚ hX · f˚(X)δ

(15.48)

˚ where the power functional notation ˚ hX was defined in (3.5).

15.3

EXAMPLES OF LABELED RFSS

The purpose of this section is to consider the following examples of labeled RFSs: 1. Section 15.3.1: Labeled i.i.d.c. RFSs. 2. Section 15.3.2: Labeled Poisson RFSs. 3. Section 15.3.3: Labeled multi-Bernoulli (LMB) RFSs. 4. Section 15.3.4: Generalized labeled multi-Bernoulli (GLMB) RFSs. 15.3.1

Labeled i.i.d.c. RFSs

The multitarget distribution of a labeled i.i.d.c. RFS ˚ Ξ is defined to be ˚X X ˚ ˚ f˚˚ ˚ ˚L · p(|X|) · s Ξ (X) = δL(|X|), X

(15.49)

where the notation L(n) was defined in (15.37), the power functional notation sX was defined in (3.5), and where: • p(n) is a cardinality distribution—that is, a probability distribution on target number n. • s(x) is a probability density on x—that is, the spatial distribution of a target track.

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• For any subsets L, L′ ⊆ L, the Kronecker delta δL,L′ is defined by δL,L′ = 1 if L = L′ and δL,L′ = 0 otherwise. ˚ = {(x1 , ℓ1 ), ..., (xn , ℓn )} with |X| ˚ = n then (15.49) can be Thus if X more concretely written as: ˚ f˚˚ Ξ (X)

= =

δ{ϖ1 ,...,ϖn },{ℓ1 ,...,ℓn } · p(n) · s(x1 ) · · · s(xn )   p(n) · s(x1 ) · · · s(xn ) if {ϖ 1 , ..., ϖn } = {ℓ1 , ..., ℓn }  0 if otherwise

(15.50) (15.51)

where the ϖ1 , ..., ϖn were defined in (15.36). Consequently, for each n > 1 we have f˚˚ Ξ ({(x1 , ℓ1 ), ..., (xn , ℓn )}) = 0 for all (ℓ1 , ..., ℓn ) except for the selections (15.52)

(ℓ1 , ..., ℓn ) = (ϖπ1 , ..., ϖπn ) where π are the n! permutations on 1, ..., n. Thus the set integral exists and is finite. 15.3.1.1

∫

˚ ˚ f˚˚ Ξ (X)δ X

The p.g.fl. of a Labeled i.i.d.c. RFS

For any label ℓ ∈ L and any test function ˚ h(x, ℓ) with 0 ≤ ˚ h(x, ℓ) ≤ 1, define the linear functional ∫ sℓ [˚ h] def. = ˚ h(x, ℓ) · s(x)dx. (15.53) ˚ Then the p.g.fl. of f˚˚ Ξ (X), as defined in (15.48), is ˚˚[˚ G Ξ h]

=

∑

p(n)

∑ n≥0

sϖi [˚ h]

(15.54)

i=1

n≥0

=

n ∏

∏

p(n)

sℓ [˚ h].

(15.55)

ℓ∈L(n)

In particular if ˚ h = 1 identically then sℓ [˚ h] = 1 for any ℓ, and so ˚˚[1] = G Ξ

∫

˚ ˚ f˚˚ Ξ (X)δ X =

∑ n≥0

p(n) = 1.

(15.56)

Exact Closed-Form Multitarget Filter

˚ Thus f˚˚ Ξ (X) is a labeled multitarget probability distribution. To see why (15.54) is true, note that ∫ ˚ ˚ ˚ ˚ ˚ ˚ G˚ hX · f˚˚ Ξ [h] = Ξ (X)δ X ∫ ∑ 1 ∑ ˚ = h(x1 , ℓ1 ) · · · ˚ h(xn , ℓn ) n! n n≥0

=

451

(15.57) (15.58)

(ℓ1 ,...,ℓn )∈L

·f˚˚ Ξ ({(x1 , ℓ1 ), ..., (xn , ℓn )})dx1 · · · dxn ∫ ∑ 1 ∑ ˚ h(x1 , ℓ1 ) · · · ˚ h(xn , ℓn ) n! n n≥0

(15.59)

(ℓ1 ,...,ℓn )∈L

·δ{ϖ1 ,...,ϖn },{ℓ1 ,...,ℓn } · p(n) · s(x1 ) · · · s(xn )dx1 · · · dxn

= =

∑ p(n) ∑ sℓ1 [˚ h] · · · sℓn [˚ h] · δ{ϖ1 ,...,ϖn },{ℓ1 ,...,ℓn }(15.60) n! n n≥0 (ℓ1 ,...,ℓn )∈L ∑ ∑ p(n) sℓ1 [˚ h] · · · sℓn [˚ h] (15.61) n≥0

{ℓ1 ,...,ℓn }⊆Fn (L)

·δ{ϖ1 ,...,ϖn },{ℓ1 ,...,ℓn } =

∑ n≥0

p(n)

n ∏

sϖi [˚ h]

(15.62)

i=1

where Fn (L), the class of finite subsets of L of cardinality n, was defined in (15.38). 15.3.1.2

PHD, p.g.f., and Cardinality Distribution of a Labeled i.i.d.c. RFS

The PHD, p.g.f., and cardinality distribution of a labeled i.i.d.c. RFS ˚ Ξ are ∑ ˚˚(x, ℓ) = s(x) D p(n) · 1L(n) (ℓ) (15.63) Ξ n≥0

˚˚(x) G Ξ

=

∑

p(n) · xn

(15.64)

n≥0

˚ p˚ Ξ (n)

=

p(n)

where 1L(n) is the indicator function of the set L(n).

(15.65)

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To prove (15.63), first note that δ sℓ′ [˚ h] = δℓ,ℓ′ · s(x). δ(x, ℓ)

(15.66)

˚˚[˚ Then note that, from (15.54), the first functional derivative of G Ξ h] is ˚˚ δG Ξ ˚ [h] δ(x, ℓ)

=

=



 n ∑ ′ · s(x) δ ℓ,ℓ  (15.67) p(n)  sℓ′ [˚ h]  ′ [˚ s h] ℓ ′ ′ n≥0 ℓ ∈L(n) ℓ ∈L(n)   ∑ ∏ 1L(n) (ℓ) s(x) p(n)  sℓ′ [˚ h] · . (15.68) sℓ [˚ h] ′ ∑



∏

n≥0

ℓ ∈L(n)

Therefore, from (4.75), ˚ ∑ Ξ ˚˚(x, ℓ) = δ G˚ D [1] = s(x) p(n) · 1L(n) (ℓ). Ξ δ(x, ℓ)

(15.69)

n≥0

Equation (15.64) follows immediately from (4.66) by substituting ˚ h = x into (15.54). Equation (15.65) follows immediately from that. Remark 65 (De-labeling) If the labels of a labeled RFS ˚ Ξ are stripped away, resulting in an unlabeled RFS Ξ, then the multitarget distribution of Ξ is the marginal distribution defined by f˚ Ξ ({x1 , ..., xn }) =

∑

f˚˚ Ξ ({(x1 , ℓ1 ), ..., (xn , ℓn )}).

(15.70)

ℓ1 ,...,ℓn ∈L

˚˚[˚ ˚ Using this equation, it is easily shown that the p.g.fl. G Ξ h] of Ξ and the p.g.fl. GΞ [h] of Ξ are related by ˚˚[h] GΞ [h] = G (15.71) Ξ for all test functions h(x) on unlabeled states x. Remark 66 (The de-labeling of a labeled i.i.d.c. RFS is i.i.d.c.) A labeled i.i.d.c. RFS is not i.i.d.c., but its corresponding delabeled RFS is i.i.d.c. If ˚ Ξ is a labeled i.i.d.c. RFS then the projection Ξ = ˚ ΞX into unlabeled state sets, as defined in (15.42), is an i.i.d.c. RFS. For, according to the previous remark and (15.55), the

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p.g.fl. of the delabeled RFS is GΞ [h]

=

˚˚[h] = G Ξ

∑

p(n)

n≥0

=

∑

p(n)

n≥0

∏

∏

sℓ [h]

(15.72)

ℓ∈L(n)

s[h] =

ℓ∈L(n)

∑

p(n) · s[h]n ,

(15.73)

n≥0

which is the p.g.fl. of an i.i.d.c. RFS. 15.3.2

Labeled Poisson RFSs

These are labeled i.i.d.c. RFSs whose cardinality distributions have the following form: ˚ |X| ˚ ˚ = δ ˚ ˚ · e−N · N f˚(X) · sXX (15.74) L(|X|),XL ˚ |X|! where N is the Poisson parameter. A labeled Poisson RFS is not Poisson, but its corresponding delabeled RFS is Poisson. 15.3.3

Labeled Multi-Bernoulli (LMB) RFSs

These are defined as follows, using the following four steps. Step 1: Let us be given a fixed number ν of target-tracks with: • Track probability densities s1 (x), ..., sν (x). • Track probabilities of existence q 1 , ..., q ν . • Track labels ϖτ 1 , ..., ϖτ ν ∈ L where the “track labeling function” τ : {1, ..., ν} → {1, 2, ...} is a one-to-one function that selects specific labels for the tracks. As with a conventional multi-Bernoulli RFS (Section 4.3.4), the number q i is interpreted as the probability that the ith track actually exists—that is, actually is a target. The primary difference between a LMB RFS and a multi-Bernoulli RFS is the fact that the track indices i = 1, ..., ν must be associated with specific labels. This is the purpose of the track labeling function τ . Step 2: Let the set of labels that have been assigned to the tracks be denoted as ν L = {ϖτ 1 , ..., ϖτ ν }. (15.75)

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ν

˚ Step 3: Note that, for any ℓ ∈ L and any labeled multitarget state set X, ˚L (that is, ℓ is not the label of any target in X), ˚ or ℓ ∈ X ˚L —in either ℓ ∈ / X which case ℓ = ϖτ i for some unique i ∈ {1, ..., ν}. Thus define the function σ : L → {0, 1, ..., ν} by { i if ℓ = ϖτ i σℓ abbr. = σ(ℓ) = . (15.76) 0 if otherwise Also define the function ˚ s(x, ℓ) on labeled states by ˚ s(x, ℓ) = sσℓ (x)

(15.77)

ν

for all x ∈ X and ℓ ∈ L. Step 4: Define the set function ω(L) by   ( ) ∏ ∏   ω(L) =  (1 − q σℓ ) q σℓ · 1 ν (ℓ)

(15.78)

L

ν

ℓ∈L

ℓ∈L−L

=

Q

∏ q σℓ · 1 ν (ℓ) L

(15.79)

1 − q σℓ

ℓ∈L

for all L ⊆ L, where 1S (ℓ) is the set indicator function for the subset S ⊆ L; and where ν ∏ Q= (1 − q i ). (15.80) i=1 ν

Note that ω(L) = 0 unless L ⊆ L. Also note that ∑ ∑ ω(L) = ω(L) = 1.

(15.81)

ν

L⊆L

L⊆L

For, ∑

ω(L)

=

Q

∑∏

ν

ν

L⊆L

ℓ∈L

q σℓ 1 − q σℓ

(15.82)

L⊆L

 =



∏ ν

ℓ∈L

=

1

σℓ



(1 − q )

∏( ν

q σℓ 1+ 1 − q σℓ

)

(15.83)

ℓ∈L

(15.84)

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where (15.83) follows from the power-functional identity, (3.7). 15.3.3.1

Definition of a LMB RFS

Given this, the probability distribution for a labeled multi-Bernoulli (LMB) RFS ˚ Ξ is defined to be [295]: ˚ f˚˚ Ξ (X)

=

˚ δ|X|,| ˚ X ˚L | · ω(XL )

∏

˚ s(x, ℓ)

(15.85)

˚ (x,ℓ)∈X

=

˚ ˚ δ|X|,| sX . ˚ X ˚L | · ω(XL ) · ˚

(15.86)

This generalizes the concept of a labeled i.i.d.c. RFS, in that the cardinality weight˚ in (15.49) is replaced by the label weighting factor ω(X ˚L ); ing factor p(|X|) and in that the spatial distribution s(x) is replaced by a labeled track distribution ˚ s(x, ℓ): ˚X ˚ X ˚ ˚ δL(|X|), ?→ δL(|X|), sX . ˚ ˚L · p(|X|) · s ˚ ˚L · ω(XL ) · ˚ X X ˚ = {(x1 , ℓ1 ), ..., (xn , ℓn )} with |X| ˚ = n, then (15.85) becomes If X ) n ∏ q σℓi sσℓi (xi ) 1{ϖτ 1 ,...,ϖτ ν } (ℓi ) · Q · . 1 − q σℓi i=1 i=1 (15.87) This is mathematically well defined. First, the factor δn,|{ℓ1 ,...,ℓn }| ensures that ˚ ˚ f˚X ˚ (X) vanishes whenever the labels of X are not distinct. Second, the product ˚ of the factors 1{ϖτ 1 ,...,ϖτ ν } (ℓi ) ensures that f˚X ˚ (X) vanishes whenever any ˚ is not one of the preassigned track labels ϖτ 1 , ..., ϖτ ν . Third and label of X ˚ actually are distinct and preassigned track labels, thereby, since all labels ℓ of X then ℓ = ϖτ i for some i = 1, ..., ν and so σℓ = i is a well defined track-index. ˚ For nonzero values of f˚ Ξ (X) we therefore get ˚ f˚˚ Ξ (X) = δn,|{ℓ1 ,...,ℓn }| ·

(

n ∏

˚ f˚˚ Ξ (X) = Q

)

n ∏ q σℓi · sσℓi (xi ) . 1 − q σℓi i=1

(

(15.88)

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15.3.3.2

The p.g.fl. of a LMB RFS

The p.g.fl. of the LMB RFS ˚ Ξ is ) ∫ ∏( σℓ σℓ ˚ ˚˚[˚ G h] = 1 − q + q h(x, ℓ) · ˚ s (x, ℓ)dx Ξ

(15.89)

ν

ℓ∈L ν ( ∏

=

i

1−q +q

i

∫

˚ h(x, ϖτ i ) · si (x)dx

)

(15.90)

i=1

where ˚ h(x, ℓ) is a test function with values in [0, 1]; and where, from (15.75), ν L = {ϖτ 1 , ..., ϖτ ν }. Setting ˚ h = 1, it is easily verified that ∫ ˚ ˚ f˚˚ (15.91) Ξ (X)δ X = 1. To prove (15.89), note from (15.87) that ∫ ˚ ˚ ˚ ˚ ˚ ˚ G˚ hX · f˚˚ Ξ [h] = Ξ (X)δ X ∫ ∑ 1 ∑ ˚ = h(x1 , ℓ1 ) · · · ˚ h(xn , ℓn ) n! n n≥0

=

(15.92) (15.93)

(ℓ1 ,...,ℓn )∈L

·f˚˚ Ξ ({(x1 , ℓ1 ), ..., (xn , ℓn )})dx1 · · · dxn ∫ ∑ 1 ∑ ˚ Q h(x1 , ℓ1 ) · · · ˚ h(xn , ℓn ) n! n n≥0 (ℓ1 ,...,ℓn )∈L ( n ) ∏ 1{ϖτ 1 ,...,ϖτ ν } (ℓi ) ·δn,|{ℓ1 ,...,ℓn }| ·

(15.94)

i=1

=

q σℓ1 sσℓ1 (x1 ) q σℓn sσℓn (xn ) · · · · dx1 · · · dxn 1 − q σℓ1 1 − q σℓn ∑ 1 ∑ Q s˜σℓ1 [˚ h] · · · s˜σℓn [˚ h] · δn,|{ℓ1 ,...,ℓn }|(15.95) n! ν n≥0

(ℓ1 ,...,ℓn )∈Ln

where s˜ [˚ h] abbr. = ℓ

∫

q σℓ sσℓ (x) ˚ h(x, ℓ) · dx. 1 − q σℓ

(15.96)

Exact Closed-Form Multitarget Filter

ν

457

ν

If Fn (L) denotes the set of all finite subsets of L of cardinality n, then ˚˚[˚ G Ξ h]

=

Q

∑

∑

s˜ℓ1 [˚ h] · · · s˜ℓn [˚ h]

(15.97)

ν

n≥0

{ℓ1 ,...,ℓn }∈Fn (L)

=

Q

∑∏ ν

s˜ℓ [˚ h].

(15.98)

ℓ∈L

L⊆L

Because of the identity for the power functional, (3.7) and since σϖτ i = i for i = 1, ..., ν then because of (15.76), we get the desired result: ˚˚[˚ G Ξ h]

=

Q

∏(

1 + s˜ℓ [˚ h]

)

(15.99)

ν

ℓ∈L

=

∏(

1 − q σℓ + q σℓ

∫

˚ h(x, ℓ) · sσℓ (x)dx

)

(15.100)

ν

=

=

ℓ∈L ν ( ∏ i=1 ν ( ∏

1 − q σϖτ i + q σϖτ i i

1−q +q

i

∫

∫

) ˚ h(x, ϖτ i ) · sσϖτ i (x)dx (15.101)

) i ˚ h(x, ϖτ i ) · s (x)dx .

(15.102)

i=1

A labeled multi-Bernoulli RFS is not multi-Bernoulli, but its corresponding delabeled RFS is multi-Bernoulli. 15.3.3.3

PHD and Cardinality Distribution of a LMB RFS

The PHD, p.g.f., and cardinality distribution of a LMB RFS ˚ Ξ are given by ˚˚(x, ℓ) D Ξ

=

˚˚(x) G Ξ

=

˚ p˚ Ξ (n)

=

1 ν (ℓ) · q σℓ · ˚ s(x, ℓ)

(15.103)

(1 − q i + q i · x)

(15.104)

L ν ∏

i=1 ( ν ∏ i=1

i

(1 − q )

)

· σν,n

(

q1 qν , ..., 1 − q1 1 − qν

)

(15.105)

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where σν,n (x1 , ..., xν ) is the elementary homogeneous symmetric function of degree n in ν variables. Equation (15.103) can be rewritten as: { i i q · s (x) if ℓ = ϖτ i , 1 ≤ i ≤ ν ˚˚(x, ℓ) = D . (15.106) Ξ 0 if otherwise To prove (15.103), note that from (15.89) ˚˚ δG Ξ ˚ [h] δ(x, ℓ)  =



∏(

ν

=

 ·

) ˚ h(x′ , ℓ′ ) · ˚ s(x′ , ℓ′ )dx′ 

∫

ν

ℓ′ ∈ L



′

ℓ′ ∈ L

∑

·

′

1 − q σℓ + q σℓ

(15.107) 

∏(

′

1 − q σℓ′

1−q

σℓ′

δℓ′ ,ℓ · q σℓ · ˚ s(x, ℓ′ ) ∫ ′ + q σℓ ˚ h(x′ , ℓ′ ) · ˚ s(x′ , ℓ′ )dx′

+q

σℓ′

∫

′

′

′

′

˚ h(x , ℓ ) · ˚ s(x , ℓ )dx

ν

ℓ′ ∈ L

′

)

 

(15.108)

1 ν (ℓ) · q σℓ · ˚ s(x, ℓ) L . ∫ 1 − q σℓ + q σℓ ˚ h(x′ , ℓ) · ˚ s(x′ , ℓ)dx′

Therefore, from (4.75), ˚ Ξ ˚˚(x, ℓ) = δ G˚ D [1] = 1 ν (ℓ) · q σℓ · ˚ s(x, ℓ). Ξ L δ(x, ℓ)

(15.109)

Equation (15.104) follows immediately from (4.66) by substituting ˚ h = x into (15.89). Equation 15.126) follows from (4.130). 15.3.4

Generalized Labeled Multi-Bernoulli (GLMB) RFSs

This section introduces a generalization of the LMB RFS, the generalized LMB (GLMB) RFS. The section is organized as follows: • Section 15.3.4.1: Definition of a GLMB RFS.

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• Section 15.3.4.2: Intuitive interpretation of a GLMB distribution. • Section 15.3.4.3: The p.g.fl. of a GLMB RFS. • Section 15.3.4.4: The PHD and cardinality distribution of a GLMB RFS. • Section 15.3.4.5: Example: labeled i.i.d.c. RFSs are GLMB RFSs. • Section 15.3.4.6: Example: LMB RFSs are GLMB RFSs. • Section 15.3.4.7: Approximate multitarget state estimation for GLMB distributions. 15.3.4.1

Definition of a GLMB RFS

Let the following be given: • A set O of indices. (This is required to achieve exact closed-form closure for the multitarget Bayes filter, and will itself evolve with time.) • For so (x, ℓ) on (x, ℓ) ∈ X × L such that ∫ o each o ∈ O, a function ˚ ˚ s (x, ℓ)dx = 1 for all ℓ ∈ L. • For each o ∈ O, a unitless set function ω o (L) defined on L ⊆ L. • ω o (L) = 0 except for a finite number of pairs (o, L) with finite L ⊆ L, in which case it is required that ∑∑

ω o (L) = 1.

(15.110)

o∈O L⊆L

Given this, the probability distribution of a GLMB RFS (also known as Vo-Vo prior) is defined to be ([295], Eq. (13)): ˚ f˚ Ξ (X)

=

δ|X|,| ˚ X ˚L |

∑ o∈O

=

δ|X|,| ˚ X ˚L |

∑

˚L ) ω o (X

∏

˚ so (x, ℓ)

(15.111)

˚ (x,ℓ)∈X ˚ ˚L ) · (˚ ω o (X s o )X .

(15.112)

o∈O

Note that this distribution is well defined with respect to units of measurement, since ˚ ˚L ) is unitless and the units of (˚ ˚ for every ω o (X so )X are the same as those of X o.

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15.3.4.2

Intuitive Interpretation of Vo-Vo Priors

˚ = {(x1 , ℓ1 ), ..., (xn , ℓn )} be a labeled multitarget state set with |X| ˚ = n, Let X ˚ and let L = XL = {ℓ1 , ..., ℓn } be the corresponding set of distinct labels. Then for each o ∈ O, the number ω o (L) is the weight of the hypothesis that: • There are n targets present with labels ℓ1 , ..., ℓn . • ˚ so (x, ℓ1 ), ...,˚ so (x, ℓn ) are their respective track distributions. The larger the value of ω o (L), the more likely it is that the corresponding hypothesis is true. The ˚ so (x1 , ℓ1 ), ...,˚ so (xn , ℓn ) are the probabilities (probability densities) that the respective kinematic states of the tracks are, respectively, x1 , ..., xn . The summation ∑ ω o ({ℓ1 , ..., ℓn }) · ˚ so (x1 , ℓ1 ) · · ·˚ so (xn , ℓn ) (15.113) o∈O

˚ is the total probability (probability density) f˚˚ Ξ (X) that targets with state set {(x1 , ℓ1 ), ..., (xn , ℓn )} are present. 15.3.4.3

The p.g.fl. of a GLMB RFS

Define the linear functional ˚ soℓ [˚ h] by ∫ ˚ soℓ [˚ h] abbr. = ˚ h(x, ℓ) · ˚ so (x, ℓ)dx.

(15.114)

˚ Then the p.g.fl. of a Vo-Vo prior f˚˚ Ξ (X) is ∑∑ ∏ ˚˚[˚ G ω o (L) ˚ soℓ [˚ h]. Ξ h] =

(15.115)

o∈O L⊆L

ℓ∈L

Setting ˚ h = 1 we get ∫ ∑∑ ˚ ˚ ˚ [1] = f˚˚ ω o (L) = 1 Ξ (X)δ X = G˚ Ξ

(15.116)

o∈O L⊆L

˚ where the last equation is true because of (15.110). Thus f˚ Ξ (X) is a multitarget probability distribution on labeled targets. Note that, by the definition of a GLMB RFS, both the summation and product in (15.115) are finite.

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To prove (15.115), note that ˚˚[˚ G Ξ h]

∫

=

˚ ˚ ˚ ˚ hX · f˚˚ Ξ (X)δ X

∑

∑ 1 n!

=

n≥0

(15.117) ∫

(ℓ1 ,...,ℓn

˚ h(x1 , ℓ1 ) · · · ˚ h(xn , ℓn )

·f˚˚ Ξ ({(x1 , ℓ1 ), ..., (xn , ℓn )})dx1 · · · dxn ∫ ∑ 1 ∑ ˚ h(x1 , ℓ1 ) · · · ˚ h(xn , ℓn ) n! n≥0 (ℓ1 ,...,ℓn )∈Ln ∑ ·δn,|{ℓ1 ,...,ℓn }| ω o ({ℓ1 , ..., ℓn })

=

(15.118)

)∈Ln

(15.119)

o∈O o

o

·˚ s (x1 , ℓ1 ) · · ·˚ s (xn , ℓn )dx1 · · · dxn and thus that ˚˚[˚ G Ξ h]

=

∑∑ 1 n!

∑

˚ soℓ1 [˚ h] · · ·˚ soℓn [˚ h]

(15.120)

(ℓ1 ,...,ℓn )∈Ln ·δn,|{ℓ1 ,...,ℓn }| · ω o ({ℓ1 , ..., ℓn }) o∈O n≥0

=

∑∑

∑

˚ soℓ1 [˚ h] · · ·˚ soℓn [˚ h]

(15.121)

o∈O n≥0 {ℓ1 ,...,ℓn }∈Fn (L)

=

·ω o ({ℓ1 , ..., ℓn }) ∑∑ ∏ ω o (L) ˚ soℓ [˚ h]. o∈O L⊆L

15.3.4.4

(15.122)

ℓ∈L

PHD and Cardinality Distribution of a GLMB RFS

The PHD of a GLMB RFS ˚ Ξ is given by:

˚˚(x, ℓ) D Ξ

=

∑ o∈O

=

∑ o∈O

  (

∑ L⊆L

∑ L∋ℓ



o

ω (L) · 1L (ℓ) · ˚ so (x, ℓ)

ω o (L)

)

·˚ so (x, ℓ).

(15.123)

(15.124)

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The p.g.f. of ˚ Ξ is: ˚˚(x) = G Ξ

∑∑

ω o (L) · x|L| .

(15.125)

o∈O L⊆L

The cardinality distribution of ˚ Ξ is ([295], Eq. (16)): ˚ p˚ Ξ (n) =

∑

∑

ω o (L)

(15.126)

o∈O L∈Fn (L)

where Fn (L) was defined in (15.38). Note that the expected number of targets in ˚ Ξ is ∑∫ ∑∑ ˚˚ = G ˚(1) (1) = ˚˚(x, ℓ)dx = N D ω o (L) · |L|. (15.127) ˚ Ξ Ξ Ξ o∈O L⊆L

ℓ∈L

To prove (15.123), first note that δ ˚ so′ [˚ h] = δℓ′ ,ℓ · ˚ so (x, ℓ′ ). δ(x, ℓ) ℓ

(15.128)

˚˚[˚ Then note that the first functional derivative of G Ξ h] in (15.115) is ˚˚ δG Ξ ˚ [h] δ(x, ℓ)

=

∑∑

ω o (L)

(

o∈O L⊆L

˚ soℓ′ [˚ h]

)

(15.129)

ℓ′ ∈L

) ∑ δℓ′ ,ℓ · ˚ so (x, ℓ′ ) · ˚ soℓ′ [˚ h] ℓ′ ∈L ( ) ∑∑ ∏ 1L (ℓ) · ˚ so (x, ℓ) o o ˚ ω (L) ˚ sℓ′ [h] · (15.130) ˚ soℓ [˚ h] o∈O L⊆L ℓ′ ∈L (

=

∏

and therefore, from (4.75), ˚ ∑∑ Ξ ˚ ˚˚(x, ℓ) = δ G˚ D [h] = ω o (L) · 1L (ℓ) · ˚ so (x, ℓ). Ξ δ(x, ℓ) o∈O L⊆L

(15.131)

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Equation (4.66) follows immediately from (15.115) by substituting ˚ h = x into (15.115). Equation (15.126) follows from (4.62): [ ] 1 dn ˚ ˚ p˚ G˚(x) (15.132) Ξ (n) = n! dxn Ξ x=0 [ ] 1 ∑∑ o = ω (L) · n!C|L|,n · x|L|−n (15.133) n! o∈O L x=0 ∑ ∑ = ω o (L). (15.134) o∈O L∈Fn (L)

15.3.4.5

Labelled i.i.d.c. RFSs Are GLMB RFSs

From (15.54) the p.g.fl. of a labeled i.i.d.c. process is ∑

˚ G˚ Ξ [h] =

p(n)

n (∫ ∏

) ˚ h(x, ϖi ) · s(x)dx .

(15.135)

i=1

n≥0

Let |O| = 1 and define ω(L)

=

p(|L|) · δL,L(|L|)

(15.136)

˚ s(x, ℓ)

=

s(x)

(15.137)

where L(n) = {ϖ1 , ..., ϖn } was defined in (15.37). Then from (15.115) we see that the p.g.fl. of the corresponding GLMB RFS is identical to the p.g.fl. of the labeled i.i.d.c. RFS: ∑ ∏ ˚˚[˚ G ω(L) ˚ sℓ [˚ h] (15.138) Ξ h] = L⊆L

=

∑

ℓ∈L

p(|L|) · δL,L(|L|)

L⊆L

∏ (∫

˚ h(x, ℓ) · ˚ s(x, ℓ)dx

)

(15.139)

ℓ∈L

and so ˚˚[˚ G Ξ h]

=

∑ ∑

p(n) · δL,{ϖ1 ,...,ϖn })

n≥0 |L|=n

·

∏ (∫ ℓ∈L

˚ h(x, ℓ) · s(x)dx

)

(15.140)

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and so ˚˚[˚ G Ξ h]

=

∑ n≥0

=

∑ n≥0

15.3.4.6

(∫

∏

p(n)

˚ h(x, ℓ) · s(x)dx

)

(15.141)

ℓ∈{ϖ1 ,...,ϖn }

p(n)

n ∏

sϖi [˚ h].

(15.142)

i=1

LMB RFSs Are GLMB RFSs

Let |O| = 1. Then it is clear that the distribution of a LMB RFS, (15.85), is a special case of the distribution of a GLMB RFS, (15.111), with |O| = 1. 15.3.4.7

Approximate Multitarget State Estimation for GLMB Distributions

The MaM multitarget state estimator (5.10) and the JoM multitarget state estimator (5.9) can be applied to GLMB distributions. However, they will usually not be computationally tractable. In their stead, Vo and Vo proposed the following intuitively appealing estimator.5 Let us be given a GLMB distribution ∑ ∏ o ˚ ˚ =δ˚ ˚ f˚˚ ( X) ω ( X ) ˚ so (x, ℓ). (15.143) L Ξ |X|,|XL | o∈O

˚ (x,ℓ)∈X

Then: ˆ so that ω oˆ(L) ˆ is maximal: 1. Step 1: Choose oˆ and L ˆ = arg max ω o (L). (ˆ o, L)

(15.144)

o,L

ˆ = {ℓˆ1 , ..., ℓˆnˆ } where n ˆ is the estimate of the number 2. Step 2: Let L ˆ = |L| of tracks. 3. Step 3: For i = 1, ..., n ˆ , define the state estimates of the tracks to be the ˆ means of the track distributions corresponding to the labels in L: ∫ x ˆi = x · ˚ soˆ(x, ℓˆi )dx. (15.145) 5

As of the time of writing, it is not known whether or not this estimator is Bayes-optimal or even approximately so.

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15.4

465

MODELING FOR THE VO-VO FILTER

The purpose of this section is to begin discussion of the Vo-Vo exact closed-form multitarget filter by specifying its models. The section is organized as follows: 1. Section 15.4.1: Labeling conventions for the Vo-Vo filter. 2. Section 15.4.2: An overview of the Vo-Vo filter. 3. Section 15.4.3: Basic motion and measurement models. 4. Section 15.4.4: Motion and measurement models for joint multitarget detection, tracking, and identification/classification. 5. Section 15.4.5: The multitarget likelihood function for the labeled standard multitarget measurement model. 6. Section 15.4.6: The multitarget Markov density for the labeled version of the standard multitarget motion model. 7. Section 15.4.7: The multitarget Markov density for the labeled version of the modified standard multitarget motion model. 15.4.1

Labeling Conventions

Thus far, the space L of all possible track labels has been treated abstractly as an unspecified, countably infinite set. For the Vo-Vo filter, it is assumed to have the specific form L = I+ × N (15.146) where I+ = {0, 1, ...} is the set of nonnegative integers and N = {1, 2, ...} is the set of natural numbers. Thus the set of all possible labeled target states ˚ x is ˚ X = X × I+ × N.

(15.147)

We must first define some particular label spaces and state spaces (Section 15.4.1.1), and then describe the restrictions that they implicitly impose on the labeled multitarget Bayes filter (Section 15.4.1.2). 15.4.1.1

Special Label and State Spaces

The following definitions are required in what follows, for k = 0, 1, ...:

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• Possible labels of all target-tracks at time tk : L0:k = {0 ≤ l ≤ k} × N.

(15.148)

• Possible labels of new tracks at time tk+1 : LB k+1 = {k + 1} × N

(15.149)

in which case it follows that L0:k+1 = L0:k ⊎ LB k+1 .

(15.150)

• State space of all possible labeled tracks at time tk : ˚ X0:k = X × L0:k .

(15.151)

• State-space of all possible newly appearing labeled tracks at time tk+1 : ˚ XB = X ⊎ LB k+1

(15.152)

˚ X0:k+1 = ˚ X0:k ⊎ ˚ XB .

(15.153)

in which case

The evolution of track labels from one time-step to the next is specified as follows. Let (xk|k , l, i) be the state of a track at time tk with label ℓ = (l, i). If the target does not disappear at time tk+1 , its state will be (xk+1|k , l, i)—that is, its label does not change. Thus, in general, if (x, l, i) ∈ ˚ X0:k then, at time tk , the target with this label: • Originally appeared at time tl . • Was at that time assigned identifying index i. • Currently has the kinematic state x. We therefore have the following strictly nested sequence of labeled state spaces: ˚ X0:0 ⊂ ˚ X0:1 ⊂ ... ⊂ ˚ X0:k ⊂ ˚ X0:k+1 ⊂ ...

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Remark 67 (Labeling and statistical independence) At first, it might appear that the necessity of labeling new tracks results in a theoretical problem: namely, that it is not possible for new target tracks to be statistically independent of existing ones. That is, suppose that at time tk , Lk|k ⊆ L is the finite subset of labels for the currently-existing tracks. Then, at time tk+1 , we must choose labels for the new tracks. If we are to distinguish between tracks, then the labels for these new tracks cannot be in Lk|k and so they must be in L − Lk|k . Ergo, one might conclude that the labeling of new tracks is a procedure that is statistically dependent on prior knowledge of the labels of existing tracks. The labeling convention proposed by Vo and Vo avoids this seeming difficulty. At time tk , the labels of the existing tracks are in L0:k , whereas the labels of the new tracks are in LB k+1 . Thus it is indeed the case that the new labels will be in L − Lk|k . However, they are not chosen by first determining Lk|k and then choosing labels in L − Lk|k . Rather, selection of the new labels requires no prior knowledge of Lk|k . The only information employed is the fact that the new tracks must exist at time tk+1 but cannot exist prior to that—and therefore must reside in LB k+1 . 15.4.1.2

Properties of the Labeled Multitarget Bayes Filter

The Vo-Vo filter is an exact, closed-form solution of the multitarget Bayes filter for labeled targets: ... →

˚ (k) ) f˚k|k (X|Z

→

˚ (k) ) f˚k+1|k (X|Z

→

˚ (k+1) ) f˚k+1|k+1 (X|Z

→ ...

defined by the time-update and measurement-update equations ∫ ˚ (k) ) = ˚X ˚′ ) · f˚k|k (X ˚′ |Z (k) )δ X ˚′(15.154) f˚k+1|k (X|Z f˚k+1|k (X| ˚ (k+1) ) f˚k+1|k+1 (X|Z f˚k+1 (Zk+1 |Z (k) )

˚ · f˚k+1|k (X|Z ˚ (k) ) f˚k+1 (Zk+1 |X) = (15.155) f˚k+1 (Zk+1 |Z (k) ) ∫ ˚ · f˚k+1|k (X|Z ˚ (k) )δ X ˚ (15.156) = f˚k+1 (Zk+1 |X)

˚ X ˚′ ⊆ ˚ and where X, X. Because of the definition of a labeled multitarget state, the ˚L | ̸= |X| ˚ then following must be true: if |X ˚ (k) ) = f˚k+1|k+1 (X|Z ˚ (k+1) ) = f˚k+1|k (X| ˚X ˚′ ) = 0. f˚k+1|k (X|Z That is,

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˚ (k) ) of a physically unrealizable state • The probability (density) f˚k+1|k (X|Z ˚ is nil. set X In addition, ˚ f˚k+1 (Z|X) ˚X ˚′ ) f˚k+1|k (X|

=

κk+1 (Z) ˚ = ˚ bk+1|k (X)

˚L | ̸= |X| ˚ |X ′ ˚L | ̸= |X ˚′ |. if |X

if

(15.157) (15.158)

˚L | ̸= |X| ˚ ˚ is The first equation follows from the fact that if |X then X physically unrealizable, and thereby cannot generate measurements. Thus the only measurements are those due to the clutter process κk+1 (Z). The second equation ˚′ is unrealizable then it cannot persist into the next follows from the fact that if X time-step. Thus the only targets in the next time-step are those due entirely to the ˚ target appearance process ˚ bk+1|k (X). In addition, for any k ≥ 0, ˚ (k+1) ) = 0 unless X ˚ ⊆ ˚ ˚ • f˚k+1|k+1 (X|Z X0:k+1 . For if otherwise, X contains a target that cannot yet exist at time tk+1 . ˚X ˚′ ) = 0 unless X ˚ ⊆˚ ˚ contains a • f˚k+1k (X| X0:k+1 . For if otherwise, X target that cannot exist at time tk+1 ˚X ˚′ ) = ˚ ˚ if it is not the case that X ˚′ ⊆ ˚ • f˚k+1k (X| bk+1|k (X) X0:k . For if ′ ˚ otherwise, X contains a target that cannot exist at time tk . ˚X ˚′ ) = 0 unless • If there is no model for target appearance, then f˚k+1k (X| ˚⊆˚ ˚ contains a target that did not originate with X X0:k . For if otherwise, X ′ ˚ a target in X . ˚X ˚′ ) = 0 unless X ˚L ⊆ X ˚′ . For, the only possible • Furthermore, f˚k+1k (X| L ′ ˚ are those targets in X ˚ that survived from time tk to time targets in X tk+1 . ˚ = κk+1 (Z) if it is not the case that X ˚ ⊆ ˚ • f˚k+1 (Z|X) X0:k+1 . For if ˚ otherwise, X contains a target that cannot exist at time tk+1 and thus it cannot generate measurements. 15.4.2

Overview of the Vo-Vo Filter

The purpose of this section is to provide a “road map” of the Vo-Vo filter, before we plunge into its technical details in the sections that follow.

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469

First, suppose that the initial labeled multitarget distribution is a LMB distribution: ˚ ˚ = δ ˚ ˚ · ω0|0 (X ˚L ) · (˚ f˚0|0 (X) s0|0 )X (15.159) |X|,|XL | where ω0|0 (L) = 0 for all but a finite number of finite L ⊆ L0:0 . ˚X ˚′ ) Second, suppose that the labeled multitarget Markov density f˚k+1|k (X| has the following generalized form (to be described more fully in Section 15.4.7): • Persisting targets are governed by the labeled version of the modified standard multi-Bernoulli motion model; and this means that the target appearance process is a LMB RFS. Then (as will be shown in Section 15.5.3) the predicted distribution is LMB: ∫ ˚ ˚X ˚′ ) · f˚0|0 (X ˚′ )δ X ˚′ f˚1|0 (X|Z) = f˚k+1|k (X| (15.160) =

˚ ˚ δ|X|,| s1|0 )X ˚ X ˚L | · ω1|0 (XL ) · (˚

(15.161)

for some ˚ s1|0 (x, ℓ) and some ω1|0 (L) with ω1|0 (L) = 0 for all but a finite number of finite L ⊆ L1:0 . Third, suppose that the multitarget likelihood function is the labeled version of the standard multitarget measurement model (to be described more fully in Section 15.4.5). Let Z1 be the measurement set collected at time t1 and let m1 = |Z1 |. Then (as will be shown in Section 15.5.2) the Bayes-updated distribution is a GLMB distribution: ˚ (1) ) f˚1|1 (X|Z

= =

˚ · f˚1|0 (X) ˚ f˚1 (Z1 |X) ˚ ) · f˚1|0 (Y ˚ )δ Y ˚ f˚1 (Z1 |Y ∑ ˚ ˚1 X ˚ θ1 ˚ δ|X|,| ω1|1 (XL ) · (˚ sθ1|1 ) ˚ X ˚L | ∫

(15.162) (15.163)

˚ θ1 ∈TZ1 ˚1 ˚ ˚ θ1 θ1 for some ˚ sθ1|1 (x, ℓ) and some ω1|1 (L) with, for each ˚ θ1 , ω1|0 (L) = 0 for all but a finite number of finite L ⊆ L1:0 . Also, the summation is taken over all measurement-to-track associations—that is, all functions ˚ θ1 : L0:1 → {0, 1, ..., m1 } such that ˚ θ1 (ℓ) = ˚ θ1 (ℓ′ ) implies ℓ = ℓ′ . Fourth, apply the next time-update step, in which case we will get a GLMB distribution of the form ∑ ˚ ˚1 X ˚ θ1 ˚ ˚ (1) ) = δ ˚ ˚ f˚2|1 (X|Z ω2|1 (XL ) · (˚ sθ2|1 ) (15.164) |X|,|XL | ˚ θ1 ∈TZ1

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˚

θ1 with, for each ˚ θ1 , ω2|1 (L) = 0 for all but a finite number of finite L ⊆ L2:0 . Fifth, let Z2 be the measurement set collected at time t2 and let m2 = |Z2 |. Then the result of the next Bayes’ rule update will be a GLMB distribution ∑ ˚ ˚1 ,˚ ˚ θ1 ,˚ θ2 ˚ θ2 X ˚ (2) ) = δ ˚ ˚ f˚2|2 (X|Z ω2|2 (XL ) · (˚ sθ2|2 ) (15.165) |X|,|XL | (˚ θ1 ,˚ θ2 )∈TZ1 ×TZ2 ˚ ˚1 ,˚ θ2 θ1 ,˚ θ2 for some ˚ sθ2|2 (x, ℓ) and ω2|2 (L) and where, for each ˚ θ1 , ˚ θ2 , it must be the ˚ ˚

θ 1 ,θ 2 case that ω2|2 (L) = 0 for all but a finite number of finite L ⊆ L2:0 . Also, ˚ θ2 : L0:2 → {0, 1, ..., m2 } is such that ˚ θ2 (ℓ) = ˚ θ2 (ℓ′ ) implies ℓ = ℓ′ . Proceeding in this fashion, after k + 1 recursions of the time- and measurement-update steps, we will end up with GLMB distributions of the form

˚ (k+1) ) f˚k+1|k+1 (X|Z

=

δ|X|,| ˚ X ˚L | ∑ ·

(15.166) ˚ ˚ ˚ θ1 ,...,˚ θk+1 ˚ θ1 ,...,˚ θk+1 X ωk+1|k+1 (XL ) · (˚ sk+1|k+1 )

(˚ θ1 ,...,˚ θk+1 )

˚ (k) ) f˚k+1|k (X|Z

=

δ|X|,| ˚ X ˚L | ∑ ˚ ˚1 ,...,˚ ˚ θ1 ,...,˚ θk ˚ θk X (XL ) · (˚ sθk+1|k ) · ωk+1|k

(15.167)

(˚ θ1 ,...,˚ θk )

where, for j = 1, ..., k, the functions ˚ θj : L0:j → {0, 1, ..., |Zj |} are such that ˚ ˚ ′ ′ ˚ ˚ θj (ℓ) = θ(ℓ ) implies ℓ = ℓ . Because ω θ1 ,...,θk (L) = 0 for all but a finite k|k

˚ (k) ) = 0 number of finite L ⊆ Lk:0 , for each ˚ θ1 , ..., ˚ θk , it follows that f˚k|k (X|Z ˚ ⊆ Xk:0 . unless X Equations (15.166) and (15.167) can be intuitively interpreted as follows: ˚ = n and L = X ˚L = • Interpretation of the measurement-update: Let |X| ˚ θ ,...,˚ θ

1 k+1 {ℓ1 , ..., ℓn }. Then ωk+1|k+1 (L) is the weight of the hypothesis that:

– There are n tracks with distinct labels ℓ1 , ..., ℓn . ˚ θ ,...,˚ θ

˚ θ ,...,˚ θ

1 1 k+1 k+1 – Their track distributions are ˚ sk+1|k+1 (x, ℓ1 ), ...,˚ sk+1|k+1 (x, ℓn ).

– These distributions arose as a result of the time history ˚ θ1 , ..., ˚ θk , ˚ θk+1 of measurement-to-track associations, including the latest association ˚ θk+1 .

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471

˚ θ ,...,˚ θ

1 k+1 – For each i = 1, ..., n, ˚ sk+1|k+1 (x, ℓi ) will be of two types (see (15.253)):

∗ The distribution of a track that was not detected, and which therefore arose from the previous time history ˚ θ1 , ..., ˚ θk . ∗ The distribution of a track that was detected, and which therefore arose from the current time history ˚ θ1 , ..., ˚ θk , ˚ θk+1 . ˚ = n and L = X ˚L = {ℓ1 , ..., ℓn }. • Interpretation of the time-update: Let |X| ˚

˚

1 ,...,θk Once again, ˚ sθk+1|k (x, ℓi ) will be of two types (see (15.281)):

– The distribution of a track that persisted from the previous time tk , and which therefore arose from the previous time history ˚ θ1 , ..., ˚ θk . – the distribution of a newly appearing track, and which will therefore be independent of ˚ θ1 , ..., ˚ θk . Now let D be the space of all Vo-Vo priors. At time tk these distributions are parametrized by the space P of a finite list of parameters of the form ˚

p

˚

θ1 ,...,θk ( ωk|k ({ℓ1 , ..., ℓn }),

=

˚

˚

˚

˚

(15.168)

θ1 ,...,θk ˚ sk|k (x, ℓ1 ), ..., 1 ,...,θk ˚ sθk|k (x, ℓn ) )k,n,ℓ1 ,...,ℓn ,˚ θ1 ,...,˚ θk .

It then follows that the GLMB distributions constitute an exact closed-form solution of the labeled multitarget Bayes filter, in the sense of Section 15.1.2. Consequently, the labeled multitarget Bayes filter can be replaced by a filter that propagates these parameters rather than their associated GLMB distributions: ... →

pk|k

→

pk+1|k

→

pk+1|k+1

→ ...

If one further employs the multitarget state-estimator of Section 15.3.4.7, the ˚k+1|k+1 can be constructed directly from the parameter multitarget state estimate X pk+1|k+1 . Thus this filter has the further form

... →

pk|k

→

pk+1|k

→

˚k+1|k+1 X ↑ pk+1|k+1

→ ...

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15.4.3

Basic Motion and Measurement Models

The Vo-Vo filter requires the following basic models: • Probability of target survival: ˚ pS (x′ , ℓ′ ) abbr. = ˚ pS,k+1|k (x′ , ℓ′ ). Since a target cannot persist if it has not yet appeared, for completeness we must define ˚ pS,k+1|k (x′ , l′ , i′ ) = 0

l′ > k.

if

(15.169)

• Single-target Markov transition density: f˚k+1|k (x, ℓ|x′ , ℓ′ ) = fk+1|k (x|x′ , ℓ′ ) · δℓ,ℓ′

(15.170)

where fk+1|k (x|x′ , ℓ′ ) is the conventional single-target Markov density for a target with label ℓ′ and where the Kronecker delta δℓ,ℓ′ specifies that transitioning targets retain their labels in the manner described at the end of Section 15.4.1.1.6 • Single-target probability of detection: ˚ pD (x, ℓ) abbr. = ˚ pD,k+1|k (x, ℓ). Since no measurement can be generated by a target that has not yet appeared, for completeness we must define ˚ pD,k+1|k (x, l, i) = 0

if

l > k + 1.

(15.171)

• Single-target likelihood function: ˚ Lz (x, ℓ) abbr. = f˚k+1 (z|x, ℓ).7 • Poisson clutter process with clutter rate λ abbr. = λk+1 and clutter spatial distribution ck+1 (z) and intensity function κ(z) abbr. = λk+1 · ck+1 (z): κk+1 (Z) = e−λ

∏

κ(z) = e−λ · κZ .

z∈Z

6

Since f˚k+1 (x, ℓ|x′ , ℓ′ ) will not appear in formulas except in the products ˚ pS (x′ , ℓ′ ) · f˚k+1 (x, ℓ|x′ , ℓ′ )

7

it is not necessary to define f˚k+1 (x, ℓ|x′ , l′ , i′ ) for l′ > k. Since f˚k+1 (z|x, ℓ) will not appear in formulas except in products ˚ pD (x, ℓ) · f˚k+1 (z|x, ℓ) it is not necessary to define f˚k+1 (z|x, l, i) for l > k + 1.

(15.172)

Exact Closed-Form Multitarget Filter

15.4.4

473

Motion and Measurement Models with Target ID

Vo and Vo have shown that the Vo-Vo filter is easily extended to incorporate target identity [307]. As a consequence, this filter provides an exact closed-form and Bayes-optimal approach to joint target detection, tracking, localization, and identification/classification. The purpose of this section is to briefly describe this approach. Let T = {τ1 , ..., τN } be a finite set of target-identity types. Replace the kinematic state space X with the joint space X × T, so that the labeled state space is X × T × L. Also, the sensor measurement space will have the form Z × F, where Z is the kinematic measurement space and F is a space of feature vectors ϕ. Given this, the basic models in the previous section take the following forms: • Probability of target survival: ˚ pS (x′ , τ ′ , ℓ′ ) abbr. =˚ pS,k+1|k (x′ , τ ′ , ℓ′ ). • Single-target Markov transition density: f˚k+1|k (x, τ, ℓ|x′ , τ ′ , ℓ′ ) = =

′

′

′

(15.173) ′

′

′

fk+1|k (x|x , τ, τ , ℓ ) · pk+1|k (τ |x , τ , ℓ ) · δℓ,ℓ′ fk+1|k (x|x′ , τ ′ , ℓ′ ) · pk+1|k (τ |τ ′ ) · δℓ,ℓ′ .

(15.174)

If target identity does not change with time, then pk+1|k (τ |τ ′ ) = δτ,τ ′ . • Single-target probability of detection: ˚ pD (x, τ, ℓ) abbr. =˚ pD,k+1|k (x, τ, ℓ). • Single-target likelihood function: ˚ Lz,ϕ (x, τ, ℓ)

(15.175)

=

f˚k+1 (z, ϕ|x, τ, ℓ) f˚k+1 (z|x, τ, ℓ, ϕ) · f˚k+1 (ϕ|x, τ, ℓ)

=

f˚k+1 (z|x, τ, ℓ) · f˚k+1 (ϕ|x, τ, ℓ).

(15.177)

=

(15.176)

If target features do not depend on target kinematics, then f˚k+1 (ϕ|x, τ, ℓ) = f˚k+1 (ϕ|τ, ℓ) (see Remark 33 in Section 9.5.8.1). • Poisson clutter process: κk+1 (Z) = e−λ

∏ z∈Z

κ(z) = e−λ · κZ .

(15.178)

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15.4.5

Advances in Statistical Multisource-Multitarget Information Fusion

The Labeled Multitarget Likelihood Function

The purpose of this section is to: 1. Provide the specific formula for the multitarget likelihood function for the labeled version of the standard multitarget measurement model. 2. Derive the GLMB reformulation of it that is necessary for achieving exact closed-form closure with respect to the multitarget version of Bayes’ rule. ˚ = {(x1 , ℓ1 ), ..., (xn , ℓn )} with |X| = n and let |Z| = m. Let X Then from (7.21) it follows that the multitarget likelihood function for the labeled standard multitarget measurement model is ˚ f˚k+1 (Z|X)  =

e

(15.179) 

1 − δ|X|,| ˚ X ˚L |   ˚  +δ ˚ X pD )X ˚L | · (1 − ˚ κ ·  ∑ ∏ |X|,|  ˚ pD (xi ,ℓi )·f˚k+1 (zθ(ℓ) |xi ,ℓi ) · θ i:θ(i)>0 (1−˚ pD (xi ,ℓi ))·κ(zθ(i) )

−λ Z

∏ where as usual κZ = z∈Z κ(z), and where the summation is taken over all functions θ : {1, ..., n} → {0, 1, ..., m} such that θ(i) = θ(i′ ) > 0 implies i = i′ . ˚ is not physically realizable—that is, if |X| ˚ ̸= | X ˚L |—then (15.179) If X reduces to ˚ = e−λ κZ . f˚k+1 (Z|X) (15.180) That is, there can be no target-generated measurements, and so the only measurements are those due to clutter. ˚ is physically realizable, then (15.179) reduces to a distinct-label version If X of the likelihood function for the standard multitarget measurement model, (7.21): ˚ ˚ = e−λ κZ (1 − ˚ f˚k+1 (Z|X) pD )X

∑

∏

θ

i:θ(i)>0

˚ pD (xi , ℓi ) · f˚k+1 (zθ(ℓ) |xi , ℓi ) . (1 − ˚ pD (xi , ℓi )) · κ(zθ(i) )

(15.181) ˚∩˚ ˚ contains a target with a state (xj , lj , ij ) Note that if X X0:k+2 ̸= ∅ then X with lj > k + 1. Thus ˚ pD (xj , lj , ij ) = 0 and so the product in (15.179) vanishes and so ˚ = e−λ κZ . f˚k+1 (Z|X) (15.182)

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475

Now define ˚ ˚ LθZ (x, ℓ) = δ0,˚ pD (x, ℓ)) + (1 − δ0,˚ θ(ℓ) · (1 − ˚ θ(ℓ) ) ·

˚ pD (x, ℓ) · f˚k+1 (z˚ θ(ℓ) |x, ℓ) κ(z˚ θ(ℓ) )

(15.183) where ˚ θ : L → {0, 1, ..., m} is any function such that (1) ˚ θ(ℓ) = ˚ θ(ℓ′ ) > 0 implies ℓ = ℓ′ and (2) ˚ θ(ℓ) > 0 for only a finite number of ℓ ∈ L. Then (15.179) can be equivalently written in GLMB-like form:8   ∏ ∑ ˚ ˚ ˚ = e−λ κZ ·1 − δ ˚ ˚ + δ ˚ ˚ ˚L ) ˚ f˚k+1 (Z|X) λθk+1 (X LθZ (x, ℓ) |X|,|XL | |X|,|XL | ˚ θ

˚ (x,ℓ)∈X

(15.184) where9 ˚ ˚ λθk+1 (L) =

{ ∏

ℓ∈L0:k+1 −L δ˚ θ(ℓ),0

0

if if

L ⊆ L0:k+1 is finite . otherwise

(15.185)

˚L | = |X| ˚ = n, in which To see why (15.184) is true, first assume that |X case ℓ1 , ..., ℓn are distinct. Then (15.184) becomes ˚ = e−λ κZ f˚k+1 (Z|X)

∑

˚ ˚ ˚L ) λθk+1 (X

˚ θ

n ∏

˚ ˚ LθZ (xi , ℓi ).

(15.186)

i=1

Second, note that, because of (15.185), the summation in (15.186) is finite. For, ˚ ˚L ) ̸= 0—that is, for all ˚ it is actually taken over all ˚ θ such that λθk+1 (X θ that ˚ ˚ have the following additional property: θ(ℓ) = 0 for all ℓ not in XL . There is a one-to-one correspondence between such ˚ θ and the finite set of all functions ˜ = θ(ℓ ˜ ′ ) implies ℓ = ℓ′ . Third, notice that ˚L → {0, 1, ..., m} such that θ(ℓ) θ˜ : X ˜ ˚ LθZ (x, ℓ)

8

9

It is not true that

=

{

∑ ∑

1 − pD (x, ℓ)

if

pD (x,ℓi )·f˚k+1 (zθ(ℓ) |x,ℓ) ˜ κ(zθ(ℓ) ) ˜

if

˚ θ ˚ θ λk+1 (L)

˜ =0 θ(ℓ) ˜ >0 . θ(ℓ)

(15.187)

= 1, so the expression in (15.184) is, strictly speaking, not ˚ need not be a GLMB distribution in the GLMB. It need not be so, however, since fk+1 (Z|X) ˚ variable X. ˚ Note: It will always be clear from context that ˚ λθk+1 (L) does not mean the same thing as the clutter rate λk+1 of the Poisson clutter process. L

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Advances in Statistical Multisource-Multitarget Information Fusion

Thus (15.186) becomes

˚ f˚k+1 (Z|X)

=

∑

e−λ κZ

θ˜



˜ i )=0 i:θ(ℓ

e

(1 − pD (xi , ℓi ))

(15.188)

κ(zθ(ℓ ˜ i))

˜ i )>0 i:θ(ℓ

=



∏

pD (xi , ℓi ) · f˚k+1 (zθ(ℓ ˜ i ) |xi , ℓi )

∏

·



−λ Z

κ ·

(

n ∏

(1 − pD (xi , ℓi ))

)

(15.189)

i=1

·

∑ θ˜

 

∏ ˜ i )>0 i:θ(ℓ

pD (xi , ℓi ) · f˚k+1 (zθ(ℓ ˜ i ) |xi , ℓi ) (1 − ˚ pD (xi , ℓi )) · κ(zθ(ℓ ˜ i))

 

and so ˚ f˚k+1 (Z|X)

˚

=

e−λ κZ · (1 − ˚ pD )X (15.190)   ∑ ∏ pD (xi , ℓi ) · f˚k+1 (zθ(ℓ ˜ i ) |xi , ℓi )   · (1 − ˚ pD (xi , ℓi )) · κ(zθ(ℓ ˜ i)) θ˜

˜ i )>0 i:θ(ℓ

˚

=

e−λ κZ · (1 − ˚ pD )X (15.191)   ∑ ∏ pD (xi , ℓi ) · f˚k+1 (zθ(i) |xi , ℓi )   · (1 − ˚ pD (xi , ℓi )) · κ(zθ(i) ) θ

i:θ(i)>0

which is identical to (15.181). 15.4.6

The Labeled Multitarget Markov Density—Standard Version

The purpose of this section is to: 1. Provide the specific formula for the multitarget Markov density for the labeled version of the standard multitarget motion model. 2. Provide the specific formula for the special case of this density in which there is no target appearance process.

Exact Closed-Form Multitarget Filter

477

3. Derive the GLMB reformulation of this latter Markov density (necessary for achieving exact closed-form closure with respect to the multitarget prediction integral). Let ˚ X ˚′ X

=

{(x1 , ℓ1 ), ..., (xn , ℓn )}

(15.192)

=

{(x′1 , ℓ′1 ), ..., (x′n′ , ℓ′n′ )}

(15.193)

with |X| = n and |X ′ | = n′ . Then from (7.66) it follows that the Markov density for the labeled version of the standard multitarget motion model is ˚X ˚′ ) f˚k+1|k (X| 

=

e

B −Nk+1|k

 ˚  bX k+1|k · 

(15.194) 

1 − δn′ ,|X ˚′ | L

˚′

+δn′ ,|X p S )X ˚′ | · (1 − ˚ ·

∑ ∏ θ

i:θ(i)>0

L ˚ pS (x′i ,ℓ′i )·f˚k+1 (xθ(i) |x′i ,ℓ′i )·δℓ ′ θ(i) ,ℓi (1−˚ pS (x′i ,ℓ′i ))·bk+1|k (xθ(i) )

  

where the summation is taken over all functions θ : {1, ..., n′ } → {0, 1, ..., n} such that θ(i) = θ(i′ ) implies i = i′ . Suppose that no new targets appear, and thus that the only targets are those that survive from one time-step to the next. Then because of (7.69), we can equivalently write (15.194) as follows. If n ≤ n′ then − ˚X ˚′ ) f˚k+1|k (X|

˚′

=

δX,∅ pS )X (15.195) ˚ · (1 − δn′ ,|X ˚′ | ) + δn′ ,|X ˚′ | · (1 − ˚ L

L

n ∑ ∏ ˚ pS (x′τ i , ℓ′τ i ) · f˚k+1 (xi |x′τ i , ℓ′τ i ) · δi,τ i · 1−˚ pS (x′τ i , ℓ′τ i ) τ i=1

where the summation is taken over all τ : {1, ..., n} → {1, ..., n′ } τ i = τ i′ implies i = i′ . Otherwise, if n > n′ , − ˚X ˚′ ) = δ ˚ . f˚k+1|k (X| X,∅

such that

(15.196)

Given this, (15.195) can be written in a GLMB-like form, as follows. If ˚ ≤ |X ˚′ | then |X| ∏ − ˚X ˚′ ) = δ ˚ ·(1− δ ˚′ ˚′ )+ δ ˚ ˚ ·β ˚′ (X ˚L ) ˜ ˚ (x′ , ℓ′ ) f˚k+1|k (X| M X,∅ |X |,|X | |X|,|XL | X X L

L

˚′ (x′ ,ℓ′ )∈X

(15.197)

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Advances in Statistical Multisource-Multitarget Information Fusion

where10 βL′ (L)

=

∏

(15.198)

1L′ (ℓ)

ℓ∈L

˜ ˚ (x′ , ℓ′ ) M X

=

′ (1 − 1X pS (x′ , ℓ)) (15.199) ˚ (ℓ )) · (1 − ˚ ∑L ′ ′ ′ ′ δℓ,ℓ′ · ˚ pS (x , ℓ ) · fk+1|k (x|x , ℓ ). + ˚ (x,ℓ)∈X

˚ > |X ˚′ | then Otherwise, if |X| − ˚X ˚′ ) = δ ˚ . f˚k+1|k (X| X,∅

(15.200)

˚ ≤ |X ˚′ | and then that |X ˚′ | = | X ˚′ |. To prove (15.197), first assume that |X| L Then (15.195) becomes n ∑ ∏ ˚ pS (x′τ i , ℓ′τ i ) · f˚k+1 (xi |x′τ i , ℓ′τ i ) · δi,τ i . 1−˚ pS (x′τ i , ℓ′τ i ) τ i=1 (15.201) Because of the factor δi,τ i , the only term in the summation that survives is that corresponding to the unique function τ such that τ i = i for all i = 1, ..., n. Thus (15.201) becomes ˚′ − ˚X ˚′ ) = (1 − ˚ f˚k+1|k (X| pS )X

˚′ ˚X ˚′ ) = (1 − ˚ f˚k+1|k (X| p S )X

n ∏ ˚ pS (x′i , ℓ′i ) · f˚k+1 (xi |x′i , ℓ′τ i ) . 1−˚ pS (x′i , ℓ′i ) i=1

(15.202)

10 That is, βL′ (L) = 0 unless L ⊆ L′ . The notation βL′ (L) in (15.198) follows from the fact that βL′ (L) is the belief-mass function of the deterministic RFS Λ whose instantiations are the subsets of L′ . In this case the belief-mass function is ∏ βΛ (L) = Pr(Λ ⊆ L) = Pr(L′ ⊆ L) = 1L′ ⊆L = 1L′ (ℓ). ℓ∈L

Exact Closed-Form Multitarget Filter

479

˚L ⊆ X ˚′ —that is, the label-set for Under the same conditions and assuming that X L surviving targets remains unchanged— (15.197) becomes ′

− ˚X ˚′ ) f˚k+1|k (X|

=

n ∏

˜ ˚ (x′i , ℓ′i ) M X

(15.203)

i=1

=

(

=

(

n ∏ i=1 n ∏ i=1

)

˜ ˚ (x′i , ℓ′i )  M X )

˜ ˚ (x′i , ℓ′i )  M X



′

n ∏ i=n+1

˜ ˚ (x′i , ℓ′i ) (15.204) M X 

′

n ∏ i=n+1

˜ ˚ (x′i , ℓ′i ) . (15.205) M X

But n ∏

˜ ˚ (x′i , ℓ′i ) M X

i=1 n ( ∏

=

i=1 n ∏

=

(15.206)

′ (1 − 1X pS (x′i , ℓi )) ˚L (ℓi )) · (1 − ˚ ∑n ′ ′ + j=1 δℓj ,ℓ′i · ˚ pS (xi , ℓi ) · fk+1|k (xj |x′i , ℓ′i )

)

( ) ˚ pS (x′i , ℓ′i ) · fk+1|k (xi |x′i , ℓ′i )

(15.207)

i=1

and ′

n ∏

′

˜ ˚ (x′i , ℓ′i ) = M X

i=n+1

n ∏

(1 − ˚ pS (x′i , ℓi ))

(15.208)

i=n+1

and so ˚X ˚′ ) = f˚k+1|k (X|

(

 )  n′ n ∏ ∏ ( ) ˚ pS (x′i , ℓ′i ) · fk+1|k (xi |x′i , ℓ′i ) · (1 − ˚ pS (x′i , ℓ′i ))

i=1



i=n+1

 ( ) n ′ ′ ′ ′ ∏ ˚ p (x , ℓ ) · f (x |x , ℓ ) S i k+1|k i i i i =  (1 − ˚ pS (x′i , ℓi )) · 1−˚ pS (x′i , ℓi ) i=1 i=1 ′

n ∏

˚′ X

= (1 − ˚ pS )

n ∏ ˚ pS (x′ , ℓ′ ) · f˚k+1 (xi |x′ ) i

i=1

which, as claimed, is just (15.202).

i

1−˚ pS (x′i , ℓ′i )

i

(15.209)

480

15.4.7

Advances in Statistical Multisource-Multitarget Information Fusion

Labeled Multitarget Markov Density—Modified

The multitarget Markov density for the standard multitarget motion model is based on the presumption that the target appearance RFS is Poisson. For the modified standard multitarget motion model of (15.23), the target appearance RFS is LMB in the sense of (15.85): ˚ ˚ bk+1|k (X)

=

B ˚ δ|X|,| ˚ X ˚L | · ωk+1|k (XL )

∏

˚ sB k+1|k (x, ℓ) (15.210)

˚ (x,ℓ)∈X

=

˚ B X ˚ δ|X|,| sB ˚ X ˚L | · ωk+1|k (XL ) · (˚ k+1|k ) .

(15.211)

The purpose of this section is to derive the formula for the multitarget Markov density that incorporates this more general target appearance model. The standard multitarget Markov density is also based on the presumption that the persisting-target process is the same as that for the standard multitarget motion model with no target appearances, that is, (15.197): ∏

− ˚X ˚′ ) = δ ˚ ·(1−δ ˚′ ˚′ )+δ ˚ ˚ ·β ˚′ (X ˚L ) f˚k+1|k (X| X,∅ |X |,|X | |X|,|XL | X L

L

˜ ˚ (x′ , ℓ′ ). M X

˚′ (x′ ,ℓ′ )∈X

(15.212) Given that target appearances occur independently of existing targets, the total Markov transition density is given by the convolution rule, (4.18): ˚X ˚′ ) = f˚k+1|k (X|

∑

˚ ˚−W ˚ ) · f˚− (W ˚ |X ˚′ ). bk+1|k (X k+1|k

(15.213)

˚ ⊆X ˚ W

Equation (15.213) can be greatly simplified as follows. Using the notation ˚ can be partitioned into of Section 15.2.1, the time-updated multitarget state X − + ˚ ˚ surviving targets X and appearing targets X : ˚=X ˚− ⊎ X ˚+ X

(15.214)

where ˚− = X ˚∩˚ X X0:k ,

˚+ = X ˚∩˚ X XB k+1 .

(15.215)

Given this, (15.213) reduces to the following simple factored form: ˚− | X ˚′ ). ˚X ˚′ ) = ˚ ˚+ ) · f˚− (X f˚k+1|k (X| bk+1|k (X k+1|k

(15.216)

Exact Closed-Form Multitarget Filter

481

To demonstrate (15.216), we want to show that the only nonzero term in the ˚ =X ˚− and thus summation on the right side of (15.213) is the one for which W − + ˚ ˚ ˚ ˚ ˚ X − W = X − X = X . To see this, note from the discussion in Section 15.4.1 − ˚ |X ˚′ ) = 0 unless W ˚ ⊆ X ˚− and thus W ˚ = W ˚ −, W ˚ + = ∅. that f˚k+1|k (W ˚−W ˚ ) = 0 unless X ˚−W ˚ ⊆X ˚+ . Given this, Similarly, ˚ bk+1|k (X ˚+ ⊇ X ˚−W ˚ = (X ˚− − W ˚ − ) ⊎ (X ˚+ − W ˚ + ) = (X ˚− − W ˚ −) ⊎ X ˚+ . (15.217) X ˚− − W ˚ − = ∅ since it cannot be in X ˚+ , and thus From this it follows that X ˚ =W ˚− = X ˚− , as desired. W Thus the total Markov density has the form ˚X ˚′ ) f˚k+1|k (X| =

(15.218)

B ˚+ ) ωk+1|k (X L

δ |X ˚B |,|X ˚B | · L   ∏  · sB k+1|k (x, ℓ) ˚B (x,ℓ)∈X

·

( +δ|X ˚S |,|X ˚S | L

δX ˚S ,∅ · (1 − δ|X ˚′ |,|X ˚′ | ) L ˚S ) · ∏ ′ ′ ˚′ M ˜ ˚S (x′ , ℓ′ ) · β ˚′ (X XL

L

(x ,ℓ )∈X

)

X

or ˚X ˚′ ) f˚k+1|k (X|

=

(

˚+ B B X ˚+ δ |X ˚B |,|X ˚+ | · ωk+1|k (XL ) · (sk+1|k ) L ( ) ˚′ ˚− ˜X · δ |X ˚− |,|X ˚− | · βX ˚′ ( X L ) · M X − ˚ L

)

(15.219)

L

or ˚X ˚′ ) f˚k+1|k (X|

=

( ) ˚+ + B X ˚+ δ|X|,| (15.220) ˚ X ˚L | · ωk+1|k (XL ) · (sk+1|k ) ( ) ˚′ ˚− ˜X · βX ˚′ (XL ) · MX ˚− . L

15.5

CLOSURE OF MULTITARGET BAYES FILTER

The purpose of this section is to demonstrate that the set of all Vo-Vo priors (GLMB multitarget distributions) provides an exact closed-form solution of the multitarget Bayes filter, in the sense described in Section 15.1.2. The section is organized as follows:

482

Advances in Statistical Multisource-Multitarget Information Fusion

1. Section 15.5.1: A “road map” of the derivations of the time-update and measurement-update equations. 2. Section 15.5.2: A demonstration that GLMB distributions have exact closedform closure under the multitarget Bayes filter measurement-update step (multitarget Bayes’ rule). 3. Section 15.5.3: A demonstration that GLMB distributions have exact closedform closure under the multitarget Bayes filter time-update (multitarget prediction integral). 15.5.1

A “Road Map” for the Derivations

The purpose of this section is to provide a “road map” of the proof that the family of GLMB distributions provides an exact closed-form solution of the multitarget Bayes filter. This roadmap is has two parts, one for the time-update (Section 15.5.1.1) and one for the measurement-update (Section 15.5.1.2). Both derivations require the following lemma (Lemma 3 of [295]): Lemma 1 Suppose that ω(L) ̸= 0 for only a finite number of finite subsets L ⊆ L. ∫ Let ˚ s(x, ℓ) be any function such that ˚ s(x, ℓ)dx = 1 for all ℓ; and let ˚ h(x, ℓ) ∫ ˚ be any function such that h(x, ℓ) · ˚ s(x, ℓ)dx exists for all ℓ. Then: ∫

˚ ˚ ˚s)X ˚ δ|X|,| δX = ˚ X ˚L | · ω(XL ) · (h˚

∑

ω(L)

L⊆L

∏∫

˚ h(x, ℓ) · ˚ s(x, ℓ)dx. (15.221)

ℓ∈L

To prove (15.221), note that ∫ ˚ X ˚ ˚ ˚ (˚ hX ˚ s) · δ|X|,| (15.222) ˚ X ˚L | · ω(XL )δ X ∫ ∑ 1 ∑ ˚ h(x1 , ℓ1 ) · ˚ s(x1 , ℓ1 ) · · · ˚ h(xn , ℓn ) = n! n n≥0

(ℓ1 ,...,ℓn )∈L

·˚ s(xn , ℓn ) · δn,|{ℓ1 ,...,ℓn }| · ω({ℓ1 , ..., ℓn })dx1 · · · dxn

=

∑ 1 n! n≥0

∑

n (∫ ∏

˚ h(x, ℓi ) · ˚ s(x, ℓi )dx

(ℓ1 ,...,ℓn )∈Ln i=1

·δn,|{ℓ1 ,...,ℓn }| · ω({ℓ1 , ..., ℓn })

)

(15.223)

Exact Closed-Form Multitarget Filter

=

∑

∑

ω(L)

n≥0 L∈Fn (L)

=

∑

ω(L)

L⊆L

15.5.1.1

∏ (∫

483

˚ h(x, ℓ) · ˚ s(x, ℓ)dx

)

(15.224)

ℓ∈L

∏ (∫

) ˚ h(x, ℓ) · ˚ s(x, ℓ)dx .

(15.225)

ℓ∈L

Time Update Derivation: Road Map

A major reason that the GLMB distributions (Vo-Vo priors) provide a computationally tractable closed-form solution to the Bayes filter is the following fact: • the convolutional formula for the labeled multitarget Markov density (15.213), ˚X ˚′ ) = f˚k+1|k (X|

∑

˚ ˚−W ˚ ) · f˚− (W ˚ |X ˚′ ), bk+1|k (X k+1|k

(15.226)

˚ ⊆X ˚ W

reduces to the simple factored form of (15.216): ˚X ˚′ ) = ˚ ˚+ ) · f˚− (X ˚− | X ˚′ ). f˚k+1|k (X| bk+1|k (X k+1|k

(15.227)

with specific models substituted, (15.227) becomes (15.220): ˚+ ˚′ B X ˚X ˚′ ) = δ ˚ ˚ ·ω B ˚+ ˚− ˜ X f˚k+1|k (X| ·βX ˚′ (XL )· MX ˚− . (15.228) |X|,|XL | k+1|k (XL )·(sk+1|k ) L

Given this, let the prior distribution be GLMB: ˚ =δ˚ ˚ f˚k|k (X) |X|,|XL |

∑

˚

o ˚L ) · (˚ ωk|k (X sok|k )X .

(15.229)

o∈Ok|k

Substituting (15.228) and (15.229) into the prediction integral, ∫ ˚ = f˚k+1|k (X| ˚X ˚′ ) · f˚k|k (X ˚′ )δ X ˚′ , f˚k+1|k (X)

(15.230)

we get ˚ f˚k+1|k (X)

˚+

=

B X ˚+ δ|X|,| sB ˚ X ˚L | · ωk+1|k (XL ) · (˚ k+1|k ) ∫ ˚′ ˚ ˚′ ˚′ ˚− ˜X · βX ˚′ ( X L ) · M X ˚− · fk|k (X )δ X . L

(15.231)

484

Advances in Statistical Multisource-Multitarget Information Fusion

An application of Lemma 1—that is, (15.221)—to the integral on the right leads to (see (15.304)) ∑

˚ = δ ˚ ˚ · ωB ˚+ f˚k+1|k (X) k+1|k (XL ) |X|,|XL |

′

˚+

′

˚−

o X ˚− ) · (˚ ω ˜ k+1|k (X sB · (ˆ s o )X k+1|k ) L

o′ ∈Ok|k

(15.232) which can then be rewritten as (see (15.275)) ∑

˚ =δ˚ ˚ f˚k+1|k (X) |X|,|XL |

˚

o ˚L ) · (˚ ωk+1|k (X sok+1|k )X .

(15.233)

o∈Ok+1|k

15.5.1.2

Measurement Update Derivation: Road Map

We are given the multitarget likelihood function of (15.186): 

˚ = e−λ κZ · 1 − δ ˚ ˚ + δ ˚ ˚ f˚k+1 (Z|X) |X|,|XL | |X|,|XL |

∑ ˚ θ∈TZ



˚ ˚ ˚ ˚ ˚L ) · (˚ λθk+1 (X LθZ )X  .

(15.234) Substitute this and the GLMB predicted distribution ∑

˚

o ˚L ) · (˚ ωk+1|k (X sok+1|k )X

(15.235)

˚ · fk+1|k (X) ˚ f˚k+1 (Zk+1 |X) ˚ f˚k+1|k+1 (X|Z) = . f˚k+1 (Z)

(15.236)

˚ =δ˚ ˚ f˚k+1|k (X) |X|,|XL |

o∈Ok+1|k

into multitarget Bayes’ rule:

Applying Lemma 1—i.e, (15.221)—to the normalization factor f˚k+1 (Z) =

∫

˚ · f˚k+1|k (X)δ ˚ X ˚ f˚k+1 (Zk+1 |X)

(15.237)

we get f˚k+1 (Z) = e−λ κZ

∑∑∑ L⊆L ˚ θ o∈O

˚

θ o ωk+1 (L) · ωk+1|k (L)

∏ ℓ∈L

˚

˚θ ˚ so,ℓ k+1|k [LZ ] (15.238)

Exact Closed-Form Multitarget Filter

485

and thus the posterior distribution becomes

=

˚ f˚k+1|k+1 (X|Z) (15.239) ∑ ∑ ˚ ˚ ˚ o ˚θ ˚ ˚ δ|X|,| sok+1|k ˚ LθZ )X ˚ X ˚L | · ˚ o∈O λk+1 (XL ) · ωk+1|k (XL ) · (˚ θ . ∑ ∑ ∑ ∏ ′ ′ ˚′ ,ℓ θ′ o′ ′ ˚˚ ˚ sok+1|k [˚ LθZ ] ˚ L′ ⊆L o′ ∈O λk+1 (XL ) · ωk+1|k (L ) ℓ′ ∈L′ ˚ θ′

After suitable algebraic grouping, we end up with a GLMB distribution: ∑ ˚ ˚ o,˚ θ θ X ˚ ˚L ) · (˚ f˚k+1|k+1 (X|Z) = δ|X|,| ωk+1|k+1 (X so, ˚ X ˚L | k+1|k+1 ) . (o,˚ θ)∈Ok+1|k+1

(15.240) 15.5.2

Closure Under Measurement Update with Respect to Vo-Vo Priors

Let Zk+1 be the new measurement set. Then Vo and Vo prove the following result ([295], Proposition 7). Let the predicted multitarget distribution be GLMB as in (15.186): ∑ ˚ o ˚ =δ˚ ˚ ˚L ) · (˚ f˚k+1|k (X) ωk+1|k (X sok+1|k )X (15.241) |X|,|XL | o∈Ok+1|k o where ωk+1|k (L) = 0 except for a finite number of finite L ⊆ L0:k+1 . Let the labeled multitarget likelihood for Zk+1 be defined as in (15.186):

˚ f˚k+1 (Zk+1 |X) 

=

 e−λ κZk+1 · 1 − δ|X|,| ˚ X ˚L | + δ|X|,| ˚ X ˚L |

∑ ˚ θ∈TZk+1

(15.242) 

˚ ˚ ˚ ˚ ˚L ) · (˚ λθk+1 (X LθZk+1 )X 

where TZk+1 is the set of all functions ˚ θ : L0:k+1 → {0, 1, ..., |Zk+1 |} such that ˚ θ(ℓ) = ˚ θ(ℓ′ ) implies ℓ = ℓ′ ; and where ∏ ˚ ˚ λθk+1 (L) = δ˚ (15.243) θ(ℓ),0 ℓ∈L0:k+1 −L ˚ ˚ LθZ (x, ℓ)

=

(15.244)

δ0,˚ pD (x, ℓ)) θ(ℓ) · (1 − ˚ ˚ pD (x, ℓ) · f˚k+1 (z˚ θ(ℓ) |x, ℓ) +(1 − δ0,˚ θ(ℓ) ) ·

. κ(z˚ θ(ℓ) )

486

Advances in Statistical Multisource-Multitarget Information Fusion

Let the posterior multitarget distribution be ˚ ˚ ˚ ˚ k+1 ) = fk+1 (Zk+1 |X) · fk+1|k (X) f˚k+1|k+1 (X|Z f˚k+1 (Zk+1 )

(15.245)

with normalization factor f˚k+1 (Zk+1 ) =

∫

˚ · f˚k+1|k (X)δ ˚ X. ˚ f˚k+1 (Zk+1 |X)

(15.246)

˚ Then f˚k+1|k+1 (X|Z) is a GLMB distribution of the form ˚ f˚k+1|k+1 (X|Z)

=

∑

δ|X|,| ˚ X ˚L |

˚

o,θ ˚L ) (15.247) ωk+1|k+1 (X

(o,˚ θ)∈Ok+1|k+1

·

∏

˚

θ ˚ so, k+1|k+1 (x, ℓ)

˚ (x,ℓ)∈X

=

(15.248)

δ|X|,| ˚ X ˚L | ∑ ·

˚

˚

˚

o,θ θ X ˚L ) · (˚ (X so, ωk+1|k+1 k+1|k+1 )

(o,˚ θ)∈Ok+1|k+1

where: • Measurement updated index set: (15.249)

Ok+1|k+1 = Ok+1|k × TZk+1 . • Measurement updated weight functions: ( ˚ ) o ˚ λθk+1 (L) · ωk+1|k (L) ∏ θ ˚˚ · ℓ∈L ˚ so,ℓ k+1|k [LZk+1 ] o,˚ θ ωk+1|k+1 (L) = ( ∑ ) ∑ ∑ θ′ ′ ˚˚ ˚ L′ ⊆L o′ ∈Ok+1|k λk+1 (L ) θ′ ′ ′ ∏ ′ ˚′ ·ω o (L′ ) ′ ′ ˚ so ,ℓ [˚ Lθ ] k+1|k

ℓ ∈L

k+1|k

(15.250)

Zk+1

where θ ˚˚ ˚ so,ℓ k+1|k [LZk+1 ]

=

∫

˚ ˚ LθZk+1 (x, ℓ) · ˚ sok+1|k (x, ℓ)dx.

(15.251)

Exact Closed-Form Multitarget Filter

487

• Measurement updated spatial distributions: ˚ θ ˚ so, k+1|k+1 (x, ℓ)

˚ ˚ sok+1|k (x, ℓ) · ˚ LθZk+1 (x, ℓ)

=

(15.252)

θ ˚˚ ˚ so,ℓ k+1|k [LZk+1 ]

δ0,˚ pD (x, ℓ)) · ˚ sok+1|k (x, ℓ) θ(ℓ) · (1 − ˚ = δ0,˚ so,ℓ θ(ℓ) · ˚ k+1|k [1

˚ ˚ so,ℓ [˚ p L k+1|k D z˚

−˚ pD ] + (1 − δ0,˚ θ(ℓ) ) · ˚ pD (x,ℓ)·˚ Lz˚

(1 − δ0,˚ θ(ℓ) ) ·

θ(ℓ)

κ(z˚ θ(ℓ) )

(x,ℓ)·˚ sok+1|k (x,ℓ)

κ(z˚ θ(ℓ) )

+ δ0,˚ so,ℓ θ(ℓ) · ˚ k+1|k [1

(15.253) ]

θ(ℓ)

˚ ˚ so,ℓ [˚ p L k+1|k D z˚

−˚ pD ] + (1 − δ0,˚ θ(ℓ) ) ·

. ]

θ(ℓ)

κ(z˚ θ(ℓ) )

Equations (15.248) through (15.252) are proved as follows. First, abbreviate Z = Zk+1 and compute f˚k+1 (Z): ∫ ˚ · f˚k+1|k (X)δ ˚ X ˚ f˚k+1 (Z) = f˚k+1 (Z|X) (15.254)   ∫ ∑ ˚ ˚ ˚ θ θ X −λ Z  ˚ ˚L ) · (˚ = e κ λk+1 (X LZ )  (15.255) ˚ θ

(

· δ|X|,| ˚ X ˚L |

∑

o ˚L ) · (˚ ωk+1|k (X sok+1|k )

˚ X

)

˚ δX

o∈O

=

e

−λ Z

κ

∑∑∫

˚ ˚ LθZ )X · δ|X|,| (sok+1|k ˚ ˚ X ˚L |

(15.256)

˚ θ o∈O ˚ ˚L ) · ω o ˚ ˚ ·˚ λθk+1 (X k+1|k (XL )δ X.

Now apply Lemma 1—that is, (15.221)—in which case (15.256) becomes ∑∑∑ ˚ ∏ o,ℓ ˚ θ o f˚k+1 (Z) = e−λ κZ ωk+1 (L) · ωk+1|k (L) ˚ sk+1|k [˚ LθZ ] (15.257) L⊆L ˚ θ o∈O

where θ ˚˚ ˚ so,ℓ k+1|k [LZ ]

=

∫

ℓ∈L

˚ ˚ LθZ (x, ℓ) · ˚ sok+1|k (x, ℓ)dx.

(15.258)

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Advances in Statistical Multisource-Multitarget Information Fusion

So the posterior distribution is

=



˚ f˚k+1|k+1 (X|Z) ( ) ∑ ∑ θ ˚˚ ˚ δ|X|,| ˚ X ˚L | · ˚ o∈O λk+1 (XL ) θ ˚ ˚ o ˚L ) · (˚ ·ωk+1|k (X sok+1|k ˚ LθZ )X ) ( ∑ ∑ ∑ θ′ ˚˚ ˚ ˚ L′ ⊆L o′ ∈O λk+1 (XL ) θ′ ′ ′ ∏ ˚′ ,ℓ o′ ·ωk+1|k (L′ ) ℓ′ ∈L′ ˚ sok+1|k [˚ LθZ ]

 ˚ ˚ ˚L ) · ω o ˚ λθk+1 (X k+1|k (XL ) ) ˚  (∏  θ X ˚˚ (˚ sok+1|k L Z) θ ˚˚ · so,ℓ ˚ o,ℓ θ ℓ∈L ˚ k+1|k [LZ ] · ∏ ˚ sk+1|k [LZ ] ℓ∈L ˚ ( ∑ ) = δ|X|,| ˚ X ˚L | · ∑ ∑ ˚ θ′ ′ ˚ λ (L ) ′ ′ ′ ˚ L ⊆L θ ∏ o ∈O o′k+1 ′ ,ℓ′ ˚˚ o′ ·ωk+1|k (L′ ) ℓ′ ∈L′ ˚ sk+1|k [LθZ ]

=

∑ ∑ ˚ θ

δ|X|,| ˚ X ˚L |

(15.259)

o∈O

∑∑

(15.260)

˚

o,θ ˚L ) ωk+1|k+1 (X

(15.261)

˚ θ o∈O

·

˚ ˚ sok+1|k (x, ℓ) · ˚ LθZ (x, ℓ)

∏

θ ˚˚ ˚ so,ℓ k+1|k [LZ ] ∑ ∑ o,˚ ˚ θ θ ˚L ) · ˚ δ|X|,| ωk+1|k+1 (X so, ˚ X ˚L | k+1|k+1 (x, ℓ) ˚ (x,ℓ)∈X

=

(15.262)

˚ θ o∈O

where (

˚

o,θ ωk+1|k+1 (L)

=

) ˚ ˚ ˚L ) · ω o λθk+1 (X k+1|k (L) ∏ θ ˚˚ · ℓ∈L ˚ so,ℓ k+1|k [LZ ] ( ∑ ) ∑ ∑ ˚ θ′ ′ ˚ λ (L ) ′ ′ ′ ˚ k+1 L ⊆L θ ∏ o ∈O o′ ,ℓ′ ˚˚ ′ o′ ·ωk+1|k (L′ ) ℓ′ ∈L′ ˚ sk+1|k [LθZ ]

(15.263)

˚

˚ θ ˚ so, k+1|k+1 (x, ℓ)

˚ sok+1|k (x, ℓ) · ˚ LθZ (x, ℓ) =

θ ˚˚ ˚ so,ℓ k+1|k [LZ ]

.

(15.264)

Exact Closed-Form Multitarget Filter

15.5.3

489

Closure Under Time Update with Respect to Vo-Vo Priors

Vo and Vo prove the following result ([295], Proposition 8). Let the prior multitarget distribution be GLMB as in (15.186): ∑ ˚ o ˚ =δ˚ ˚ ˚L ) · (˚ f˚k|k (X) ωk|k (X sok|k )X (15.265) |X|,|XL | o∈Ok|k o where ωk|k (L) = 0 except for a finite number of finite L ⊆ L0:k ; and where ∑ ∑ o ωk|k (L) = 1. (15.266) L⊆L0:k o∈Ok|k

Let the labeled multitarget Markov density be defined as in (15.220): ˚+ ˚′ B X ˚X ˚′ ) = δ ˚ ˚ ·ω B ˚+ ˚− ˜ X f˚k+1|k (X| ·βX ˚′ (XL )· MX ˚− (15.267) k+1|k (XL )·(sk+1|k ) |X|,|XL | L

where ˚− X ˚+ X

= =

˚∩X ˚0:k (persisting targets in X) ˚ X B ˚∩X ˚k+1 (appearing targets in X) ˚ X

B ωk+1|k (L)

=

LMB weight function for target appearances (15.270)

sB k+1|k (x, ℓ)

=

βL′ (L)

=

LMB spatial density for target appearances ∏ 1L′ (ℓ)

(15.268) (15.269)

(15.271) (15.272)

ℓ∈L

˜ ˚− (x′ , ℓ′ ) M X

=

′ (1 − 1X pS (x′ , ℓ′ )) (15.273) ˚− (ℓ )) · (1 − ˚ L ∑ + δℓ,ℓ′ · ˚ pS (x′ , ℓ′ ) · fk+1|k (x|x′ , ℓ′ ) ˚− (x,ℓ)∈X

where, note, βL′ (L) = 0 unless L ⊆ L′ . Let the predicted multitarget distribution be defined by the prediction integral ∫ ˚ ˚ ˚X ˚′ ) · f˚k|k (X ˚′ )δ X ˚′ . fk+1|k (X) = f˚k+1|k (X| (15.274) ˚ Then f˚k+1|k (X|Z) is a GLMB distribution of the form ∑ ˚ o ˚ =δ˚ ˚ ˚L ) · (˚ f˚k+1|k (X) ωk+1|k (X sok+1|k )X |X|,|XL | o∈Ok+1|k

(15.275)

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Advances in Statistical Multisource-Multitarget Information Fusion

where: • Time updated index set: (15.276)

Ok+1|k = Ok|k . • Time updated weight functions: o B o ωk+1|k (L) = ωk|k (L ∩ Lk+1 ) · ω ˜ k|k (L ∩ L0:k )

where for finite J ⊆ L0:k , ( o ω ˜ k+1|k (J )

=

∏

˚ soℓ [˚ pS ]

(15.277)

)

(15.278)

ℓ∈J

·

(

∑

βL (J ) ·

o ωk|k (L)

L

˚ soℓ [˚ pS ] ˚ soℓ [1 − ˚ pS ]

∏

˚ soℓ [1

−˚ pS ]

)

ℓ∈L−J

=

∫

˚ pS (x, ℓ) · ˚ sok|k (x, ℓ)dx

(15.279)

=

∫

(1 − ˚ pS (x, ℓ)) · ˚ sok|k (x, ℓ)dx.

(15.280)

• Time updated spatial distributions: ˚ sok+1|k (x, ℓ) = 1L0:k (ℓ) · sˆo (x, ℓ) + (1 − 1L0:k (ℓ)) · ˚ sB k+1|k (x, ℓ) (15.281) where sˆo (x, ℓ)

=

˚x ] ˚ soℓ [˚ pS M

=

˚x ] ˚ soℓ [˚ pS M o ˚ sℓ [˚ pS ]  ∑  ∫ pS (x′ , ℓ) ˚˚ (x,ℓ)∈X  ·fk+1|k (x|x′ , ℓ)  dx′ ·˚ sok|k (x′ , ℓ)

and where the expression ∑ ˚ pS (x, ℓ) · fk+1|k (x|x′ , ℓ) · ˚ sok|k (x′ , ℓ) ˚ (x,ℓ)∈X

is a shorthand way of writing the expression ˚ pS (x′ , ℓ) · fk+1|k (xi |x′ , ℓ) · ˚ sok|k (x′ , ℓ)

(15.282)

(15.283)

Exact Closed-Form Multitarget Filter

491

˚ where xi corresponds to the unique value of i such that (xi , ℓ) ∈ X. To prove (15.275) through (15.283), note that the predicted labeled multitarget distribution is given by the prediction integral: ∫ ˚ = ˚X ˚′ ) · f˚k|k (X ˚′ )δ X ˚′ f˚k+1|k (X) f˚k+1|k (X| (15.284) ˚+

=

B X ˚+ δ|X|,| sB (15.285) ˚ X ˚L | · ωk+1|k (XL ) · (˚ k+1|k ) ∫ ˚′ ˚− ˜X · ωX ˚′ ( X L ) · M X ˚− L   ∑ ′ ′ ′ ˚ o ˚L′ ) · β ˚′ (X ˚− ) · (˚ ˚′ ·  δ |X ωk|k (X sok|k )X  δ X ˚′ |,|X ˚′ | L X L

L

o′ ∈Ok|k

˚+

=

B X ˚+ δ|X|,| sB (15.286) ˚ X ˚L | · ωk+1|k (XL ) · (˚ k+1|k ) ∫ ∑ ′ ˚′ ˚′ o′ ˚′ ˚− ˜ ˚− )X · δ |X sok|k M δX . ˚′ |,|X ˚′ | · ωk|k (XL ) · βX ˚′ (XL ) · (˚ X L

L

o′ ∈Ok|k

By Lemma 1—that is, (15.221)—the integral becomes ∫ ′ ˚′ ˚′ o′ ˚′ ˚− ˜ ˚− )X δ |X sok|k M δ X (15.287) ˚′ |,|X ˚′ | · ωk|k (XL ) · βX ˚′ (XL ) · (˚ X L L ∑ ∏ ′ o′ ˚− ) ˜ ˚− ] = ωk|k (L) · βL (X (˚ sok|k )ℓ′ [M L X ℓ′ ∈L′

˚′ L⊆X L

where ′ ˜ ˚− ] = (˚ sok|k )ℓ′ [M X

and where the product ′

∏

∫

′ ˜ ˚− (x′ , ℓ′ ) · ˚ M sok|k (x′ , ℓ′ )dx′ X

′ ˜ ˚− ] sok|k )ℓ′ [M ℓ′ ∈L (˚ X

is finite since, by assumption,

o ωk|k (L) vanishes for nonfinite L. Thus (15.287) becomes ∫ ′ ˚′ ˚′ o′ ˚′ ˚− ˜ ˚− ) X sok|k M δ |X δX ˚′ |,|X ˚′ | · ωk|k (XL ) · βX ˚′ (XL ) · (˚ X L L   ∑ ∏ ∏ ′ ′ o ˜ ˚− ]  = ωk|k (L) ·  (˚ sok|k )ℓ′ [M X ˚′ ⊇L⊇X ˚− L:X L L

˚− ℓ′ ∈ X L

(15.288)

˚− ℓ′ ∈L−X L

(15.289) 

′ ˜ ˚− ]  (˚ sok|k )ℓ′ [M X

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Advances in Statistical Multisource-Multitarget Information Fusion

∑

=

′

o ωk|k (L)

(15.290)

˚′ ⊇L⊇X ˚− L:X L L

(

∏

·

˚− ℓ′ ∈ X L



·

pS (x′ , ℓ′ ) ˚ δℓ,ℓ′ · ˚ (x,ℓ)∈X ′ ′ o′ ·fk+1|k (x|x , ℓ ) · ˚ sk|k (x′ , ℓ′ )dx′ ∫

∏

) 

′

(1 − ˚ pS (x′ , ℓ′ )) · ˚ sok|k (x′ , ℓ′ )dx′ 

˚− ℓ′ ∈L−X L

∑

=

∫

∑

′

o ωk|k (L)

(15.291)

˚′ ⊇L⊇X ˚− L:X L L

( ∫∑

∏

·

˚− ℓ′ ∈ X L



·

∫

∏ ˚− ℓ′ ∈L−X L

pS (x′ , ℓ′ ) · fk+1|k (x|x′ , ℓ′ ) ˚˚ (x,ℓ′ )∈X ′ ·˚ sok|k (x′ , ℓ′ )dx′

)



′

(1 − ˚ pS (x′ , ℓ′ )) · ˚ sok|k (x′ , ℓ′ )dx′  .

Let ′

˚x ] ˚ soℓ′ [˚ pS M  ∫ ∑  =

˚ (x,ℓ′ )∈X

(15.292) ′



˚ pS (x′ , ℓ′ ) · fk+1|k (x|x′ , ℓ′ ) · ˚ sok|k (x′ , ℓ′ ) dx′

and ′ ˚ soℓ′ [˚ pS ]

=

′

sˆo (x, ℓ′ )

=

′

˚ soℓ′ [1 − ˚ pS ]

=

∫

′

˚ pS (x′ , ℓ′ ) · ˚ sok|k (x′ , ℓ′ )dx

′ ˚x ] ˚ soℓ′ [˚ pS M ′ o ˚ s ′ [˚ pS ] ∫ ℓ ′ (1 − ˚ pS (x′ , ℓ′ )) · ˚ sok|k (x′ , ℓ′ )dx′ .

(15.293) (15.294) (15.295)

Exact Closed-Form Multitarget Filter

493

Then (15.291) becomes ∫ ′ ˚′ ˚′ o′ ˚′ ˚− ˜ ˚− )X δ |X sok|k M δ X (15.296) ˚′ |,|X ˚′ | · ωk|k (XL ) · βX ˚′ (XL ) · (˚ X L L   ∑ ∏ ′ ′ o ˚x ] = ωk|k (L)  ˚ soℓ′ [˚ pS M ˚′ ⊇L⊇X ˚− L:X L L



·

∏

˚− ℓ′ ∈ X L

˚− ℓ′ ∈L−X L

˚ soℓ′ [1 − ˚ pS ]

∑

=

·

=

∏

∑ 

=





·

˚− (x′ ,ℓ′ )∈X



′

sˆo (x, ℓ′ )) · ˚ soℓ′ [˚ pS ] (15.297)



˚ soℓ′ [1 − ˚ pS ] 

∏

′

˚− ℓ′ ∈L−X L

˚− ℓ′ ∈ X L

·

′

o ˚− ) ·  ωk|k (L) · βL (X L

∏



∏

′

˚′ L⊆X L



o ωk|k (L) 

′

˚− ℓ′ ∈L−X L

·



′

˚′ ⊇L⊇X ˚− L:X L L





′



soℓ′ [1 − ˚ p S ] 

∏ ˚− ℓ′ ∈ X L



′

soℓ′ [˚ pS ]

∏

′

˚− (x′ ,ℓ′ )∈X

(15.298) 

sˆo (x, ℓ)



′

˚ soℓ′ [˚ p S ]

∑

(15.299)

′

o ˚− ) ωk|k (L) · βL (X L

˚− ℓ′ ∈L−X L

˚′ L⊆X L

∏ ˚− (x′ ,ℓ′ )∈X

∏

′



sˆo (x′ , ℓ′ )

′



˚ soℓ′ [1 − ˚ pS ]

494

Advances in Statistical Multisource-Multitarget Information Fusion

 

=

∏ ˚− ℓ′ ∈ X L



·

˚ soℓ′ [˚ p S ]

∑

(15.300)

′

˚− ) · ω o (L) β L (X k|k L

∏ ˚− ℓ′ ∈L−X L

˚′ L⊆X L



·



′

∏ ˚− (x′ ,ℓ′ )∈X



′

˚ soℓ′ [1 − ˚ pS ]



′

sˆo (x′ , ℓ′ )

and so ∫

′ ˚′ ˚′ o′ ˚′ ˚− ˜ ˚− )X δ |X sok|k M δ X (15.301) ˚′ |,|X ˚′ | · ωk|k (XL ) · βX ˚′ (XL ) · (˚ X L

L

′

=

′

o ˚− ) · (ˆ ω ˜ k+1|k (X so ) L

˚− X

where ′

o ω ˜ k+1|k (J ) =

(

∏ ℓ′ ∈J

 ) ∑ ∏ ′ ′ o ˚ soℓ′ [˚ pS ]  βL (J ) · ωk|k (L) ˚ soℓ′ [1 − ˚ pS ] . ′

˚′ L⊆X L

ℓ′ ∈L−J

(15.302) Note that this vanishes for all but a finite number of finite J ⊆ L, because the same o′ is true of ωk|k (J ). Thus the entire predicted multitarget distribution in (15.286) becomes ˚ f˚k+1|k (X)

˚+

=

B X ˚+ δ|X|,| sB ˚ X ˚L | · ωk+1|k (XL ) · (˚ k+1|k ) ∑ ′ ˚− o′ ˚− ) · (ˆ · ω ˜ k+1|k (X s o )X L

(15.303)

o′ ∈Ok|k

=

B ˚+ δ|X|,| (15.304) ˚ X ˚L | · ωk+1|k (XL ) ∑ ′ + ′ ˚− ˚ o X ˚− ) · (˚ · ω ˜ k+1|k (X sB · (ˆ so )X . k+1|k ) L o′ ∈Ok|k

Now, as in (15.278), define ˚ sok+1|k (x, ℓ) = 1L0:k (ℓ) · sˆo (x, ℓ) + (1 − 1L0:k (ℓ)) · ˚ sB k+1|k (x, ℓ)

(15.305)

Exact Closed-Form Multitarget Filter

and note that ∫

495

˚ sok+1|k (x, ℓ)dx = 1L0:k (ℓ) + 1 − 1L0:k (ℓ) = 1.

(15.306)

Then ∏

˚

(˚ sok+1|k )X =

˚ sok+1|k (x, ℓ)

(15.307)

˚ (x,ℓ)∈X

 

=

˚− (x,ℓ)∈X

 =



∏



˚ sok+1|k (x, ℓ)  

∏ ˚− (x,ℓ)∈X

sˆo (x, ℓ) 

˚−

=



∏ ˚+ (x,ℓ)∈X

˚ sok+1|k (x, ℓ)

(15.308)



∏ ˚+ (x,ℓ)∈X

 ˚ sB k+1|k (x, ℓ)

(15.309)

˚+

X (ˆ so )X · (˚ sB . k+1|k )

(15.310)

Thus (15.304) becomes ˚ f˚k+1|k (X)

=

∑

B ˚+ δ|X|,| ˚ X ˚L | · ωk|k (XL )

o′ ˚− ) ω ˜ k+1|k (X L

(15.311)

o′ ∈Ok|k ′

˚

·(˚ sok+1|k )X =

∑

δ|X|,| ˚ X ˚L |

′ ˚ o′ ˚L ) · (˚ ωk+1|k (X sok+1|k )X

(15.312)

o′ ∈Ok|k

where ′

o ωk+1|k (L)

Now, since

∫

′

=

B o ωk+1|k (L ∩ Lk+1 ) · ω ˜ k|k (L ∩ L0:k )

=

B ωk+1|k (L

− L0:k ) ·

o′ ω ˜ k|k (L

∩ L0:k ).

(15.313) (15.314)

˚ X ˚ = 1 by construction, (15.116) tells us that f˚k+1|k (X)δ ∑

∑

L⊆L0:k+1 o′ ∈Ok|k

and so we are done.

′

o ωk+1|k (L) = 1

(15.315)

496

15.6

Advances in Statistical Multisource-Multitarget Information Fusion

IMPLEMENTATION OF THE VO-VO FILTER: SKETCH

One could try to implement the multitarget Bayes filter using the time-update and measurement-update steps of Sections 15.5.3 and 15.5.2, respectively. However, Vo and Vo have shown that it is computationally advantageous to reformulate these steps by replacing GLMB distributions with “δ-GLMB distributions.” The implementation approach is discussed in detail in [296]. This section presents a sketch, organized as follows: 1. Section 15.6.1: δ-GLMB distributions. 2. Section 15.6.2: δ-GLMB version of the Vo-Vo filter. 3. Section 15.6.3: An exact L1 characterization of the effect of pruning δGLMB components. 15.6.1

δ-GLMB Distributions

As was explained in Section 15.4.2, after k recursions of the time-update and measurement-update steps of Sections 15.5.3 and 15.5.2, we will end up with GLMB distributions of the form ∑ ˚ θ1 ,...,˚ θk ˚ ˚ (k) ) = δ ˚ ˚ f˚k|k (X|Z ωk|k (X L ) (15.316) |X|,|XL | (˚ θ1 ,...,˚ θk ) ˚ ˚ θ1 ,...,˚ θk X ·(˚ sk|k )

where, for j = 1, ..., k, the functions ˚ θj : L0:j → {0, 1, ..., |Zj |} are such that ′ ′ ˚ ˚ θj (ℓ) = θ(ℓ ) implies ℓ = ℓ . Now, (15.316) can be rewritten as ˚ (k) ) f˚k|k (X|Z

=

(15.317)

δ|X|,| ˚ X ˚L | ∑

·

J,˚ θ1 ,...,˚ θk ωk|k

· δJ,X ˚L

(J,˚ θ1 ,...,˚ θk )∈F(L0:k )×Ak|k ˚

˚

˚

1 ,...,θk ·(˚ sθk+1|k+1 )X

=

∑

δ|X|,| ˚ X ˚L |

(J,αk )∈F(L0:k )×Ak|k ˚

X k ·(˚ sα k+1|k+1 )

J,αk ωk|k · δJ,X ˚L (15.318)

Exact Closed-Form Multitarget Filter

497

where F(L0:k ) is the class of all finite subsets of L0:k and where Ak|k

=

αk J

= ⊆

J,˚ θ1 ,...,˚ θk ωk|k

=

TZ1 × ... × TZk , (˚ θ1 , ..., ˚ θk ), L0:k ˚

(15.319) (15.320) (15.321)

˚

θ1 ,...,θk ωk|k (J ).

(15.322)

This leads to the following definition, in which the set-parameter J is absorbed into the index set Ok|k ([295], Definition 9). A δ-GLMB RFS is a GLMB RFS with the following specific form: 1. The index space O has the form O = F(L) × A and o = (J, α) ∈ O, where A is a set of association-sequences α. 2. The GLMB weight ω o (L) = ω J,α (L) has the form ω J,α (L) = ω J,α · δJ,L . 3. The GLMB density so (x, ℓ) = sJ,α (x, ℓ) ˚ sJ,α (x, ℓ) = sα (x, ℓ).

does not depend on

J:

Intuitively speaking: • The pair (J, αk ) is a hypothesis about how the measurements in the sequence Z1 , ..., Zk have been successively assigned to those tracks whose labels are in J ⊆ L0:k . J,αk • The number ωk|k is the degree of confidence in this hypothesis.

Vo and Vo have shown that the class Dδ of δ-GLMB distributions also solves the multitarget Bayes filter in exact closed form. The demonstration of this fact will not be reproduced here. Interested readers are referred to [295]. This seemingly minor reformulation results in a significant computational savings. When implementing the time-updates and measurement-updates of Sec˚ ˚1 ,...,˚ θ1 ,...,˚ θk θk (L) and ˚ sθk|k (x, ℓ) tions 15.5.3 and 15.5.2, we must construct ωk|k for every choice of (L, ˚ θ1 , ..., ˚ θk ). This requires storage and computation of |F(L0:k ) × TZ1 × ... × TZk | and |F(L0:k ) × TZ1 × ... × TZk | items, respectively. For the δ-GLMB formulation of the Vo-Vo filter we still must store and compute the L,˚ θ1 ,...,˚ θk quantities ωk|k for every choice of (L, ˚ θ1 , ..., ˚ θk ). But we need store and ˚1 ,...,˚ θk compute the sθk+1|k+1 (x, ℓ) only for every choice of ˚ θ1 , ..., ˚ θk —which involves only |TZ1 × ... × TZk | items.

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δ-GLMB Version of the Vo-Vo Filter

The δ-GLMB version of the Vo-Vo filter propagates labeled multi-Bernoulli mixtures of the form ∑ ˚1 ,...,˚ ˚ J,˚ θ1 ,...,˚ θk θk X ˚ (k) ) = δ ˚ ˚ f˚k|k (X|Z ωk|k · δJ,X sθk+1|k+1 ) . (15.323) ˚L · (˚ |X|,|XL | (J,˚ θ1 ,...,˚ θk )

As time progresses the number of components in this mixture greatly increases, and so components must be pruned to keep it down to a computationally feasible level. After each measurement-update, a certain number Mk|k of components ˚

˚

J,θ1 ,...,θk are discarded and the weights ωk|k are renormalized. This elimination process is accomplished using Murty’s algorithm, which can determine the Mk|k most significant components without evaluating the entire set of weights. Since additional components are added because of the target appearance model, additional elimination of components is required during the time-update step. Multitarget state estimation is accomplished using the heuristic approach described in Section 15.3.4.7. For greater detail, see [296].

15.6.3

Characterization of Pruning

Because the number of terms in a δ-GLMB distribution grows super-exponentially with time, it is necessary to prune small-weight terms. Similar pruning is required in tracking algorithms such as MHT and JPDA, but the effect of hypothesis-pruning on the probability law of the multitarget state is unknown. By way of contrast, the effect of pruning terms from δ-GLMB distributions can be characterized not only exactly, but by a simple formula. The purpose of this section is to summarize this (somewhat amazing) result. Let us be given an unnormalized δ-GLMB distribution ∑

˚ =δ˚ ˚ f˚O (X) |X|,|XL |

˚

ω J,α · δJ,X s α )X ˚L · (˚

(15.324)

(J,α)∈O

defined for a given set O of indices. Suppose that we wish to eliminate the terms in this sum corresponding to a subset O′ ⊆ O of indices. Let ∥f˚∥1 =

∫

˚ X ˚ |f˚(X)|δ

(15.325)

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499

˚ on X∞ . Then the error caused by the denote the L1 norm on functions f˚(X) truncation is given by ([296], Proposition 5): ∥f˚O − f˚O′ ∥1 =

∑

ω J,α .

(15.326)

(J,α)∈O−O′

That is, the L1 norm between the pruned and unpruned distributions is just the sum of the weights of the pruned terms. Furthermore, the error between the corresponding normalized distributions is bounded as follows: ? ? ? f˚ f˚O′ ? ∥f˚O ∥1 − ∥f˚O′ ∥1 ? O ? − . (15.327) ? ? ≤2 ? ∥f˚O ∥1 ∥f˚O′ ∥1 ? ∥f˚O ∥1 1

15.7

PERFORMANCE RESULTS

Vo and Vo report performance evaluations of Gaussian mixture and sequential Monte Carlo (SMC) implementations of the δ-GLMB version of the Vo-Vo filter. These are now described. 15.7.1

Gaussian Mixture Implementation of Vo-Vo Filter

In this implementation, the single-target motion and measurement models are ˚1 ,...,˚ θk linear-Gaussian and the spatial distributions sθk|k (x, ℓ) are approximated as Gaussian mixtures [305]. The authors have given this implementation of the δGLMB version of the Vo-Vo filter the name “para-Gaussian multi-target filter.” The test scenario includes up to 10 appearing and disappearing targets, which follow linear trajectories in a rectangular region 2000 meters on a side. The targets are observed by a single linear-Gaussian sensor with constant probability of detection of 0.98, and with uniformly distributed Poisson clutter having a constant clutter rate λ = 60. The authors reported that the filter initiated and terminated tracks with a very small delay, and accurately estimates target number and the target states. The filter also exhibited only a very small number of spurious or dropped tracks, especially considering the fairly large clutter rate. The authors also compared the new filter with a GM-CPHD filter. The GM implementation of the Vo-Vo filter was significantly better at estimating target

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number than the GM-CPHD filter, while also doing a somewhat better job of estimating target states. 15.7.2

Particle Implementation of the Vo-Vo Filter

In this implementation [295], the single-target motion and measurement models ˚ θ1 ,...,˚ θk are nonlinear and the sk|k (x, ℓ) are approximated as Dirac mixtures and propagated using particle methods. Up to 10 appearing and disappearing targets are present at any given time, and follow curvilinear trajectories within a half-disc of 2000 meter radius. The targets are observed by a single range-bearing sensor located at the origin. Clutter is uniformly distributed and Poisson, with constant clutter rate λ = 20. The probability of detection pD (x) is state-dependent and is circular-Gaussian in shape, peaking at 0.98 at the origin and tapering to 0.92 at the edge of the surveillance region. The single-target motion model is a nonlinear coordinated-turn model. The target appearance process is a labeled Poisson process. The track distributions are approximated and propagated using particle methods. The authors report that this implementation accurately estimated the target states, while initiating and terminating tracks with a small delay. There were a small number of dropped tracks and false tracks. Most crucially, no track-switching was observed—meaning that track labels were consistently estimated and propagated throughout the entire scenario. The authors also compared their filter with an SMC-CPHD filter. The SMC implementation of the SMC-δ-GLMB filter estimated target number significantly more accurately than the SMC-CPHD filter, and with significantly smaller variance. The two filters were also compared using the OSPA metric (Section 6.2.2), using 100 Monte Carlo trials. The OSPA error of the SMC-δ-GLMB filter was less than half that of the SMC-CPHD filter. However, this performance gain was achieved at the cost of an order of magnitude increase in computational load.

Part III

RFS Filters for Unknown Backgrounds

501

Chapter 16 Introduction to Part III In Section 7.2, the following multitarget measurement model was considered: all measurements

target measurements

clutter measurements

? ?? ? Σk+1

? ?? ? = Υk+1 (x1 ) ∪ ... ∪ Υk+1 (xn ) ∪

? ?? ? Ck+1

(16.1)

where • X = {x1 , ..., xn } with |X| = n is the multitarget state of the targets present at time tk+1 . • Υk+1 (x) is the RFS of measurements generated by a target with state x. • Ck+1 is the clutter RFS. • Υk+1 (x1 ), ..., Υk+1 (xn ), Ck+1 are assumed to be statistically independent. The “standard” multitarget measurement model was further distinguished by the following additional two requirements: Ck+1 is Poisson and Υk+1 (x) is Bernoulli. That is, the p.g.fl. of Υk+1 (x) is GΥk+1 (x) [g] = 1 − pD (x) + pD (x)

503

∫

g(z) · fk+1 (z|x)dz

(16.2)

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where pD (x) is the probability of detection and fk+1 (z|x) is the sensor likelihood function. Equivalently, the probability distribution of Υk+1 (x) is  1 − pD (x) if Z=∅  pD (x) · fk+1 (z|x) if Z = {z} . fΥk+1 (x) (Z) = (16.3)  0 if otherwise In what follows, the term measurement background will refer to: • The clutter background, as specified by Ck+1 or its probability distribution κk+1 (Z); together with • The background detection profile, as specified by the state-dependent probability of detection pD (x). All major multitarget detection and tracking algorithms are based on the assumption that both of these models are known a priori. Typically, these models have two forms: • Explicit clutter and detection-profile models, such as those presumed for all of the RFS multitarget tracking algorithms considered thus far (such as the CPHD and CPHD filters). • Implicit background models, such as those used in traditional multitarget tracking algorithms such as multihypothesis trackers (MHTs). At a purely theoretical level, MHTs are usually based on the assumption that probability of detection is constant and that the clutter process is a spatially uniform Poisson RFS. But at a practical implementation level, an MHT’s a priori clutter model and detection profile models are usually implicit, in the form of track-initiation and track-termination rules. For example: – Track initiation: Assume that the probability of detection is large. If the clutter rate is small, then a newly appearing track can be declared quickly. This is because any new measurements can be presumed to be probably target-generated. If the clutter rate is large, however, then track initiation must be accomplished more cautiously, requiring possibly many time-steps before a new track can be declared. – Track termination: Assume that the clutter rate is small. If in addition the probability of detection is small, then caution is required before eliminating a track. This is because it is possible that, at least momentarily, no measurements are being collected from it. If the probability

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of detection is large, then a track can be eliminated nearly as soon as one stops observing it. In real-world applications, a priori background models (whether explicit or implicit) are often not available. The clutter background will typically be both unknown and unpredictably varying over time. The detection profile will typically be even more unpredictable and dynamic, since it often varies with a target’s bodyframe orientation and surface characteristics. In such cases, trackers with a priori models will typically exhibit degraded performance, because of mismatch between the presumed and the actual background models. What does one do, then, when the background is unknown and/or dynamic? The traditional approach, “clutter rejection,” falls roughly into two categories: • Background modeling: Physics-based modeling of the physical environment is employed to predict what clutter background one would expect to observe at any given moment. Typically, this is a very expensive and time-consuming approach, and must be accomplished anew for each new sensor. It also requires very accurate a priori environmental information, such as threedimensional terrain maps. • Empirical training: Learning algorithms are used to statistically characterize the background using a set of training data. This approach is based on the assumption that the training data is statistically diverse enough to address all of the scenarios that might be encountered at all times. Neural nets are one commonly employed technique. Another one is “clutter mapping,” in which the region of interest is subdivided into disjoint cells and the clutter density or its inverse is estimated in each cell using histogram or other methods [216]. The purpose of the chapters in Part III is to describe PHD, CPHD, and multi-Bernoulli filters that do not fall into either of these categories. They are “background-agnostic” in the following sense: • They implicitly estimate the clutter and/or detection background, on-the-fly, while simultaneously detecting and tracking the targets that may be obscured within the background.

16.1

INTRODUCTION

This introductory chapter is organized as follows:

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1. Section 16.2: Overview of the approach: unknown backgrounds.

multitarget Bayes filters for

2. Section 16.3: General and specific models for unknown clutter and unknown probability of detection. 3. Section 16.4: The organization of Part III.

16.2

OVERVIEW OF THE APPROACH

In the approach taken throughout Part III, the unknown background is taken into account in three ways: 1. Modeling an unknown detection profile (see Section 16.3.1 for more details): single-target states x are replaced by augmented states ˚ x = (a, x), where 0 ≤ a ≤ 1 is the unknown probability of detection of x. Thus (16.1) takes the form Σk+1 = Υk+1 (a1 , x1 ) ∪ ... ∪ Υk+1 (an , xn ) ∪ Ck+1

(16.4)

where the RFS Υk+1 (a, x) is Bernoulli with GΥk+1 (a,x) [g] = 1 − a + a

∫

g(z) · fk+1 (z|x)dz

(16.5)

and where, as usual, fk+1 (z|x) is the sensor likelihood function. 2. Modeling an unknown clutter process (see Section 16.3.2 for more details): Clutter measurements are presumed to be caused by an unknown number ν of unknown clutter generators. Like targets, clutter generators are characterized by states c belonging to a clutter state space C. Just as a target with state x has an associated Bernoulli measurement-generation RFS Υk+1 (x), so a clutter generator with state c has an associated measurement-generation RFS Ck+1 (c). In this case, (16.1) takes the form Σk+1 = Υk+1 (x1 ) ∪ ... ∪ Υk+1 (xn ) ∪ Ck+1 (˚ c1 ) ∪ ... ∪ Ck+1 (˚ cν ) (16.6) where the notation ˚ c will be explained momentarily.

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3. Modeling an unknown detection profile and unknown clutter process: In this most general case, (16.1) takes the form Σk+1 = Υk+1 (a1 , x1 ) ∪ ... ∪ Υk+1 (an , xn ) ∪ Ck+1 (˚ c1 ) ∪ ... ∪ Ck+1 (˚ cν ). (16.7) In 2009, Mahler proposed that the clutter generator measurement-generation processes Ck+1 (˚ c) be Poisson [155], with ˚ c = (c, c) and with unknown Poisson parameter c > 0. Equation (16.1) then takes the form Σk+1 = Υk+1 (a1 , x1 ) ∪ ... ∪ Υk+1 (an , xn ) ∪ Ck+1 (c1 , c1 ) ∪ ... ∪ Ck+1 (c1 , cν ) (16.8) where the p.g.fl. of the RFS Ck+1 (c, c) has the form ( ∫ ) κ GCk+1 (c,c) [g] = exp c (g(z) − 1) · fk+1 (z|c)dz (16.9) with unknown clutter rate c > 0 and clutter-generator spatial distribution κ fk+1 (z|c). This approach is the subject of Section 16.3.3. Chen Xin, Kirubarajan, et al. [37], [39] in 2009 and Mahler [189], [199], [194] in 2010 independently proposed that Ck+1 (˚ c) be Bernoulli. There are two possibilities. First, one can assume that the probability of detection c = pκD (c) of ˚ c = (c, c) is known a priori, ∫ κ GCk+1 (c) [g] = 1 − pκD (c) + pκD (c) g(z) · fk+1 (z|c)dz (16.10) with clutter probability of detection pκD (c) and clutter likelihood function κ fk+1 (z|c); and, more restrictively, that it is constant: ∫ κ κ κ GCk+1 (c) [g] = 1 − pD + pD g(z) · fk+1 (z|c)dz. (16.11) This latter case will be described in more detail in Section 16.3.5. Second, one can assume that the probability of detection for c is an unknown quantity 0 ≤ c ≤ 1. In this case the clutter state has the form ˚ c = (c, c) and (16.1) takes the form Σk+1 = Υk+1 (a1 , x1 ) ∪ ... ∪ Υk+1 (an , xn ) ∪ Ck+1 (c1 , c1 ) ∪ ... ∪ Ck+1 (cν , cν ) (16.12)

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with GCk+1 (c,c) [g] = 1 − c + c

∫

κ g(z) · fk+1 (z|c)dz.

(16.13)

This case is described in more detail in Section 16.3.4. Given these background models, in principle it becomes possible to detect and track multiple targets in unknown backgrounds. Let ˚ X be the space of augmented target-states (a, x) and ˚ C the space of augmented clutter-generator states (c, c) (where either c > 0 in the Poisson case or 0 ≤ c ≤ 1 in the Bernoulli case). Let ¨ denote a finite subset of the joint target-clutter state space ¨ ⊆X X ¨ =˚ X X ⊎˚ C.

(16.14)

Then given the models, the optimal solution to the unknown-background problem is the following generalization of the multitarget Bayes recursive filter: ¨ (k) ) → fk+1|k (X|Z ¨ (k) ) → fk+1|k+1 (X|Z ¨ (k+1) ) → ... ... → fk|k (X|Z where

16.3

¨ (k) ) fk+1|k (X|Z

=

¨ (k+1) ) fk+1|k+1 (X|Z

=

fk+1 (Z|Z (k) )

=

∫

¨ X ¨ ′ ) · fk|k (X ¨ ′ |Z (k) )δ X ¨ ′ (16.15) fk+1|k (X|

¨ · fk+1|k (X|Z ¨ (k) ) fk+1 (Zk+1 |X) (16.16) fk+1 (Zk+1 |Z (k) ) ∫ ¨ · fk+1|k (X|Z ¨ (k) )δ X. ¨ (16.17) fk+1 (Z|X)

MODELS FOR UNKNOWN BACKGROUNDS

The purpose of this section is to describe, in greater detail, the unknown-background models just introduced. The section is organized as follows: 1. Section 16.3.1: A general model for unknown detection profile. 2. Section 16.3.2: A general model for unknown clutter. 3. Section 16.3.3: A Poisson-mixture model for unknown clutter. 4. Section 16.3.4: A general multi-Bernoulli model for unknown clutter. 5. Section 16.3.5: A simplified multi-Bernoulli model for unknown clutter.

Introduction to Part III

16.3.1

509

A Model for Unknown Detection Profile

The following simple example demonstrates that it is possible, at least in principle, to recursively estimate the probability of detection pD (x) at the state x of a given target track. Suppose that a single sensor observes a single static target with state x0 ,1 and that there is no clutter. Suppose that, over k time-steps, the sensor collects the measurement sets Z1 , ..., Zk , where by assumption |Zk | = 0 or |Zk | = 1. Let k ∑ νk = |Zk | (16.18) l=1

be the cumulative number of target detections at time tk . Then the probability of detection at x0 is, approximately, νk pD (x0 ) ∼ . = k

(16.19)

The same reasoning holds if the target is dynamic and pD is known to be approximately constant in the region of interest. At any instant, we can estimate both the state-vector xk|k of the target and the probability of detection pD (xk|k ) of the track. The reasoning still holds if there are multiple targets that are not too close to each other (with respect to sensor resolution). Stated differently: • The probability of detection at a track is an unknown that can be statistically estimated using a recursive filter. This fact can be expressed more formally as follows [190]. Replace the kinematic state x by the augmented state ˚ x = (a, x)

(16.20)

where 0 ≤ a ≤ 1 is the unknown probability of detection of x. Thus the space of augmented target states is ˚ X = [0, 1] × X (16.21) where [0, 1] is the unit interval. The integral of a function f˚(˚ x) with arguments in this space is ∫ ∫ ∫ 1 ˚ f (˚ x)d˚ x= f˚(a, x)dadx. (16.22) 0

1

The term “static” means that the state x of the target does not vary with time.

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The multitarget state space ˚ X∞ is the hyperspace of all finite subsets of ˚ X. A multitarget state set has the form ˚ = {˚ X x1 , ...,˚ xn } = {(a1 , x1 ), ..., (an , xn )}. The corresponding set integral has the form ∫ ∑ 1 ∫ ˚ ˚ ˚ f (X)δ X = f˚({˚ x1 , ...,˚ xn })d˚ x1 · · · d˚ xn . n!

(16.23)

(16.24)

n≥0

Because the state space has been changed from X to ˚ X, we must correspondingly change any modeling formulas, occurring earlier in the book, that involve state variables. Thus the usual probability of detection pD (x) and usual single-target likelihood function Lz (x) are replaced by the augmented probability of detection and augmented likelihood function ˚ pD (˚ x) ˚ Lz (˚ x)

= ˚ pD (a, x) def. =a = ˚ Lz (a, x) def. = Lz (x).

(16.25) (16.26)

(Here it is being assumed, as an approximation, that a target will generate the same measurement, regardless of its detectability.) Given this, the multitarget measurement model of (16.4) can be written as Σk+1 = Υk+1 (˚ x1 ) ∪ ... ∪ Υk+1 (˚ xn ) ∪ Ck+1

(16.27)

where Ck+1 is the a priori clutter process, and where Υk+1 (˚ x) is Bernoulli with ∫ GΥk+1 (a,x) [g] = 1 − a + a g(z) · fk+1 (z|x)dz. (16.28) From (4.126), the p.g.fl. of Σk+1 is ˚

˚ = (1 − ˚ Lg )X · Gκk+1 [g] Gk+1 [g|X] pD + ˚ pD ˚

(16.29)

where the functional-power notation hX was defined in (3.5); where Gκk+1 [g] is the p.g.fl. of Ck+1 ; and where ∫ ∫ ˚ ˚ ˚ Lg (˚ x) = Lg (a, x) = g(z) · Lz (a, x)dz = g(z) · Lz (x)dz. (16.30)

Introduction to Part III

16.3.2

511

A General Model for Unknown Clutter

This has the form of (16.8): Σk+1 = Υk+1 (˚ x1 ) ∪ ... ∪ Υk+1 (˚ xn ) ∪ Ck+1 (˚ c1 ) ∪ ... ∪ Ck+1 (˚ cν ).

(16.31)

Here, ˚ xi = (ai , xi ) with 0 ≤ ai ≤ 1, and Υk+1 (˚ x) is Bernoulli with p.g.fl. GΥk+1 (a,x) [g] = 1 − a + a

∫

g(z) · fk+1 (z|x)dz.

(16.32)

Also, ˚ ci = (ci , ci ) where ci ∈ C, and where there are four cases of interest: 1. Case 1: Ck+1 (˚ c) is Poisson with unknown clutter rate c > 0: ˚κ

GCk+1 (c,c) [g] = eLg−1 (c.c) where ˚ Lκg−1 (c, c) = c

∫

κ (g(z) − 1) · fk+1 (z|c)dz.

(16.33)

(16.34)

2. Case 2: Ck+1 (˚ c) is Bernoulli with unknown probability of detection 0 ≤ c ≤ 1: ∫ κ GCk+1 (c,c) [g] = 1 − c + c g(z) · fk+1 (z|c)dz, (16.35) in which case the clutter probability of detection and clutter-generator likelihood function are ˚ pκD (c, c) κ fk+1 (z|c, c)

= =

c κ ˚ Lκz (c, c) = Lκz (c) = fk+1 (z|c).

(16.36) (16.37)

3. Case 3: Ck+1 (˚ c) = Ck+1 (c) is Bernoulli with known probability of detection pκD (c): GCk+1 (c) [g] = 1 − pκD (c) + pκD (c)

∫

κ g(z) · fk+1 (z|c)dz.

(16.38)

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4. Case 4: Ck+1 (˚ c) = Ck+1 (c) is Bernoulli with known and constant probability of detection c = pκD , and where the clutter spatial distribution is independent of c and thus is known a priori: κ fk+1 (z|c) = ck+1 (z).

(16.39)

In this case GCk+1 (c) [g] = 1 − pκD + pκD

∫

g(z) · ck+1 (z)dz

(16.40)

and thus only the clutter rate λk+1 is unknown and must be determined. In all of these cases, let us package the unknowns—those involving both targets and the clutter generators—into an unknown state set of the form ¨ X

=

˚⊎C ˚ {¨ x1 , ..., x ¨n+ν } = X

(16.41)

= =

{˚ x1 , ...,˚ xn } ⊎ {˚ c1 , ...,˚ cν } {˚ x1 , ...,˚ xn ,˚ c1 , ...,˚ cν }

(16.42) (16.43)

where x ¨ denotes an element of the joint target-clutter state space ¨ =˚ X X ⊎˚ C

(16.44)

and where ‘⊎’ denotes disjoint union. Thus x ¨ =˚ x or x ¨ =˚ c. Remark 68 (Notational convention) Using a slight abuse of notation, if a = pD (x) or c = pκD (c), then it will be understood that ˚ x = (pD (x), x) ?→ x respectively ˚ c = (pκD (c), c) ?→ c. that is, ˚ x is identified with x and ˚ c is identified with c. In this case,  c if prob. det. is unknown  pκD (c) if prob. det. is known ˚ pκD (c, c) = (16.45)  pκD if prob. det. is known and constant κ fk+1 (z|c, c)

=

κ fk+1 (z|c).

(16.46)

Introduction to Part III

The integral ¨ is space X

∫

513

f¨(¨ x)d¨ x of a function f¨(¨ x) on the joint target-clutter state ∫

f¨(¨ x)d¨ x=

∫

∫

f¨(˚ x)d˚ x+ ˚ X

f¨(˚ c)d˚ c

˚ C

with the following special cases (assuming that the target probability of detection pD (x) is known): • Poisson clutter: ∫

f¨(¨ x)d¨ x=

∫

f¨(x)dx + X

∫ ∫

∞

f¨(c, c)dcdc. 0

C

• Bernoulli clutter, unknown clutter probability of detection: ∫

f¨(¨ x)d¨ x=

∫

f¨(x)dx + X

∫ ∫ C

1

f¨(c, c)dcdc.

(16.47)

0

• Bernoulli clutter, known clutter probability of detection: ∫

f¨(¨ x)d¨ x=

∫

f¨(x)dx +

∫

f¨(c)dc.

(16.48)

∑ 1 ∫ f ({¨ x1 , ..., x ¨n })d¨ x1 · · · d¨ xn . n!

(16.49)

X

C

The corresponding set integrals have the form ∫

¨ X ¨ = f (X)δ

n≥0

¨ (k) ) is the distribution of the RFS Ξ ¨ k||k then If fk|k (X|Z ˚ ¨ k|k ∩ ˚ Ξk|k = Ξ X

(16.50)

˚k|k = Ξ ¨ k||k ∩ ˚ Ψ C

(16.51)

is the RFS of targets and

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is the RFS of clutter generators. According to the discussion in Section 3.5.3, the ˚k|k are the marginals distributions of ˚ Ξk|k and Ψ ˚ (k) ) fk|k (X|Z

˚ (k) ) fk|k (C|Z

=

∫

˚ ⊎ C|Z ˚ (k) )δ C ˚ f¨k|k (X

=

∫

˚ C|Z ˚ (k) )δ C ˚ fk|k (X,

=

∫

˚ ⊎ C|Z ˚ (k) )δ X ˚ f¨k|k (X

=

∫

˚ C|Z ˚ (k) )δ X. ˚ fk|k (X,

(16.52)

(16.53)

Given all of this, the p.g.fl. of the measurement model of (16.31) is: ˚

˚

¨ = (1 − ˚ Gk+1 [g|X] pD + ˚ pD ˚ Lg )X · (1 − ˚ pκD + ˚ pκD ˚ Lκg )C .

(16.54)

¨ factors as This is because, given the independence assumptions, Gk+1 [g|X] ¨ Gk+1 [g|X]

= =

= 16.3.3

˚ ⊎ C] ˚ Gk+1 [g|X Gk+1 [g|˚ x1 ] · · · Gk+1 [g|˚ xn ] ·Gk+1 [g|˚ c1 ] · · · Gk+1 [g|˚ cν ] ˚ ˚ X pκD + ˚ (1 − ˚ pD + ˚ pD ˚ Lg ) · (1 − ˚ pκD ˚ Lκg )C .

(16.55) (16.56)

(16.57)

Unknown-Clutter Models: Poisson-Mixture

This is the measurement model for the Poisson-mixture clutter-agnostic PHD filter to be described in Section 18.10. It was introduced by Mahler in [179], Section 12.11, and is a generalization of a Bayesian static data-clustering approach due to Cheeseman [33], [34]. In this case the clutter RFS in (16.8) has the form ˚ = Ck+1 (c1 , c1 ) ∪ ... ∪ Ck+1 (cν , cν ) Ck+1 (C)

(16.58)

˚ = {(c1 , c1 ), ..., (cν , cν )}. Since the Poisson RFSs Ck+1 (c1 , c1 ),..., where C ˚ is itself a Poisson RFS. AccordCk+1 (cν , cν ) are independent, Ck+1 (C) ing to (16.33), the respective PHDs (intensity functions) of the clutter generaκ κ tors are c1 · fk+1 (z|c1 ), ..., cν · fk+1 (z|cν ) and their respective p.g.fl.’s are κ κ ˚ ˚ Lg−1 (c1 ,c1 ) Lg−1 (cν ,cν ) κ ˚ e , ..., e , where L (c, c) was defined in (16.34). Then the g−1

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PHD (intensity function) of the total clutter process is κ κ c1 · fk+1 (z|c1 ) + ... + cν · fk+1 (z|cν ) κ κ ˚ ˚ = Lz (c1 , c1 ) + ... + Lz (cν , cν )

˚ κk+1 (z|C)

(16.59)

=

(16.60)

where ˚ Lκz (c, c) was defined in (16.37). Thus the p.g.fl. of the total measurement model is ˚ ⊎ C] ˚ Gk+1 [g|X

˚ · Gk+1 [g|C] ˚ Gk+1 [g|X] ( )X ˚ ˚κ ˚ 1−˚ pD + ˚ pD ˚ Lg · (eLg−1 )C

= =

(16.61) (16.62)

where ˚ Gk+1 [g|C]

˚κ

˚

(eLg−1 )C =

=

∏

˚κ

eLg−1 (c,c)

(16.63)

˚ (c,c)∈C

˚ Lκg−1 (c, c)

c·

=

∫

κ (g(z) − 1) · fk+1 (z|c)dz.

(16.64)

Remark 69 It should also be noted (see Section 18.10) that the Poisson-mixture model—as well as the Poisson-mixture CPHD filter—can be easily generalized to clutter generators whose p.g.fl.’s have the form Gκk+1 [g|c, c]

=

Gκk+1

(

1−c+c

∫

g(z) ·

κ fk+1 (z|c)dz

)

,

where Gκk+1 (z) is some p.g.f. 16.3.4

Unknown-Clutter Models: General Bernoulli

This is the case considered in (16.35). It is the measurement model for the “κagnostic” CPHD filter of Section 18.5. By analogy with the unknown-pD model in (16.8), the state representation of a clutter generator has the form ˚ c = (c, c) where 0 ≤ c ≤ 1 is the unknown probability of detection of the clutter generator with kinematic state c. Thus ˚ pκD (˚ c) κ ˚ Lz (˚ c)

= =

c κ Lκz (c) = fk+1 (z|c).

(16.65) (16.66)

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The entire clutter intensity function κk+1 (z) is unknown and must be determined. According to (16.54), its p.g.fl. is ( )X )C˚ ˚ ( ˚ ⊎ C] ˚ = 1−˚ Gk+1 [g|X pD + ˚ pD ˚ Lg · 1−˚ pκD + ˚ pκD ˚ Lκg

(16.67)

˚ = {˚ ˚ = {(c1 , c1 ), ..., (cν , cν )} where X x1 , ...,˚ xn } = {(a1 , x1 ), ..., (an , xn )} and C and ˚ ˚ = (1 − ˚ Gk+1 [g|C] pκD + ˚ pκD ˚ Lκg )C (16.68) is the p.g.fl. of the multi-Bernoulli clutter process. 16.3.5

Unknown-Clutter Models: Simplified Bernoulli

This was the case considered in (16.40). It is used for the “λ-agnostic” CPHD filter of Section 18.4. By assumption, the clutter RFS in (16.8) has the form Ck+1 = Ck+1 (c1 ) ∪ ... ∪ Ck+1 (cν )

(16.69)

where the RFSs Ck+1 (c1 ), ..., Ck+1 (cν ) are Bernoulli and independent, with stateindependent clutter probability of detection and state-independent clutter likelihood function: pκD (c) Lκz (c)

= =

pκD ck+1 (z).

(16.70) (16.71)

The p.g.fl. of Ck+1 (c) is therefore independent of the state c: Gk+1 [g|c] = 1 − pκD + pκD · ck+1 [g], where ck+1 [g] =

∫

g(z) · ck+1 (z)dz.

(16.72)

(16.73)

The p.g.fl. of the total measurement process, (16.54), becomes ( )X ˚ |C| ¨ = 1−˚ Gk+1 [g|X] pD + ˚ pD ˚ Lg · (1 − pκD + pκD ck+1 [g])

(16.74)

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where X = {˚ x1 , ...,˚ xn } = {(a1 , x1 ), ..., (an , xn )} and C = {c1 , ..., cν } and ¨ =X ˚ ⊎ C and where the p.g.fl. of the clutter RFS is X Gk+1 [g|C] = (1 − pκD + pκD · ck+1 [g])|C| .

(16.75)

Note that the exponent on the right side of this equation is a scalar |C| rather than a finite set C.

16.4

ORGANIZATION OF PART III

Part III is organized as follows: 1. Chapter 17: PHD filters, CPHD filters, and multi-Bernoulli filters that can operate with unknown probability of detection. 2. Chapter 18: PHD filters, CPHD filters, and multi-Bernoulli filters that can operate in unknown clutter. In regard to the RFS filters described in these chapters, the following disclaimers should be emphasized: • The detection profile must not change too rapidly in comparison to the measurement-update rate. • The clutter statistics must not change too rapidly in comparison to the measurement-update rate. Furthermore, all of these filters must simultaneously accomplish at least two or more of the following difficult tasks, using the same measurement-stream: 1. Detect targets. 2. Track targets. 3. Implicitly or explicitly estimate pD . 4. Implicitly or explicitly estimate clutter rate λ or the clutter intensity function κ. Despite these difficulties, simulations have shown that these filters do seem to perform reasonably well (although with decreasing performance as they are required to simultaneously accomplish more and more tasks).

Chapter 17 RFS Filters for Unknown pD 17.1

INTRODUCTION

The model for an unknown target-detection profile was described in Section 16.3.1. As noted there, if an RFS-based multitarget detection and tracking filter employs the standard multitarget measurement model, then the unknown detection profile model can be used to convert it into a filter that does not require a priori knowledge of the probability of detection (see Section 17.1.1). The purpose of this chapter is to describe the following filters and their practical implementations: • pD -agnostic PHD filter (“pD -PHD filter” for short): the classical PHD filter, generalized to address unknown probability of detection. • pD -agnostic CPHD filter (“pD -CPHD filter” for short): the classical CPHD filter, generalized to address unknown probability of detection. • pD -agnostic CBMeMBer filter (“pD -CBMeMBer filter” for short): the CBMeMBer filter, generalized to address unknown probability of detection. The remainder of this Introduction is organized as follows: 1. Section 17.1.1: Overview of the approach—converting RFS tracking filters into filters that do not require priori knowledge of the probability of detection. 2. Section 17.1.2: Defining motion models for probability of detection—that is, the Markov transition density for the probability of detection. 3. Section 17.1.3: A summary of the major Lessons Learned in the chapter.

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4. Section 17.1.4: Organization of the chapter. 17.1.1

Converting RFS Filters into pD -Agnostic Filters

The conversion requires only a simple substitution of variables. Specifically: • Whenever x occurs in a formula, substitute (a, x). • Whenever pD (x) occurs, substitute a. • Whenever Lz (x) occurs, do not change it. ∫ ∫ ∫1 • Whenever an integral ·dx occurs, substitute the integral ·dadx. 0 Thus, for example, the measurement-update and time-update formulas for the classical PHD filter are (see (8.50) through (8.52), and (8.15) through (8.16)): Dk+1|k+1 (x) Dk+1|k (x)

=

τk+1 (z)

=

Dk+1|k (x)

=

1 − pD (x) +

∑

pD (x) · Lz (x) κk+1 (z) + τk+1 (z)

(17.1)

z∈Zk+1

∫

pD (x) · Lz (x) · Dk+1|k (x)dx

(17.2)

bk+1|k (x) (17.3) ) ∫ ( pS (x′ ) · fk+1|k (x|x′ ) + · Dk|k (x′ )dx′ . +bk+1|k (x|x′ )

Making the indicated substitutions renders the PHD filter pD -agnostic: ˚k+1|k+1 (a, x) D ˚k+1|k (a, x) D τk+1 (z)

=

1−a+

∑

a · Lz (x) κk+1 (z) + τk+1 (z)

(17.4)

z∈Zk+1

=

∫ ∫

1

˚k+1|k (a, x)dadx a · Lz (x) · D

(17.5)

0

˚k+1|k (a, x) D

= ˚ bk+1|k (a, x) (17.6) ) ∫ ∫ 1( ˚ pS (a′ , x′ ) · f˚k+1|k (a, x|a′ , x′ ) + +˚ bk+1|k (a, x|a′ , x′ ) 0 ˚k|k (a,′ x′ )da′ dx′ . ·D

Some thought is, of course, required to suitably interpret items such as ˚ bk+1|k (a, x), ˚ pS (a′ , x′ ), and ˚ bk+1|k (a, x|a′ , x′ ). Since probability of detection

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521

should not influence the appearance of new targets, one must have ˚ bk+1|k (a, x) ′

′

˚ bk+1|k (a, x|a , x )

= =

(17.7)

bk+1|k (x) ′

(17.8)

bk+1|k (x|x ).

Similarly, ˚ pS (a′ , x′ ) = pS (x′ )

(17.9)

since probability of detection should have no bearing on whether or not an existing target disappears. For the Markov transition density, we assume that: f˚k+1|k (a, x|a′ , x′ ) = fk+1|k (a|a′ ) · fk+1|k (x|x′ ).

(17.10)

This is an approximation, because the value of the probability of detection a will in general be correlated with the kinematic target-state x. With these changes, the time-update equation for the pD -PHD filter becomes ˚k+1|k (a, x) D

17.1.2

=

bk+1|k (x) (17.11) ) ∫ ∫ 1( ′ ′ ′ pS (x ) · fk+1|k (a|a ) · fk+1|k (x|x ) + +bk+1|k (x|x′ ) 0 ˚k|k (a,′ x′ )da′ dx′ . ·D

A Motion Model for Probability of Detection

In (17.11), the Markov transition fk+1|k (a|a′ ) must be defined. In Appendix F it is shown that it can be defined implicitly as

βuk+1|k ,vk+1|k (a) =

∫

1

fk+1|k (a|a′ ) · βuk|k ,vk|k (a′ )da′

(17.12)

0

where βu,v (a) denotes a beta distribution with parameters u, v; and where uk+1|k v

k+1|k

θk|k

= = =

uk|k · θk|k v

k|k

· θk|k

1 · uk|k + v k|k

(17.13) (17.14) (

k|k

k|k

u ·v 1 · 2 −1 (uk|k + v k|k )2 σk+1|k

)

(17.15)

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and where the desired variance of the time-updated probability of detection, ( ) 1 uk|k · v k|k 2 ( )( ), σk+1|k = + ε · (17.16) uk|k + v k|k uk|k + v k|k uk|k + v k|k + 1 is set by choosing a value of ε with 0 ≤ ε ≤ 1. 2 2 It is always the case that σk+1|k ≥ σk|k —that is, that the uncertainty in a never decreases during a time-update. The following two values of ε describe the 2 extremes of σk+1|k : ε ε

17.1.3

= =

0: 1:

2 2 σk+1|k = σk|k 2 σk+1|k = (

u

(no increase)

k|k

uk|k

·v

(17.17)

k|k

+ v k|k

)2 (largest increase).

(17.18)

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter: • A simple approach for addressing unknown probability of detection is to replace the single-target state x with an augmented state ˚ x = (a, x) where 0 ≤ a ≤ 1 is the unknown probability of detection of the unknown track x (Section 17.1.1). • A suitable Markov motion model for a allows us to model the increase in uncertainty in the time interval between measurement-updates (Section 17.1.2). • Any computationally tractable RFS multitarget detection and tracking filter can be converted to a tractable filter that operates with unknown probability of detection (Section 17.1.1). • PHD and CPHD filters with unknown probability of detection can be implemented in exact closed form using beta-Gaussian mixture (BGM) techniques, in which the statistical behavior of a is modeled by a beta distribution βu,v (a) (Sections 17.4 and 17.4): νk|k

˚k|k (a, x) = D

∑ i=1

k|k

wi

k|k

· βuk|k ,vk|k (a) · NP k|k (x − xi ). i

i

i

(17.19)

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523

• Garbage in, garbage out: For good performance to be possible in an RFS pD -agnostic multitarget detection and tracking filter, the detection profile must be slowly-varying in comparison to the measurement-update rate. 17.1.4

Organization of the Chapter

The chapter is organized as follows: 1. Section 17.2: The pD -CPHD filter—a version of the CPHD filter that does not require a priori knowledge of the detection profile. 2. Section 17.3: The beta-Gaussian mixture (BGM) approximation of a PHD Dk|k (a, x). 3. Section 17.4: BGM implementation of the pD -PHD filter. 4. Section 17.5: BGM implementation of the pD -CPHD filter. 5. Section 17.6: The pD -CBMeMBer filter—a version of the CBMeMBer filter that does not require a priori knowledge of the detection profile. 6. Section 17.7: Implementations of pD -agnostic RFS filters.

17.2

THE PD -CPHD FILTER

The pD -CPHD filter is derived from the usual CPHD filter equations using the procedure outlined in Section 17.1.1. The section is organized as follows: 1. Section 17.2.1: Modeling assumptions for the pD -CPHD filter. 2. Section 17.2.2: Time update equations for the pD -CPHD filter. 3. Section 17.2.3: Measurement update equations for the pD -CPHD filter. 4. Section 17.2.4: Multitarget state estimation for the pD -CPHD filter. 17.2.1

pD -CPHD Filter Models

The following models are used in the pD -CPHD filter: • Probability of target survival: pS (x′ ). • Target Markov density: fk+1|k (x|x′ ).

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• Markov density for probability of detection: fk+1|k (a|a′ )—as defined in (17.12) through (17.16). ∫ def. B • PHD of the target appearance RFS: bk+1|k (x) with Nk+1|k = bk+1|k (x)dx. • Cardinality distribution and p.g.f. of the target appearance RFS: pB k+1|k (n), ∑ ∑ B B B B with Nk+1|k = n≥0 n · pk+1|k (n); and Gk+1|k (x) = n≥0 pk+1|k (n) · xn . • Probability of detection at a track-state x: an unknown state variable 0 ≤ a ≤ 1, the explicit estimation of which is optional (as desired or not). abbr,

• Sensor likelihood function: Lz (x) = fk+1 (z|x). • Clutter cardinality distribution: pκk+1 (m). • Clutter spatial distribution: ck+1 (z). 17.2.2

pD -CPHD Filter Time Update

The time-update equations for the pD -CPHD filter are: • Predicted spatial distribution:  ˚ sk+1|k (a, x)

ψk

 ∫ ∫1 bk+1|k (x) + Nk|k p (x′ ) 0 S   ·fk+1|k (a|a′ ) ′ ′ ′ ′ ′ ·fk+1|k (x|x ) · ˚ sk|k (a , x )da dx = (17.20) B Nk+1|k + Nk|k · ψk ∫ ∫ 1 = ˚ sk|k [˚ pS ] = pS (x) · ˚ sk|k (a, x)dadx (17.21) 0

∫

B where Nk+1|k = bk+1|k (x)dx. Expressed in terms of PHDs rather than spatial distributions, this becomes:

˚k+1|k (a, x) D

=

bk+1|k (x) ∫ ∫ 1 + pS (x′ ) · fk+1|k (a|a′ )

(17.22)

0

ψk

=

˚k|k (a′ , x′ )da′ dx′ ·fk+1|k (x|x′ ) · D ∫ ∫ 1 1 ˚k|k (a, x)dadx. (17.23) pS (x) · D Nk|k 0

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• Predicted cardinality distribution and p.g.f.: Gk+1|k (x)

=

pk+1|k (n)

=

pk+1|k (n|n′ )

GB k+1 (x) · Gk|k (1 − ψk + ψk · x) ∑ pk+1|k (n|n′ ) · pk|k (n′ )

n′ ≥0 n ∑ = pB k+1|k (n i=0

(17.24) (17.25) ′

− i) · Cn′ ,i · ψki (1 − ψk )n −i . (17.26)

• Predicted expected number of targets: B Nk+1|k = Nk+1|k + Nk|k · ψk .

17.2.3

(17.27)

pD -CPHD Filter Measurement Update

Let a new measurement set Zk+1 with |Zk+1 | = m be collected. Then the measurement-update equations for the pD -CPHD filter are: • Measurement updated cardinality distribution and p.g.f.: pk+1|k+1 (n)

=

Gk+1|k+1 (x)

=

ℓZ (n) · pk+1|k (n) ∑ k+1 (17.28) l≥0 ℓZk+1 (l) · pk+1|k (l) ( ∑m j ) κ j=0 x · (m − j)! · pk+1 (m − j) ·G(j) (x · ϕk ) · σj (Zk+1 ) ( ∑m ) (17.29) κ i=0 (m − i)! · pk+1 (m − i) ·G(i) (ϕk ) · σi (Zk+1 )

where ( ∑ min{m,n} ℓZk+1 (n) =

(m − j)! · pκk+1 (m − j) j=0 ·j! · Cn,j · ϕn−j · σj (Zk+1 ) k ( ∑m ) κ l=0 (m − l)! · pk+1 (m − l) (l) ·σl (Zk+1 ) · Gk+1|k (ϕk )

) .

(17.30)

• Measurement updated spatial distribution or PHD: ˆ Z (a, x) · ˚ ˚ sk+1|k+1 (a, x) = L sk+1|k (a, x) k+1

(17.31)

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or ˚k+1|k+1 (a, x) = ˚ ˚k+1|k (a, x) D LZk+1 (a, x) · D

(17.32)

where ˆ Z (a, x) L k+1 

1

=

˚ LZk+1 (a, x)

=



ND

(1 − a) · L Zk+1

(17.33) 

 ∑m a·Lz (x) D Nk+1|k+1 + j=1 ck+1j(zj ) · LZk+1 (zj )   ND − a) · L Zk+1 1  ∑ (1 a·L  (17.34) D z (x) m Nk+1|k + j=1 ck+1j(zj ) · LZk+1 (zj ) ( ∑m

) − j)! · pκk+1 (m − j) (j+1) ·σj (Zk+1 ) · Gk+1|k (ϕk ) ( ∑m ) κ l=0 (m − l)! · pk+1 (m − l) (l) ·σl (Zk+1 ) · Gk+1|k (ϕk ) j=0 (m

ND

L Zk+1

=

(17.35)

D

( ∑m−1

(17.36) )

∑

(17.37)

LZk+1 (zj ) (m − i − 1)! · pκk+1 (m − i − 1) (i+1) ·σi (Zk+1 − {zj }) · Gk+1|k (ϕk ) ( ∑m ) κ l=0 (m − l)! · pk+1 (m − l) (l) ·σl (Zk+1 ) · Gk+1|k (ϕk ) i=0

=

(l)

Gk+1|k (ϕk )

=

pk+1|k (n) · l! · Cn,l · ϕn−l k

n≥l (j+1)

(17.38)

Gk+1|k (ϕk ) =

∑ n≥j+1

pk+1|k (n) · (j + 1)! · Cn,j+1 · ϕn−j−1 k

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where ϕk

∫ ∫

1

(1 − a) · ˚ sk+1|k (a, x)dadx (17.39) ( ) τˆk+1 (z1 ) τˆk+1 (zm ) σm,i , ..., (17.40) ck+1 (z1 ) ck+1 (zm )   τˆ? τˆk+1 (z1 ) k+1 (zj ) , ..., ck+1 (zj )  (17.41) σm−1,i  ck+1 (z1 ) τˆ (z m) k+1 , ..., ck+1 (zm )

=

0

σi (Zk+1 )

=

σi (Zk+1 − {zj })

=

= ˚ sk+1|k [˚ pD ˚ Lz ] ∫ = a · Lz (x) · ˚ sk+1|k (a, x)dadx.

τˆk+1 (z)

(17.42) (17.43)

• Measurement updated expected number of targets: ND

Nk+1|k+1 = ϕk · L Zk+1 +

m ∑ τˆk+1 (zi )

ck+1 (zi )

D

· LZk+1 (z).

(17.44)

i=1

17.2.4

pD -CPHD Filter Multitarget State Estimation

At time tk+1 we are given a measurement-updated cardinality distribution pk+1|k+1 (n) and a measurement-updated PHD Dk+1|k+1 (a, x) or spatial distribution sk+1|k+1 (a, x). We are to estimate the number and states of the targets. Two approaches are considered here: target-only estimation and joint target-pD estimation. 17.2.4.1

Method 1: Target-Only Estimation

Assume that the target states are to be estimated, but not their probabilities of detection. We can then integrate a out as a nuisance variable: Dk+1|k+1 (x) =

∫

1

˚k+1|k+1 (a, x)da. D

(17.45)

0

Then we can apply the usual state-estimation procedure for the CPHD filter as in Section 8.5.5. That is, determine the MAP estimate of target number: n ˆ = arg sup pk+1|k+1 (n) n≥0

(17.46)

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and determine the n ˆ largest suprema of Dk+1|k+1 (x). The state estimates are the states x ˆ1 , ..., x ˆnˆ that correspond to those suprema. The same approach applies to the pD -PHD filter of (17.4) and (17.11), except that instead of n ˆ we substitute the nearest integer to the measurement-updated expected number of targets, Nk+1|k+1 . 17.2.4.2

Method 2: Joint Target-pD Estimation

Assume that we want to know not only the states of the targets, but also the probabilities of detection at those states. In this case, determine the MAP estimate of target number: n ˆ = arg sup pk+1|k+1 (n). (17.47) n≥0

˚k+1|k+1 (a, x) and let (ˆ Then determine the n ˆ largest suprema of D a1 , x ˆ1 ), ..., (ˆ anˆ , x ˆnˆ ) be the augmented states that correspond to them. The estimated target states are x ˆ1 , ..., x ˆnˆ , and their respective probabilities of detection are a ˆ1 , ..., a ˆnˆ . The same approach applies to the pD -PHD filter of (17.4) and (17.11), except that the nearest integer to Nk+1|k+1 is used in place of n ˆ.

17.3

BETA-GAUSSIAN MIXTURE (BGM) APPROXIMATION

Gaussian mixture (GM) implementation of the pD -PHD filter and pD -CPHD filter ˚k|k (a, x) is not possible because the PHDs or spatial distributions have the form D or ˚ sk|k (a, x), rather than the usual Dk|k (x) or sk|k (x) where x is a Euclidean column vector. Consequently, some generalization must be devised. The purpose of this section is to describe such a generalization: beta-Gaussian mixture (BGM) approximation. The section is organized as follows: 1. Section 17.3.1: Overview of the approach. 2. Section 17.3.2: Beta-Gaussian mixtures (BGMs). 3. Section 17.3.3: Pruning BGMs. 4. Section 17.3.4: Merging BGMs.

RFS Filters for Unknown pD

17.3.1

529

Overview of the BGM Approach

The two following items pose a challenge to the problem of generalizing the concept of a Gaussian mixture to pD -agnostic CPHD filter: • The factors a and 1 − a that occur in (17.4) and (17.34). For the sake of conceptual simplicity, suppose for the moment that targets are static (that is, their states do not change with time). Then time-updates are unnecessary and the pD -PHD and pD -CPHD filters reduce to repeated application of the measurement-update equations. As time progresses and because of (17.4) and (17.34), the formulas for the PHD or the spatial distribution will contain factors of the form ai (1 − a)j where i, j are arbitrarily large integers. A simple way to address this challenge was proposed by Mahler, Vo, and Vo in [194]. They noted that beta distributions have a similar form:

βu,v (a) = where β(u, v) =

∫

au−1 (1 − a)v−1 β(u, v)

(17.48)

1

au−1 (1 − a)v−1 da

(17.49)

0

is the beta function. (See Appendix E for a discussion of beta distributions and their properties.) The Gaussian mixture approximation is made possible by the fact that Gaussian distributions are algebraically closed under multiplication: NP1 (x − x1 ) · NP2 (x − x2 ) E −1 E −1 e

= = =

NP1 +P2 (x2 − x1 ) · NE (x − e) (17.50) P1−1 + P2−1 , (17.51) P1−1 x1 + P2−1 x2 . (17.52)

Beta distributions are also algebraically closed under multiplication: βu1 ,v1 (a) · βu2 ,v2 (a)

=

β(u1 + u2 − 1, v1 + v2 − 1) β(u1 , v1 ) · β(u2 , v2 ) ·βu1 +u2 −1,v1 +v2 −1 (a).

(17.53)

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In particular, by (E.6) and (E.7) in Appendix E, we have the following two special cases: a · βu,v (a)

= =

β(u + 1, v) · βu+1,v (a) β(u, v) u · βu+1,v (a) u+v

(17.54) (17.55)

and (1 − a) · βu,v (a)

= =

17.3.2

β(u, v + 1) · βu,v+1 (a) β(u, v) v · βu,v+1 (a). u+v

(17.56) (17.57)

Beta-Gaussian Mixtures (BGMs)

˚ x) be a PHD on the space These considerations suggest the following. Let D(a, N ˚ x) is [0, 1] × R . Then a beta-Gaussian mixture (BGM) approximation of D(a, an approximation of the form ˚ x) ∼ D(a, =

ν ∑

wi · βui ,vi (a) · NPi (x − xi )

(17.58)

i=1

where wi ≥ 0 for all i = 1, ..., ν. Remark 70 (Implicit assumptions) Implicit in this approximation is the assumption that pD (xi ) is statistically independent of xi , which is not true in general. However, the pD -PHD and pD -CPHD filters themselves depend on the implicit assumption that the data rate is rapid enough that pD (x) changes relatively slowly in both time and space. It follows from (17.50) that the product of two BGMs is also a BGM. From (17.55) and (17.57) it follows that the integrals ∫

1

˚ x)da, a · D(a, 0

∫

1

˚ x)da (1 − a) · D(a, 0

˚ x) is a BGM. can be evaluated in exact closed form if D(a, It follows that if the PHD is approximated as a BGM, then:

(17.59)

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531

• The time-update and measurement-update equations for the pD -PHD and pD -CPHD filters can be evaluated in exact closed form, provided that we impose a few relatively minor restrictions (see Sections 17.4.1 and 17.5.1). Thus we write νk|k

˚k|k (a, x) D

=

∑

k|k

k|k

· βuk|k vk|k (a) · NP k|k (x − xi )

wi

i

i

(17.60)

i

i=1 νk+1|k

˚k+1|k (a, x) D

=

∑

k+1|k

· βuk+1|k vk+1|k (a)

wi

i

(17.61)

i

i=1 k+1|k

·NP k+1|k (x − xi

)

i

for all k ≥ 0. It follows that the Time propagation of the PHDs is equivalent to the Time propagation of families of the form k|k

k|k

k|k

k|k

k|k

(ℓi , wi , ui , vi , Pi

k|k ν

k|k , xi )i=1

k|k

where, in addition to the other items in the family, ℓi is the track label of the ith BGM component. This is the basis for the BGM implementations of the pD -PHD filter and pD -CPHD filter equations to be described shortly. Briefly stated, the measurementupdate equations for these filters are constructed by making the substitutions k|k

pD,k (xi )

ui

?→

k|k

ui

(17.62) k|k

+ vi k|k

1 − pD,k (xi )

vi

?→

k|k

ui 17.3.3

.

(17.63)

k|k

+ vi

Pruning BGM Components

As time progresses, the number of components in a BGM approximation of a PHD tends to increase without bound. As with Gaussian mixtures, various techniques must be employed to merge similar components and prune insignificant components. For computational reasons, it is better to prune before merging. Doing so avoids the computational cost associated with merging BGM components that will end up being pruned anyway.

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Pruning BGM components is similar to pruning GM components. Suppose that we are to prune components from the measurement-updated BGM system k+1|k+1

(wi

k+1|k+1

, ui

k+1|k+1

, vi

k+1|k+1

, Pi

k+1|k+1 νk+1|k+1 )i=1

, xi

∑νk+1|k+1 k+1|k+1 with Nk+1|k+1 = i=1 wi . Set a pruning threshold τprune , identify those components for which k+1|k+1

wi

< τprune ,

(17.64)

and then eliminate them. This results in a pruned system k+1|k+1

(w ˇi

k+1|k+1

,u ˇi

k+1|k+1

, vˇi

ˇk+1|k+1 k+1|k+1 k+1|k+1 ν , Pˇi ,x ˇi )i=1

with νˇk+1|k+1 components. Let ν ˇk+1|k+1

w ˇ k+1|k+1 =

∑

k+1|k+1

w ˇi

(17.65)

i=1

be the combined weight of all components that remain. Define k+1|k+1 k+1|k+1

= Nk+1|k+1 ·

w ˆi

w ˇi w ˇ k+1|k+1

(17.66)

for all i = 1, ..., νˇk+1|k+1 . Then k+1|k+1

(w ˆi

k+1|k+1

,u ˇi

k+1|k+1

, vˇi

ˇk+1|k+1 k+1|k+1 k+1|k+1 ν , Pˇi ,x ˇi )i=1

is the final pruned BGM system. 17.3.4

Merging BGM Components

To merge BGM components, we must first specify a merging criterion. Suppose that f˚1 (a, x) f˚2 (a, x)

=

w1 · βu1 ,v1 (a) · NP1 (x − x1 )

(17.67)

=

w2 · βu2 ,v2 (a) · NP2 (x − x2 )

(17.68)

RFS Filters for Unknown pD

533

are two components. From a purely mathematical point of view, it might seem desirable to define a metric that measures the distance between the density functions f˚1 (a, x) and f˚2 (a, x). However, the goal in tracking applications is to detect and localize targets—not to determine their respective probabilities of detection. So, instead, one should determine the distance between the marginal densities f1 (x) = w1 · NP1 (x − x1 ),

f2 (x) = w2 · NP1 (x − x1 ).

(17.69)

Consequently, the merging criterion for BGMs is the same as for Gaussian mixtures (as described in Section 9.5.3). Now suppose that we are given n BGM components f˚(a, x) = w1 · βu1 ,v1 (a) · NP1 (x − x1 ) + ... + wn · βun ,vn (a) · NPn (x − xn ) that are to be merged. In Section K.22 it is shown that the merged component that has the same mean and covariance as f˚(a, x) is w0 · βu0 ,v0 (a) · NP0 (x − x0 ) where: w0

=

n ∑

wi

(17.70)

i=1

w ˆi

=

x0

=

wi w0 n ∑

(17.71) w ˆ i · xi

(17.72)

( ) w ˆi · Pi + xi xTi

(17.73)

i=1

and where P0

=

−x0 xT0 +

n ∑ i=1

=

n ∑

w ˆ i · Pi +

i=1

u0 v0

= =

θ0 µ0 θ0 (1 − µ0 )

∑

w ˆi · w ˆj · (xi − xj )(xi − xj )T

(17.74)

1≤i 0. ´ • Clutter Markov transition density—in a transition from (´ c′ , C´ ′ ) to (´ c, C), ′ ´ the matrix C does not change whereas the uncertainty in the mean ´ c′ ′ ´ increases with covariance proportional to the inverse of C : ´ c′ , C´ ′ ) = N ´ f˚k+1|k (´ c, C|´ c−´ c′ ) · δC´ ′ (C) ´ ′ )−1 (´ (˚ φ k ·C

(18.313)

´ denotes the Dirac delta function on for some φ ˚k > 0, where δC´ ′ (C) ´ information matrices C, concentrated at the information matrix C´ ′ . The approach in [39] was used to implement a NWM-κ-PHD filter. The most significant innovation of this section will be to generalize this to a NWM-κ-CPHD filter. 18.5.8.1

Normal-Wishart Mixtures (NWMs)

When the clutter likelihood function has the form of (18.242),

Lκz (c)

=

k+1 ˚ ν∑

˚ k+1 c), ek+1 · NR ˚k+1 (z − Hi i

(18.314)

i

i=1

it is possible to implement the κ-CPHD filter using BGM approximation. When the likelihood function has the form of (18.312), however, this is no longer possible. The NWM approximation was designed to address this generalized likelihood function, and is as follows. An introduction to normal-Wishart distributions can be found in Appendix G. These have the form ´ N Wd,o,o,O (´ c, C)

RFS Filters for Unknown Clutter

605

where d, o, o, O are parameters, with o > 0, d > M , o ∈ Z = RM , and O an M × M positive-definite matrix that has the same units as a measurementcovariance matrix R. Normal-Wishart distributions satisfy the following two identities. First, the following slight generalization of [39], Eq. (27): ´ · N Wd,o,o,O (´ ´ = qz,d,o,o,O · N Wd∗ ,o∗ ,o∗ ,O∗ (´ ´ Lκz (´ c, C) c, C) c, C) z z

(18.315)

where d∗ o∗ o∗z Oz∗

qz,d,o,o,O

= d+1 = ˚ ηk+1 + o ˚ ηk+1 · z + o · o = ˚ ηk+1 + o ˚ ηk+1 · o · (z − o)(z − o)T = O+ ˚ ηk+1 + o ( ∗) (˚ ηk+1 · o)M/2 · Γ d2 · (det O)d/2 ( ∗ ) = . (π · o∗ )M/2 · Γ d −M · (det Oz∗ )d∗ /2 2

(18.316) (18.317) (18.318) (18.319)

(18.320)

Second ([39], Eq. (37)), ∫ ´ c′ , C´ ′ ) · N Wd,o,o,O (´ ´ (18.321) f˚k+1|k (´ c, C|´ c′ , C´ ′ )d´ c′ dC´ ′ = N Wd,˜o,o,O (´ c, C) where φ ˚k · a . (18.322) φ ˚k + o For the sake of conceptual completeness, both identities are verified in Appendix G. Given this, a normal-Wishart mixture (NWM) approximation of the clutter ´ has the form: PHD Dk|k (´ c, C) o˜ =

νk|k

˚k|k (´ ´ = D c, C)

∑

k|k

wi

´ c, C). · N Wdk|k ,ok|k ,ok|k ,Ok|k (´ i

i

i

(18.323)

i

i=1

Because of (18.315) and (18.321), the κ-CPHD filter equations can be solved in exact closed form. It thus becomes possible to propagate the system k|k

k|k

k|k

k|k

k|k ν

k|k (ℓi , di , oi , oi , Oi )i=1

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Advances in Statistical Multisource-Multitarget Information Fusion

´ of normal-Wishart parameters rather than propagate the clutter PHD Dk|k (´ c, C) itself. This process is described in the following subsections. For notational clarity, labels ℓ will be suppressed. 18.5.8.2

NWM-κ-CPHD Filter: Models

The NWM-κ-CPHD filter requires the following models: • Target probability of survival is constant: pS (x) abbr. = pS . • Target Markov density is linear-Gaussian: fk+1|k (x|x′ ) = NQk (x − Fk x′ ).

(18.324)

• PHD for target appearance is a Gaussian mixture: B νk+1|k

bk+1|k (x) =

∑

k+1|k

bi

k+1|k

· NB k+1|k (x − bi

).

(18.325)

i

i=1

• Cardinality distribution for target appearance: pB k+1|k (n), with B νk+1|k

B Nk+1|k

=

∑ i=1

k+1|k

bi

=

∑

n · pB k+1|k (n).

(18.326)

n≥0

´ =˚ • Clutter-generator probability of survival is constant: ˚ pS (c, ´ c, C) pS . ´ =˚ • Clutter-generator probability of detection is constant: ˚ pD (c, ´ c, C) pD . • Clutter-generator Markov density: has the form ´ c′ , C´ ′ ) = N ´ f˚k+1|k (´ c, C|´ c−´ c′ ) · δC´ ′ (C) ´ ′ )−1 (´ (˚ φ k ·C

(18.327)

´ is the Dirac delta function concentrated for some φ ˚k > 0, where δC´ ′ (C) ′ ´ ´ at C . Because δC´ ′ (C) cannot model any increase in the uncertainty in ´ Chen et al. compensated for this fact by introducing a time-update for the C, parameter d of the form δ · d′ (18.328) d= δ + d′ where δ is a fading factor that causes knowledge of d′ to diminish over time (see [39], Eq. (38)).

RFS Filters for Unknown Clutter

607

• PHD for appearance of clutter generators is a NWM:1 B νk+1|k

˚ ´ = bk+1|k (´ c, C)

∑

k+1|k ˚ ´ bi · N Wδk+1|k ,uk+1|k ,uk+1|k ,U k+1|k (´ c, C). i

i

i

i

i=1

(18.329) • Cardinality distribution for clutter-generator appearance: ˚ pk+1|k (˚ n) with B νk+1|k

˚B N k+1|k =

∑

k+1|k ˚ bi =

i=1

∑

n·˚ pB n). k+1|k (˚

(18.330)

˚ n≥0

• Target probability of detection is constant: pD (x) = pD . • Target likelihood function is linear-Gaussian: Lz (x) = NRk+1 (z − Hk+1 x).

(18.331)

• Clutter likelihood function: ´ =N Lκz (´ c, C) c) ´ −1 (z − ´ (˚ ηk+1 ·C)

(18.332)

for some ˚ ηk+1 > 0. 18.5.8.3

NWM-κ-CPHD Filter: Time Update

Suppose that we are given the NWM system p¨k|k (¨ n), k|k

k|k

k|k

(ℓi , wi , Pi k|k

k|k

k|k

k|k ν

k|k , xi )i=1 ,

k|k

ν k|k ˚

k|k (˚ wi , di , oi , oi , Oi )i=1 .

1

Chen et al. construct this PHD from the new measurements. Let zj be one of the measurements at 2 time-step k + 1 and let σk+1 be the variance of the measurement noise. Then they create mk+1 2 NWM components with dj = 1.5, oj = zj , oj = 0.5, and Oj = 8d−1 · σk+1 · IM ×M , j where IM ×M is the M × M identity matrix (see [39], p. 1227, Section IV-C-1).

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Advances in Statistical Multisource-Multitarget Information Fusion

We are to update it to a NWM system p¨k+1|k (¨ n), k+1|k νk+1|k )i=1 , νk+1|k k+1|k k+1|k k+1|k k+1|k k+1|k ˚ (˚ wi , di , oi , oi , Oi )i=1 . k+1|k

(ℓi

k+1|k

, wi

k+1|k

, Pi

, xi

This is accomplished as follows. Let νk|k

Nk|k

=

∑

k|k

(18.333)

wi

i=1 ˚ νk|k

˚k|k N

=

∑

k|k

(18.334)

w ˚i .

i=1

Then • Time update for cardinality distribution (same as for BGM-κ-CPHD filter): p¨k+1|k (¨ n)

=

∑

p¨k+1|k (¨ n|¨ n′ ) · p¨k|k (¨ n′ )

(18.335)

n ¨ ′ ≥0

p¨k+1|k (¨ n|¨ n′ )

=

n ¨ ∑

p¨B n − i) · Cn¨ ′ ,i k+1|k (¨

(18.336)

i=0 ′

·ψ¨ki (1 − ψ¨k )n¨ −i ψ¨k

=

˚k|k pS · Nk|k + ˚ pS · N ˚k|k Nk|k + N

(18.337)

pB pB n). k+1|k (n) · ˚ k+1|k (˚

(18.338)

where as in (18.29), p¨B n) = k+1|k (¨

∑ n+˚ n=¨ n

B There are νk+1|k = νk|k + νk+1|k GM components for the time-updated B target PHD and ˚ νk+1|k = ˚ νk|k + ˚ νk+1|k NWM components for the time-updated clutter PHD.

RFS Filters for Unknown Clutter

609

• Time update for persisting-target GM components (same as for BGM-κCPHD filter)—for i = 1, ..., νk|k : k+1|k

ℓi

k|k

=

k+1|k wi k+1|k xi k+1|k Pi

= = =

ℓi

(18.339)

k|k pS · w i k|k Fk xi k|k Fk Pi FkT

(18.340) (18.341) + Qk

(18.342)

• Time update for appearing-target GM components (same as for BGM-κB CPHD filter)—for i = νk|k + 1, ..., νk|k + νk+1|k : k+1|k

ℓi k+1|k wi

=

new labels

(18.343)

=

k+1|k bi−νk|k

(18.344)

k+1|k

=

bi−νk|k

(18.345)

=

k+1|k Bi−νk|k .

(18.346)

k+1|k

xi k+1|k Pi

• Time update for persisting-generator NWM components—for l = 1, ..., ˚ νk|k : k+1|k

w ˚l

k|k

= ˚ pS · w ˚l

k+1|k

dl

=

k+1|k

oi

=

k+1|k

oi k+1|k Oi

= =

k|k δk · dl k|k δk + dl k|k φ ˚k · oi k|k φ ˚k + oi k|k oi Oik+k

where δk is the fading factor of (18.328).

(18.347) (18.348)

(18.349) (18.350) (18.351)

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Advances in Statistical Multisource-Multitarget Information Fusion

• Time update for appearing-generator NWM components—for l = ˚ νk|k + B 1, ..., ˚ νk|k + ˚ νk+1|k : k+1|k

k+1|k = ˚ bl−˚ νB

w ˚l

(18.352)

k+1|k

k+1|k

k+1|k

dl k+1|k oi k+1|k oi k+1|k Oi

18.5.8.4

=

δl−˚ νk|k

(18.353)

=

k+1|k ul−˚ νk|k

(18.354)

=

k+1|k ul−˚ νk|k

(18.355)

=

k+1|k Ul−˚ νk|k .

(18.356)

NWM-κ-CPHD Filter: Measurement Update

Suppose that we are given the predicted NWM system p¨k+1|k (¨ n), k+1|k νk+1|k )i=1 , νk+1|k k+1|k k+1|k k+1|k k+1|k k+1|k ˚ (˚ wi , di , oi , oi , Oi )i=1 k+1|k

(ℓi

k+1|k

, wi

k+1|k

, Pi

, xi

Given a new measurement set Zk+1 with |Zk+1 | = mk+1 , we are to update it to a NWM system p¨k+1|k (¨ n), k+1|k+1 νk+1|k+1 )i=1 , νk+1|k+1 k+1|k+1 k+1|k+1 k+1|k+1 k+1|k+1 k+1|k+1 ˚ (˚ wi , di , oi , oi , Oi )i=1 . k+1|k+1

(ℓi

k+1|k+1

, wi

k+1|k+1

, Pi

, xi

We are also to determine the measurement-updated target cardinality distribution pk+1|k+1 (n). This is accomplished as follows. Let νk+1|k

Nk+1|k

=

∑

k+1|k

(18.357)

wi

i=1 ˚ νk+1|k

˚k+1|k N

=

∑ i=1

Then:

k+1|k

w ˚i

.

(18.358)

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611

• Measurement update for joint target/clutter cardinality distribution: p¨k+1|k+1 (¨ n)

=

ℓ¨Zk+1 (¨ n) · p¨k+1|k (¨ n) ∑ ¨ ¨k+1|k (l) l≥0 ℓZk+1 (l) · p

ℓ¨Zk+1 (¨ n)

=

Cn¨ ,m · ϕ¨k

ϕ¨k

=

n ¨ −mk+1

(18.359) (18.360)

˚k+1|k (1 − pD ) · Nk+1|k + (1 − ˚ pD ) · N . (18.361) ˚k+1|k Nk+1|k + N

There are νk+1|k+1 = νk+1|k + mk+1 · νk+1|k Gaussian components for the measurement-updated target PHD and ˚ νk+1|k+1 = ˚ νk+1|k + mk+1 ·˚ νk+1|k NWM components for the measurement-updated clutter PHD. • Measurement update for undetected-target GM components (same as for the BGM-κ-CPHD filter)—for i = 1, ..., ˚ νk+1|k : k+1|k+1

ℓi

k+1|k

=

ℓi

(18.362)

=

1 − pD ˚k+1|k Nk+1|k + N

(18.363)

k+1|k+1

wi

¨ (mk+1 +1) (ϕ¨k ) G k+1|k · k+1|k+1

xi k+1|k+1 Pi

¨ (mk+1 ) (ϕ¨k ) G k+1|k

k+1|k

· wi

k+1|k

=

xi

(18.364)

=

k+1|k Pi .

(18.365)

• Measurement update for detected-target GM components (same as for the BGM-κ-CPHD filter)—for i = 1, ..., νk+1|k and j = 1, ..., mk+1 : k+1|k+1

ℓi,j

k+1|k

=

(18.366)

ℓi k+1|k+1 wi,j

pD · N R

(18.367)

k+1|k T Hk+1 k+1 +Hk+1 Pi

(zj −

k+1|k Hk+1 xi )

= κ ˆ k+1 (zj ) + τk+1 (zj ) k+1|k

k+1|k

xi,j

=

k+1|k Pi,j

=

k+1|k

xi + Kik+1 (zj − Hk+1 xi ( ) k+1|k I − Kik+1 Hk+1 Pi

)

k+1|k

· wi (18.368) (18.369)

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Advances in Statistical Multisource-Multitarget Information Fusion

where k+1|k

Kik+1 = Pi

( )−1 k+1|k T T Hk+1 Hk+1 Pi Hk+1 + Rk+1 .

(18.370)

• Measurement update for undetected-generator NWM components—for l = 1, ..., ˚ νk+1|k : ¨ (mk+1 +1) (ϕk ) · (1 − ˚ pD ) G k+1|k · (mk+1 ) ˚k+1|k ¨ Nk+1|k + N Gk+1|k (ϕk ) k+1|k

=

w ˚l

=

dl

(18.372)

k+1|k oi k+1|k oi k+1|k Oi .

(18.373)

k+1|k+1

w ˚l k+1|k+1

k+1|k

dl k+1|k+1 oi k+1|k+1 oi k+1|k+1 Oi

= = =

(18.371)

(18.374) (18.375)

• Measurement update for detected-generator NWM components—for 1, ..., ˚ νk+1|k and j = 1, ..., mk+1 :

i =

k+1|k k+1|k+1 w ˚i,j

˚ pD · w ˚i

k+1|k+1 oi,j

i

i

k+1|k

=

di

i

(18.376) (18.377)

+1

= ˚ ηk+1 +

k+1|k oi

(18.378)

k+1|k k+1|k+1

oi,j

i

κ ˆ k+1 (zj ) + τk+1 (zj )

k+1|k+1

di,j

· qzj ,dk+1|k ,ok+1|k ,ok+1|k ,Ok+1|k

=

=

˚ ηk+1 · zj + oi

k+1|k

· oi

(18.379)

k+1|k

˚ ηk+1 + oi k+1|k k+1|k+1

Oi,j

k+1|k

=

Oi

+

˚ ηk+1 · oi

(18.380)

k+1|k

˚ ηk+1 + oi k+1|k

·(zj − oi

k+1|k T

)(zj − oi

)

RFS Filters for Unknown Clutter

613

and where qzj ,dk+1|k ,ok+1|k ,ok+1|k ,Ok+1|k i i i i ( ) ) ( k+1|k M/2 k+1|k+1 (˚ ηk+1 · oi ) · ΓM di,j /2

(18.381)

k+1|k dk+1|k /2 i

=

 

·(det Oi

)

( )  k+1|k · ΓM di /2 . k+1|k+1 dk+1|k+1 /2 ·(det Oi,j ) i,j k+1|k+1 M/2

(π · oi,j

)

• Target cardinality distribution: pk+1|k+1 (n) =

n rk+1 ¨ (n) ·G k+1|k+1 (1 − rk+1 ) n!

(18.382)

where rk+1

=

¨ k+1|k+1 (¨ G x)

=

Nk+1|k+1 ˚k+1|k+1 Nk+1|k+1 + N ∑ p¨k+1|k+1 (¨ n) · x ¨n¨

(18.383) (18.384)

n ¨ ≥0 νk+1|k

Nk+1|k+1

=

∑

νk+1|k mk+1 k+1|k+1 wi

+

i=1

∑ ∑ i=1

k+1|k+1

wi,j

(18.385)

j=1

and ˚ νk+1|k

˚k+1|k+1 N

=

∑

k+1|k+1

(18.386)

w ˚l

l=1 ˚ νk+1|k ˚ ν k+1 mk+1

+

∑ ∑ ∑ i=1

18.5.8.5

l=1

k+1|k+1

w ˚i,l,j

.

j=1

NWM-κ-CPHD Filter Merging and Pruning

Merging and pruning is more complicated for NWMs than it is for Gaussian mixtures. In what follows, two possible approaches are considered: a more theoretically justified (and more computationally demanding) one; and an approximate approach along the lines of that proposed in Chen Xin et al. [39].

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Advances in Statistical Multisource-Multitarget Information Fusion

• Exact Merging of NWMs—Merging Criterion. Suppose that we are given two NMW components k|k

´ D1 (´ c, C)

=

w1

´ · N Wdk|k ,ok|k ,ok|k ,Ok|k (´ c, C)

(18.387)

´ · N Wdk|k ,ok|k ,ok|k ,Ok|k (´ c, C)

(18.388)

1

k|k

´ D2 (´ c, C)

=

w2

2

1

2

1

2

1

2

and that we want to determine if they should be merged. This can be accomplished by determining the probability (density) of overlap, ∫

´ N Wdk|k ,ok|k ,ok|k ,Ok|k (´ c, C) 1

=

1

1

(18.389)

1

´ cdC´ c, C)d´ ·N Wdk|k ,ok|k ,ok|k ,Ok|k (´ (2 2 2 M/22 ) (o1 · o2 ) · (det O1 )d1 /2 ·(det O2 )d2 /2 · ΓM (d/2) ( M (M +1)/2 ) 2 · (π · o)M/2 · (det O)d/2 ·ΓM (d1 /2) · ΓM (d2 /2)

(18.390)

where the last equation follows from (G.29) in Appendix G. Thresholding this quantity determines which components should be merged. • Exact Merging of NWMs—Merging Formulas: Suppose that it has been determined that the following NWM k|k ´ + ... + wνk|k · N W k|k k|k k|k k|k (´ ´ w1 · N Wdk|k ,ok|k ,ok|k ,Ok|k (´ c, C) c, C) d ,o ,o ,O 1

1

1

ν

1

ν

ν

ν

should be merged into a single component k|k

w0

´ · N Wdk|k ,ok|k ,ok|k ,Ok|k (´ c, C). 0

0

0

0

Set these two PHDs equal to each other and solve for d0 , o0 , o0 , O0 . Let

w0

=

ν ∑

wi

(18.391)

i=1

w ˆi

=

wi w0

(18.392)

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615

and apply (G.4) through (G.8). Then we get

o0

o0

=

=

(

ν ∑

w ˆi ·

o−1 i

)−1

i=1 ν ∑

w ˆ i · oi

(18.393)

(18.394)

i=1

d0 d0 − M2−1

=

O0

=

( )−1 w ˆi · dl − M2−1 · trOi ((∑ ) ) −1 ν tr ˆl · dl · Ol−1 l=1 w ( ν )−1 ∑ −1 d0 · w ˆ i · di · O i . ∑ν

i=1

(18.395)

(18.396)

i=1

Equation (18.395) results from (G.8), which leads to )−1 ( M −1 w ˆ i · dl − · trOi 2 i=1 ( )−1 M −1 d0 − · trO0 2 (( )−1 ) d0 · tr d0 O0−1 M −1 d0 − 2 ν ∑

= =

(18.397)

(18.398)

and then to (G.5), which finally leads to (18.395). • Approximate Merging of NWMs—Merging Criterion. Let ´ = w k|k · N ´ D1 (´ c, C) c − o1 ) · Wd1 ,O1 (C) ´ −1 (´ 1 (o1 ·C) ´ is (d1 − M − 1) · O −1 . be an NWM component. The mode of Wd1 ,O1 (C) 1 ´ is sufficiently tightly concentrated around its mode. Assume that Wd1 ,O1 (C) According to (G.9) in Appendix G, this occurs if √ d1 − M − 1 · trO1−1 (18.399) is sufficiently small. Then we can approximate ´ ∼ ´ D1 (´ c, C) c − o1 ) · Wd1 ,O1 (C). = w1 · N(o1 ·(d1 −M −1))−1 ·O1 (´

(18.400)

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Advances in Statistical Multisource-Multitarget Information Fusion

Now let ´ = w k|k · N ´ D2 (´ c, C) c − o2 ) · Wd2 ,O2 (C) ´ −1 (´ 2 (o2 ·C)

(18.401)

´ be another tightly concentrated component. Then to determine if D1 (´ c, C) ´ and D2 (´ c, C) should be merged, it is enough to determine if ˜ 1 (´ D c) = N(o1 ·(d1 −M −1))−1 ·O1 (´ c − o1 )

(18.402)

˜ 2 (´ D c) = N(o2 ·(d2 −M −1))−1 ·O2 (´ c − o2 )

(18.403)

and should be merged, using the usual merging criteria for Gaussian components described in Section 9.5.3. • Approximate Merging of NWMs—Merging Formulas. If it is determined that these components should be merged, then the associated merged component is ˜ 0 (´ D c) = w0 · NO0 (´ c − o0 ) (18.404) where w0 w ˆ1

= =

w1 + w2 w1 /w0

(18.405) (18.406)

w ˆ2 o0

= =

w2 /w0 w ˆ 1 · o1 + w ˆ 2 · o2

(18.407) (18.408)

O0

=

w ˆ1 · (o1 · (d1 − M − 1))−1 · O1 +w ˆ2 · (o1 · (d2 − M − 1))−1 · O2 +w ˆ1 · w ˆ2 · (o1 − o2 )(o1 − o2 )T .

(18.409)

´ and D2 (´ ´ are tightly Finally, note that, by assumption, D1 (´ c, C) c, C) concentrated at their respective matrix modes (o1 · (d1 − M − 1))−1 · O1 and (o2 · (d2 − M − 1))−1 · O2 . But—since they are to be merged—they must have nearly identical modes. Thus the merged component can be taken to have the form ´ =D ˜ 0 (C) ´ · Wd ,O (C) ´ ∼ ˜ 0 (C) ´ · Wd ,O (C). ´ D0 (´ c, C) =D 1 1 2 2

(18.410)

RFS Filters for Unknown Clutter

18.6

617

MULTISENSOR κ-CPHD FILTERS

The λ-CPHD and κ-CPHD filters are single-sensor filters. However, they can be extended to the multisensor case using the techniques described in Chapter 10. 18.6.1

Iterated-Corrector κ-CPHD Filter

The easiest approach is the iterated-corrector method of Section 10.5. In this case, one simply repeats the measurement-update for the λ-CPHD or κ-CPHD filter, once for each sensor. Of course, this approach inherits the limitations of the iteratedcorrector approach. 18.6.2

Parallel-Combination κ-CPHD Filter

One can also apply the approximate parallel-combination approach of Section 10.6. The purpose of this section is to briefly illustrate how this approach is applied to the κ-CPHD filter. Let the augmented target-clutter state space be 1

s

¨ = X ⊎˚ X C ⊎ ... ⊎ ˚ C

(18.411)

j j

j

j

j

where ˚ C = [0, 1] × C is the space of clutter generators ˚ c = (c, c) for the jth sensor. The integral on this space is defined as ∫ ∫ ∫ ∫ 1 1 s s f¨(¨ x)d¨ x= f¨(x)dx + 1 f¨(˚ c)d˚ c + ... + s f¨(˚ c)d˚ c (18.412) ˚ C

X

where

∫

j j

j

f¨(˚ c)d˚ c=

˚ C

∫ ∫

˚ C

1 j j j j f¨(c, c)dcdc.

(18.413)

0

The PCAM-κ-CPHD filter has the form ... →

p¨k|k (¨ n)

→

p¨k+1|k (¨ n)

... →

sk|k (x)

→

sk+1|k (x)

1

1 1

1

... → ˚ sk|k (c, c) .. . s s

p¨k+1|k+1 (¨ n)

→ ...

sk+1|k+1 (x)

→ ...

1 1

1

→ ˚ sk+1|k (c, c) .. . s

s

... → ˚ sk|k (c, c)

→ ↑↓ → ↑↓

1

1

→ ˚ sk+1|k+1 (c, c) .. .. . .

→ ...

s s

s

→ ˚ sk+1|k (c, c)

s

s

→ ˚ sk+1|k+1 (c, c)

→ ...

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Advances in Statistical Multisource-Multitarget Information Fusion

Here, the top filter propagates the cardinality distribution p¨k|k (¨ n) on the number 1

j

s

n ¨ = n + n + ... + n, where n is the number of targets and n is the number of clutter generators for the jth sensor. The second filter propagates the spatial distribution sk|k (x) on targets; and the following rows are filters that propagate j j

j

the clutter-generator distributions ˚ sk|k (c, c) of the sensors. Suppose that at time tk the s sensors collect the respective measurement 1

j

s

j

sets Z k+1 , ..., Z k+1 with |Z k+1 | = m. Then we are to construct the joint targetclutter spatial distribution s¨k+1|k+1 (¨ x) and joint target-clutter cardinality distri1

s

bution p¨k+1|k+1 (¨ n), updated using all of the measurement sets Z k+1 , ..., Z k+1 . Given this, the measurement-updated spatial distribution and cardinality distribution are direct analogs of (10.94) through (10.103). Let:

=

∫

(1 − pD (¨ x)) · s¨k+1|k (¨ x)d¨ x

=

∫

pD (¨ x) · Lj (¨ x) · s¨k+1|k (¨ x)d¨ x

j

ϕ¨k+1 j

j

τ¨k+1 (z)

j

(18.414)

j j

(18.415)

z

and s¨k+1|k+1 (¨ x)

=

p¨k+1|k+1 (¨ n)

=

1 ¨ 1 ·L (¨ x) · s¨k+1|k (¨ x) (18.416) s ¨k+1|k+1 Z k+1 ,...,Z k+1 N n ¨ p˜(¨ n) · θ¨k+1 (18.417) ˜ θ¨k+1 ) G(

RFS Filters for Unknown Clutter

619

where 1

¨ L

1

(¨ x)

=

k+1|k+1

=

s

Z k+1 ,...,Z k+1 1...s

¨ N

s

¨1 L

¨s (¨ x) · · · L

(¨ x) Z k+1 G (θ¨k+1 ) Z · 1 k+1 (18.418) s ˜ θ¨k+1 ) G( ¨ k+1|k+1 · · · N ¨ k+1|k+1 N ∫ 1 ¨ 1 (¨ L x) (18.419) ˜ (1)

Z k+1 s

¨s ···L

Z k+1

(¨ x) · s¨k+1|k (¨ x)d¨ x

1...s

θ¨k+1

¨ N

k+1|k+1

(18.420)

= 1

s

¨ k+1|k+1 · · · N ¨ k+1|k+1 N ¨k+1|k+1 N

=

p˜k+1|k+1 (¨ n)

=

˜ k+1|k+1 (¨ G x)

=

¨ ˜ (1) G k+1|k+1 (θk+1 ) ¨ ·θ ˜ k+1|k+1 (θ¨k+1 ) k+1 G 1

(18.421)

s

ℓ¨z1 (n) · · · ℓ¨zs (¨ n) · p¨k+1|k (¨ n) ∑ p˜k+1|k+1 (¨ n) · x ¨n¨

(18.422) (18.423)

n ¨ ≥0

and where j

j

ℓ¨j

(¨ n)

=

C

j

n ¨ ,m

Z k+1

¨ −m · ϕ¨nk+1

(18.424) j

j

¨j L

(¨ x)

=

Z k+1

j ¨ (m+1) (ϕ¨k+1 ) 1 − p¨D (¨ x) G k+1|k · j ¨ Nk+1|k ¨ (m) (ϕ¨k+1 ) G k+1|k

(18.425)

j

j

x) · Lj (¨ x) ∑ p¨D (¨ z

+

j

j

j

j

τ¨k+1 (z)

z∈Z k+1 j j

¨ k+1|k+1 N

=

j

¨ (m+1) (ϕ¨k+1 ) ϕ¨k+1 G j k+1|k · + m. j ¨ Nk+1|k ¨ (m) ¨ G (ϕk+1 ) k+1|k

(18.426)

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Advances in Statistical Multisource-Multitarget Information Fusion

The multisensor PCAM-κ-CPHD filter results when we define PHDs and spatial distributions for the targets and for each of the clutter spaces: Dk+1|k (x)

=

Nk+1|k

=

sk+1|k (x)

=

Dk+1|k+1 (x)

=

Nk+1|k+1

=

sk+1|k+1 (x)

=

¨ k+1|k (x) D ∫ Dk+1|k (x)dx

(18.427) (18.428)

Dk+1|k (x) Nk+1|k ¨ Dk+1|k+1 (x) ∫ Dk+1|k+1 (x)dx

(18.430)

Dk+1|k+1 (x) Nk+1|k+1

(18.432)

(18.429)

(18.431)

and j j

˚k+1|k (˚ D c)

j

=

j

˚k+1|k N

=

¨ k+1|k (˚ D c), ∫ j j j ˚k+1|k (˚ D c)d˚ c

(18.433) (18.434)

j j j

˚k+1|k (˚ D c)

j

˚ sk+1|k (˚ c)

(18.435)

= j

˚k+1|k N

j j

˚k+1|k+1 (˚ D c)

j

=

j

˚k+1|k+1 N

=

¨ k+1|k+1 (˚ D c), ∫ j j j ˚k+1|k+1 (˚ D c)d˚ c

(18.436) (18.437)

j j j

˚k+1|k+1 (˚ D c)

j

˚ sk+1|k+1 (˚ c)

=

.

(18.438)

j

˚k+1|k+1 N Given this, the conventional PCAM-CPHD formulas are rewritten so that they are expressed in terms of these PHDs and/or spatial distributions. The details will not be considered here.

RFS Filters for Unknown Clutter

621

For the purpose of multitarget state estimation, the cardinality distribution on targets is given by the obvious analog of (18.62): pk+1|k+1 (n)

=

rk+1

=

n rk+1 ¨ (n) ·G k+1|k+1 (1 − rk+1 ) n! Nk+1|k+1

(18.439)

1

. (18.440)

s

˚k+1|k+1 + ... + N ˚k+1|k+1 Nk+1|k+1 + N The estimated clutter intensity function for the jth sensor is the analog of (18.81): j j

κ ˆ k+1 (z) =

∫ ∫

1

j j j j

j

j

j

j

j

˚k+1|k+1 (c, c)dcdc c · f κk+1 (z|c) · D

(18.441)

0

with estimated clutter rate j

ˆ k+1 = λ

∫ ∫

1

j j j j j j ˚k+1|k+1 (c, c·D c)dcdc.

(18.442)

0

The estimated clutter cardinality distribution for the jth sensor is the analog of (18.93): j

˜m λ k+1|k

j

pˆk+1 (m)

= m!

j

˜ k+1|k ) ¨ (m) (1 − λ ·G k+1|k

(18.443)

j j

˜ k+1 λ

ˆ k+1 λ =

. 1

s

(18.444)

˚k+1|k+1 + ... + N ˚k+1|k+1 Nk+1|k+1 + N

18.7

THE κ-CBMEMBER FILTER

Using the techniques of this and the previous chapter, Vo, Vo, Hoseinnezhad, and Mahler have generalized the CBMeMBer filter (Chapter 13) to situations in which the clutter background and detection profile are not known [312], [313]. As with background-agnostic CPHD filters, the background-agnosic CBMeMBer filters are constructed directly from the CBMeMBer filter by:

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Advances in Statistical Multisource-Multitarget Information Fusion

• Modeling unknown probability of detection using a new state (a, x) with conventional state x and unknown probability of detection 0 ≤ a ≤ 1. • Modeling clutter as clutter generators of the form (c, c) with clutter probability of detection c and clutter likelihood function ˚ Lz (c). The first approach is straightforward: in the formulas for the CBMeMBer ∫ ∫1 filter, one simply substitutes ‘a’ wherever ‘pD (x)’ occurs, and ‘ 0 ·dadx’ ∫ wherever ‘ ·dx’ occurs. Thus in what follows, only the second approach will be described. It results in the “κ-CBMeMBer filter.” As with the CBMeMBer filter, it is expected that the κ-CBMeMBer filter will be most effective when it is implemented using particle methods, for applications involving significant motion and/or measurement nonlinearity. As with the κ-CPHD filter, the κ-CBMeMBer filter results when one replaces the conventional state x with a state x ¨, which can take two forms: x ¨ = x or x ¨ = (c, c), with associated measurement and motion models p¨D (¨ x), ¨ z (¨ ¨ x¨ (¨ L x) = fk+1 (z|¨ x), p¨S (¨ x), and M x′ ) = f¨k+1 (¨ x|¨ x′ ). The section is organized as follows: 1. Section 18.7.1: Modeling assumptions for the κ-CBMeMBer filter. 2. Section 18.7.2: Time update equations for the κ-CBMeMBer filter. 3. Section 18.7.3: Measurement update equations for the κ-CBMeMBer filter. 4. Section 18.7.4: State estimation for the κ-CBMeMBer filter. 5. Section 18.7.5: Clutter estimation for the κ-CBMeMBer filter. 18.7.1

κ-CBMeMBer Filter: Modeling

The motion and measurement models for this filter are essentially the same as those for the κ-CPHD filter (Section 18.5.1): • Targets can transition only to targets, and clutter generators only to clutter generators. • Target probability of survival: pS (x) abbr. = pS,k+1 (x). • Target Markov density: Mx (x′ ) = fk+1|k (x|x′ ). • Clutter-generator probability of survival: ˚ pS (c) abbr. = pS,k+1 (c).

RFS Filters for Unknown Clutter

623

κ • Clutter-generator Markov density: Mcκ (c′ ) = fk+1|k (c|c′ ). κ • Markov density for clutter probability of detection: Mcκ (c′ ) = fk+1|k (c|c′ ), defined as in (17.12) through (17.16).

• Target probability of detection: pD (x) abbr. = pD,k+1 (x). • Target likelihood function: Lz (x) abbr. = fk+1 (z|x). κ • Clutter-generator likelihood function: Lκz (c) abbr. = fk+1 (z|c).

• Since clutter is modeled using clutter generators, the a priori clutter intensity has value κk+1 (z) = 0. Given this, the κ-CBMeMBer filter is constructed as follows. At time tk and for i = 1, ..., ν¨k|k , the filter consists of (1) a list of joint target-clutter track distributions s¨ik|k (¨ x) where x ¨ = x or x ¨ = (c, c); (2) joint target-clutter i probabilities of existence q¨k|k ; and (3) joint target-clutter track labels ℓ¨ik|k . Define the density functions sik|k (x)

=

s¨ik|k (x)

(18.445)

˚ sik|k (c, c)

=

s¨ik|k (c, c)

(18.446)

where it must be the case that ∫ i ˚i 1 = s¨ik|k (¨ x)d¨ x = Nk|k +N k|k

(18.447)

where i Nk|k =

∫

sik|k (x)dx,

˚i = N k|k

∫ ∫

1

˚ sik|k (c, c)dcdc.

(18.448)

0 ν ¨

k|k Thus the joint probability distributions s¨1k|k (¨ x),...,¨ sk|k (¨ x) can be equivalently

ν ¨

k|k replaced by the target densities s1k|k (x),...,sk|k (x)

and the clutter densities

ν ¨k|k ˚ s1k|k (c, c),...,˚ sk|k (c, c).

Note that (18.447) can be satisfied when sik|k (x) = 0 identically or when ˚ sik|k (c, c) = 0 identically. Also, in general the labels ℓ¨ik|k have the form ℓ¨ik|k = (ℓik|k , ˚ ℓik|k ) where ℓik|k are target track labels and ˚ ℓik|k are clutter-generator track labels. Since it is unnecessary to propagate labels for the clutter generators,

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Advances in Statistical Multisource-Multitarget Information Fusion

we can eliminate the ˚ ℓik|k and propagate only the ℓik|k . Thus a multi-Bernoulli system will have the general form ν ¨

k|k i {ℓik|k , q¨k|k , sik|k (x),˚ sik|k (c, c)}i=1 .

18.7.2

κ-CBMeMBer Filter: Time Update

We are given the prior multi-Bernoulli system ν ¨k|k i T¨k|k = {ℓik|k , q¨k|k , sik|k (x),˚ sik|k (c, c)}i=1 .

(18.449)

We are to determine the time-updated multi-Bernoulli system ν ¨k+1|k i T¨k+1|k = {ℓik+1|k , q¨k+1|k , sik+1|k (x),˚ sik+1|k (c, c)}i=1 .

(18.450)

This has the form persist birth T¨k+1|k = T¨k+1|k ∪ T¨k+1|k

(18.451)

where ν ¨

persist T¨k+1|k

=

k|k {(ℓi , q¨i , si (x),˚ sB i (c, c))}i=1

birth T¨k+1|k

=

bk {(ℓB ¨iB , sB sB i ,q i (x),˚ i (c, c))}i=1

(18.452)

¨

(18.453)

and where the persisting-track components are given by ℓi

=

q¨i

=

si (x)

=

˚ sB i (c, c)

=

sik|k [pS ]

=

˚ sik|k [˚ pS ]

=

ℓik|k

(18.454) (

i q¨k|k · sik|k [pS ] + ˚ sik|k [˚ pS ]

)

(18.455)

sik|k [pS Mx ] sik|k [pS ] + ˚ sik|k [˚ pS ]

(18.456)

˚(c,c) ] ˚ sik|k [˚ pS M sik|k [pS ] + ˚ sik|k [˚ pS ]

(18.457)

where ∫

pS (x) · sik|k (x)dx ∫ ∫ 1 ˚ pS (c) · ˚ sik|k (c, c)dcdc. 0

(18.458) (18.459)

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18.7.3

625

κ-CBMeMBer Filter: Measurement Update

We are given the predicted multi-Bernoulli system ν ¨k+1|k i T¨k+1|k = {ℓik+1|k , q¨k+1|k , sik+1|k (x),˚ sik+1|k (c, c)}i=1 .

(18.460)

Suppose that a new measurement set Zk+1 = {z1 , ..., zmk+1 } is collected with |Zk+1 | = mk+1 . We are to determine the form of the measurement-updated multiBernoulli system ν ¨k+1|k+1 i T¨k+1|k+1 = {ℓik+1|k+1 , q¨k+1|k+1 , sik+1|k+1 (x),˚ sik+1|k+1 (c, c)}i=1 . (18.461) This has the form legacy meas T¨k+1|k+1 = T¨k+1|k+1 ∪ T¨k+1|k+1 (18.462)

where legacy T¨k+1|k+1 meas T¨k+1|k+1

ν ¨

=

k+1|k {(ℓ¨L ¨iL , sL sL i ,q i (x),˚ i (c, x))}i=1

(18.463)

=

mk+1 {(ℓ¨U ¨jU , sU sU j ,q j (x),˚ j (c, c))}j=1 .

(18.464)

The measurement-update equations for the legacy components are: ℓL i

=

ℓik+1|k

(18.465) (

i q¨k+1|k · 1 − sik+1|k [pD ] − ˚ sik+1|k [˚ pD ]

q¨iL

=

s¨L i (x)

=

˚ sL i (c, c)

=

)

i i ·˚ sik+1|k [˚ pD ] 1 − q¨k+1|k · sik+1|k [sD ] − q¨k+1|k

s¨ik+1|k (x) ·

1 − pD (x) i sk+1|k [pD ] − ˚ sik+1|k [˚ pD ]

(18.467)

1−c 1 − sik+1|k [pD ] − ˚ sik+1|k [˚ pD ]

(18.468)

1−

s¨ik+1|k (c, c) ·

(18.466)

where sik+1|k [pD ]

=

˚ sik+1|k [˚ pD ]

=

∫

pS (x) · sik+1|k (x)dx ∫ ∫ 1 c ·˚ sik+1|k (c, c)dcdc. 0

(18.469) (18.470)

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The measurement-update equations for the measurement-updated components are: ∗ ℓU (18.471) j = ℓk+1|k and q¨jU ∑ν¨k+1|k i=1

(18.472) i i ˚z q¨k+1|k (1−¨ qk+1|k )· sik+1|k [pD Lzj ]+˚ sik+1|k [˚ pD L j i i i (1−¨ qk+1|k ·(sk+1|k [pD ]+˚ sk+1|k [˚ pD ))2

(

= κk+1 (zj ) +

∑ν¨k+1|k i=1

)

( ) i ˚z q¨k+1|k · sik+1|k [pD Lzj ]+˚ sik+1|k [˚ pD L j ( ) i 1−¨ qk+1|k · sik+1|k [pD ]+˚ sik+1|k [˚ pD ]

and sU j (x) ∑ν¨k+1|k i=1

=

∑ν¨k+1|k i=1

(18.473) i q¨k+1|k i 1−¨ qk+1|k

i q¨k+1|k i 1−¨ qk+1|k

˚ sU j (c, c) ∑ν¨k+1|k i=1

=

∑ν¨k+1|k i=1

· sik+1|k (x) · pD (x) · Lzj (x)

( ) · sik+1|k [pD Lzj ] + ˚ sik+1|k [˚ pD ˚ L zj ] (18.474)

i q¨k+1|k i 1−¨ qk+1|k

i q¨k+1|k i 1−¨ qk+1|k

·˚ sik+1|k (c, c) · c · ˚ Lzj (c)

( ) · sik+1|k [pD Lzj ] + ˚ sik+1|k [˚ pD ˚ L zj ]

where sik+1|k [pD Lzj ]

=

˚ sik+1|k [˚ pD ˚ L zj ]

=

∫

pD (x) · Lzj (x) · sik+1|k (x)dx ∫ ∫ 1 c·˚ Lzj (c) · ˚ sik+1|k (c, c)dcdc.

(18.475) (18.476)

0

Also, ℓ¨∗j,k+1|k is the label of the track that has the largest contribution to the probability of existence q¨jU in (18.472).

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18.7.4

627

κ-CBMeMBer Filter: Multitarget State Estimation

Multitarget state estimation can be accomplished as follows. First, some notation must be established. The existence probability of the ith legacy target is

q¨iL =

( ) i q¨k+1|k · 1 − sik+1|k [pD ] − ˚ sik+1|k [˚ pD ] (18.477)

i i 1 − q¨k+1|k · sik+1|k [sD ] − q¨k+1|k ·˚ sik+1|k [˚ pD ]

and the existence probability of the ith measurement-updated target with measurement zj is ∑ν¨k+1|k

i i ˚z ) q¨k+1|k (1−¨ qk+1|k )·(sik+1|k [pD Lzj ]+˚ sik+1|k [˚ pD L j

i=1

q¨jU =

κk+1 (zj ) +

i (1−¨ qk+1|k ·(sik+1|k [pD ]+˚ sik+1|k [˚ pD ))2 ) ( i ˚z · sik+1|k [pD Lzj ]+˚ sik+1|k [˚ pD L ∑ν¨k+1|k q¨k+1|k ( )j i=1 1−¨ qi · si [p ]+˚ si [˚ p ] k+1|k

k+1|k

D

k+1|k

.

(18.478)

D

Given this, define the probabilities of existence of the legacy tracks to be qiL = sik+1|k [1] · q¨iL ;

(18.479)

and the probabilities of existence of the updated tracks to be qjU = sik+1|k [1] · q¨jU .

(18.480)

Then ν ¨k+1|k

Nk+1|k+1 =

∑ i=1

ν ¨k+1|k mk+1

qiL

+

∑ ∑ i=1

U qi,j

(18.481)

j=1

is an estimate of the total expected number of targets. Second, round Nk+1|k+1 off to the nearest integer, ν. Third, find those ν target densities sk+1|k+1 (x) i U with largest existence probabilities qk+1|k+1 or qi,j . Fourth, determine the MAP estimate of each such sk+1|k+1 (x) or, alternatively, the mean of each such sk+1|k+1 (x)/sk+1|k+1 [1]. 18.7.5

κ-CBMeMBer Filter: Clutter Estimation

Let i i ˚ qk+1|k =˚ sk+1|k [1] · q¨k+1|k

(18.482)

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be the probability of existence of the ith clutter component. Then the clutter intensity function can be heuristically estimated as ν ¨k+1|k+1

κ ˆ k+1 (z)

=

∑

i ˚ qk+1|k

∫ ∫

c · Lκz (c) · 0

i=1 ν ¨k+1|k+1

=

1

∑

i q¨k+1|k

∫ ∫

˚ sik+1|k (c, x) dcdc (18.483) ˚ sk+1|k [1]

1

c · Lκz (c) · ˚ sik+1|k (c, x)dcdc (18.484) 0

i=1

with associated clutter rate ν ¨k+1|k+1

ˆ k+1 = λ

∑

i q¨k+1|k

∫ ∫

1

c ·˚ sik+1|k (c, x)dcdc

(18.485)

0

i=1

and associated clutter cardinality distribution pκk+1 (m) =

˜m λ k+1 ˜ k+1 ) ¨ (m) (1 − λ ·G k+1|k m!

(18.486)

where ˆ k+1 λ

˜ k+1 λ

=

¨ k+1|k (z) G

=

¨k+1|k+1 N

ˆ k+1 λ = ∑νk+1|k k+1|k q¨i i=1

ν ¨k+1|k

∏ (

k+1|k

1 − q¨i

k+1|k

+ z · q¨i

(18.487) )

.

(18.488)

i=1

18.8

IMPLEMENTED CLUTTER-AGNOSTIC RFS FILTERS

Four such implementations are described: two for the λ-CPHD filter (Section 18.8.1 and Section 18.8.2), one for the κ-CBMeMBer filter (Section 18.8.3), and one for a normal-Wishart mixture (NWM) implementation of the κ-PHD filter. 18.8.1

Implemented λ-CPHD Filter

Mahler, Vo, and Vo have reported a Gaussian mixture implementation of the λCPHD filter [195], [194], as well as performance results using simulated data. These simulations were of two types:

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Scenario 1: up to 12 appearing and disappearing targets, following linear trajectories and observed by a linear-Gaussian sensor with clutter rate 50. Implementation type: beta-Gaussian mixture approach with EKF. Scenario 2: up to 10 appearing and disappearing targets, following curvilinear trajectories and observed by a range-bearing sensor with clutter rate 10. Implementation type: beta-Gaussian mixture approach with UKF. The results were as follows: Scenario 1: The λ-CPHD filter had good tracking performance, better than that of a conventional PHD filter but not as good as a conventional CPHD filter. The clutter rate was successfully estimated to be 50. Scenario 2: The λ-CPHD filter had good tracking performance, though it experienced some difficulty when targets became closely spaced. The clutter rate was successfully estimated to be 10. 18.8.2

“Bootstrap” λ-CPHD Filter

In [18], Beard, Vo, and Vo reported certain limitations of the λ-CPHD filter, and devised a heuristic means of correcting them. They noted that, on average, the performance of the λ-CPHD filter is significantly worse than that of a “matched” CPHD filter (that is, one that has been given the correct clutter rate). They attributed this to the fact that the number of actual targets cannot be estimated from the λCPHD filter’s cardinality distribution (which, recall, is a distribution on the sum of the number of actual and clutter targets).2 As a remedy, Beard et al. proposed a simple “bootstrap” procedure consisting of two parallel λ-CPHD filters operating in two stages: 1. Stage 1: The first λ-CPHD filter is used to estimate the clutter rate λ. 2. Stage 2: the second λ-CPHD filter uses this estimate to detect and track the actual targets. This approach is somewhat questionable from a theoretical point of view, since the algorithm double-counts the measurements. Nevertheless, Beard et al. reported that this approach proved to be surprisingly successful: the bootstrap λCPHD filter performed nearly as well as the matched CPHD filter. The bootstrap λ-CPHD filter employed the uniformly-distributed target-birth model of Beard, Vo, Vo, and Arulampalam [16] (described in Section 9.5.7). Beard 2

Performance might be improved by using the target cardinality distribution to estimate target number. The formula for this distribution was unknown at the time of writing of [18].

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Advances in Statistical Multisource-Multitarget Information Fusion

et al. applied a Gaussian mixture implementation of it in a scenario in which a single bearing-only sensor with pD = 0.95 is carried on a platform moving along a sinusoidal trajectory. The sensor observes five appearing, disappearing, and slowly maneuvering targets, various amounts of uniformly distributed (in the bearing variable) Poisson clutter. Two simulations were considered, one with a constant clutter rate of λ = 30 and the other with a variable clutter rate, increasing from λ = 20 to λ = 40 during the middle third of the scenario. In the first scenario, the OSPA tracking performance of the bootstrap λ-CPHD filter was as good as that of the matched CPHD filter. It also estimated the (constant) clutter rate with good accuracy. In the second scenario, the tracking performance of the bootstrap λ-CPHD filter was once again as good as that of the matched CPHD filter. The bootstrap filter also successfully estimated the variable clutter rate, but exhibited a slight (8%) upward bias in the clutter estimate during the first third of the scenario, and a slight (5%) downward bias during the final third. 18.8.3

Implemented λ-CBMeMBer Filter

Vo, Vo, Hoseinnazhad, and Mahler have reported simulation results for an implementation of the λ-CBMeMBer filter [312], [313]. This implementation, previously mentioned in Section 17.7, was also pD -agnostic, in that target probability of detection a was also assumed unknown. In this implementation, the velocities of clutter generators were ignored, so that their states had the form (c, x, y). It was assumed that the generators followed a random walk on the coordinates (x, y) with a linear-Gaussian Markov density ˚c (c′ ). Also, the clutter probability of detection c was assumed to be constant M and known. In the simulation, up to 10 appearing and disappearing targets followed curvilinear trajectories and were observed by a range-bearing sensor. The number of clutter measurements was binomially distributed with clutter rate of 10 returns per scan. The clutter was spatially distributed so that it was increasingly concentrated near the origin of the scenario, with clutter density decreasing radially from the origin. The actual targets followed a coordinated-turn model with turn rate ω, and thus had states of the form (x, y, vx , vy , ω) where ω is the turn rate. The Markov transition for the target probability of detection was chosen to be a suitable beta distribution. The authors reported acceptable tracking performance, with accurate localization and correct track initiation and termination (with some delay).

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18.8.4

631

Implemented NWM-PHD Filter

This is the PHD filter special case of the Normal-Wishart mixture (NWM) CPHD filter discussed in Section 18.5.8, and is due to Chen Xin, Kirubarajan, Tharmarasa, and Pelletier in 2009 [39], [37]. These authors also proposed the integration of a special case—a clutter estimator based on Wishart mixtures—into conventional filters such as the MHT [38]. Chen et al. tested their NWM-κ-PHD filter in two-dimensional simulations, one with a linear-Gaussian sensor and the other with a bearing-only sensor. In both cases, three appearing and disappearing targets move in time-varying, spatially nonhomogeneous clutter. The clutter process consists of several dense-clutter subregions, of which there are two types. For the first type, the clutter spatial distribution is uniform within an L-shaped region with clutter rates of either 10 or 18. For the second type, the spatial distribution is Gaussian with a clutter rate 9.6. Outside of these dense-clutter areas, clutter is uniform with small clutter rate. In all cases, the probability of detection is 0.96. For the simulation involving the linear-Gaussian sensor, the authors reported that “the performance of the [NWM-κ-PHD filter] is comparable to the performance obtained when the clutter’s true spatial distribution is perfectly known” ([39], p. 1227). For the bearing-only sensor, they reported that “the performance of the [NWM-κ-PHD filter] is comparable to that obtained when the clutter’s true spatial distribution is known...[but] brings an approximately two-scan delay for new target initialization” ([39], p. 1227). The authors attributed this delay to the fact that targets are more difficult to distinguish from clutter because of the low observability of the bearing-only measurement model.

18.9

CLUTTER-AGNOSTIC PSEUDOFILTERS

Let us begin by reviewing what has been accomplished in this chapter so far. In Section 18.2, a modeling approach for “clutter agnostic” RFS filters was described. It was based on a Bernoulli clutter-generator model and the joint target-clutter state space ¨ = X ⊎˚ X C. (18.489) Then, in Section 18.3, the time- and measurement-update equations for a general CPHD filter based on this general multi-Bernoulli model were described. Two possible special cases of the joint target/clutter motion model were pointed out:

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Advances in Statistical Multisource-Multitarget Information Fusion

1. A phenomenology-nonintermixing motion model, in which targets can transition only to targets and clutter generators only to clutter generators (Section 18.2.2). 2. A problematic phenomenology-intermixing motion model, in which clutter generators can transition to targets; and/or targets can transition to clutter generators; and in which the latter is interpreted as a model for target disappearance and the former is interpreted as a model for target appearance (Section 18.2.3). Finally, Sections 18.4 and 18.5 were devoted to clutter-agnostic CPHD filters based on the nonintermixing model: a λ-CPHD filter that can estimate the clutter cardinality distribution pκk+1 (m), and a κ-CPHD filter that can estimate both the clutter intensity function κk+1 (z) and the clutter cardinality distribution. But what if we had instead derived these filters assuming the phenomenologyintermixing motion model? The purpose of this section is to answer this question. For conceptual and notational clarity, we will concentrate on the PHD filter special case of the λ-CPHD filter. As we shall see, it—and therefore the intermixing-model analogs of the λ-CPHD and κ-CPHD filters—exhibit serious pathologies and are therefore described as “pseudofilters.” The section is organized as follows: 1. Section18.9.1: The λ-PHD pseudofilter (the intermixing model PHD pseudofilter). 2. Section 18.9.2: Pathological behavior of the λ-PHD pseudofilter. 18.9.1

The λ-PHD Pseudofilter

The purpose of this section is to present the time-update and measurement-update equations for the λ-PHD pseudofilter. 18.9.1.1

λ-PHD Pseudofilter Time Update Equations

The time-update equations for the general Bernoulli clutter-generator CPHD filter were given in (18.36) through (18.43). As with the motion models for the λ-PHD filter (Section 18.4.6), assume that: • Since target-to-clutter transitions model target disappearance, the target probability of survival is redundant and is therefore unity: pS (x′ ) = 1;

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• Since clutter-to-target transitions model target appearance, the target birth PHD is redundant and therefore vanishes: bk+1|k (x) = 0. • Since target-to-clutter transitions also model clutter-generator appearance, the clutter-generator birth PHD is redundant and thus vanishes, ˚ bk+1|k (c) = 0. • The probability that clutter generators will transition to clutter generators is constant: ˚ pT (c′ ) = ˚ pT . • The transition density from generators to targets does not depend on the generator state: ⇒ fk+1|k (x|c′ ) = sB (18.490) k+1|k (x), where sB k+1|k (x) is the spatial distribution of the appearing targets, assumed to be known a priori. Given this, (18.36) through (18.43) reduce to: Dk+1|k (x)

˚k|k · sB (1 − ˚ pT ) · N k+1|k (x) ∫ + pT (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′

=

(18.491)

and ˚k+1|k (c) D

=

∫

⇐ (1 − pT (x′ )) · fk+1|k (c|x′ ) · Dk|k (x′ )dx′ (18.492) ∫ ˚k|k (c′ )dc′ . +˚ pT f˚k+1|k (c|c′ ) · D

Integrating both sides of (18.492) results in: • λ-PHD pseudofilter time-update for number of clutter generators: ∫ ˚k+1|k = ˚ ˚k|k + (1 − pT (x′ )) · Dk|k (x′ )dx′ . N pT N (18.493) We also rewrite (18.491) as: • λ-PHD pseudofilter time-update for target PHD: ∫ ˆ Dk+1|k (x) = bk+1|k (x) + pT (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′ (18.494)

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Advances in Statistical Multisource-Multitarget Information Fusion

where ˆbk+1|k (x) = (1 − ˚ ˚k|k · sB pT ) · N k+1|k (x)

(18.495)

is interpreted as the estimated target-birth PHD, with sB k+1|k (x) being known ˚k|k being interpreted as the estimated target-birth a priori and (1 − ˚ pT ) · N rate. 18.9.1.2

λ-PHD Pseudofilter Measurement Update Equations

The measurement-update equations for the general Bernoulli clutter-generator CPHD filter were given in (18.44) through (18.53). As with the measurement models for the λ-PHD filter (Section 18.4.6), assume that: • The predicted joint target/clutter process is Poisson: ¨ k+1|k (x) = eN¨k+1|k ·(x−1) . G

(18.496)

• The clutter-generator probability of detection is known and constant: ˚ pκD (c) = ˚ pD . • The clutter spatial distribution ck+1 (z) is known, and the clutter-generator likelihood function is state-independent: ˚ Lκz (c) = ck+1 (z). Given this, (18.53) reduces to ˚k+1|k+1 (c) ∑ D ˚ pD · ck+1 (z) = 1−˚ pD + ˚ ˆ Dk+1|k (c) z∈Zk+1 λk+1 ck+1 (z) + τk+1 (z)

(18.497)

and (18.52) reduces to: • λ-PHD pseudofilter measurement-update for target PHD: ∑ Dk+1|k+1 (x) pD (x) · Lz (x) = 1 − pD (x) + ˆ Dk+1|k (x) z∈Zk+1 λk+1 ck+1 (z) + τk+1 (z)

(18.498)

where ˆ k+1 λ τk+1 (z)

˚k+1|k = ˚ pD N ∫ = pD (x) · Lz (x) · Dk+1|k (x)dx.

(18.499) (18.500)

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Since the right side of (18.497) does not involve c, we can integrated this variable out and replace it with: • λ-PHD pseudofilter measurement-update for number of clutter generators: ˚k+1|k+1 ∑ N ˚ pD · ck+1 (z) = 1−˚ pD + . ˚ ˆ Nk+1|k z∈Zk+1 λk+1 ck+1 (z) + τk+1 (z)

(18.501)

Equations (18.498) and (18.501) are identical to the measurement-update equations for the λ-PHD filter, (18.132) and (18.133). 18.9.2

Pathological Behavior of the λ-PHD Pseudofilter

The λ-PHD pseudofilter exhibits the following behaviors [153]: 1. The λ-PHD pseudofilter cannot always estimate the target-birth rate. From ˚k|k . Suppose, (18.495), the claimed estimated birth rate is (1 − ˚ pT ) · N however, that there is no clutter and therefore that there can be no clutter ˚k|k = 0 and so the estimated target-birth rate is generators. Then N incorrectly estimated as 0, regardless of its actual value. This behavior should be contrasted to that of the classical PHD filter. Even when there is no clutter, the time-update equation for the classical PHD filter is ∫ Dk+1|k (x) = bk+1|k (x) + pS (x′ ) · fk+1|k (x|x′ ) · Dk|k (x′ )dx′ (18.502) and so this filter still has a target-birth model. 2. The λ-PHD pseudofilter cannot always estimate the clutter rate. This can be demonstrated using a simple analytical counterexample. Assume (a) that pD (x) = 1 and ˚ pD = 1 (that is,both targets and clutter generators are perfectly detected); (b) pT (x) = pT is constant (that is, targets transition to targets with constant probability); and (c) the target-to-target probability pT and the generator-to-generator probability ˚ pT are “conjugate” in the sense that, at any given time tk , they satisfy the relationship pT + ˚ pT = 1.

(18.503)

Then the λ-PHD pseudofilter’s estimate of the clutter rate is always a fixed fraction ˚ pT of the current number of measurements, regardless of what the

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Advances in Statistical Multisource-Multitarget Information Fusion

clutter rate actually is: ˆ k+1 = ˚ λ pT · mk+1 .

(18.504)

This equation is proved as follows. From (18.498) and (18.493) it follows ˚k|k = mk+1 . But from (18.493) and the assumption ˚ that Nk|k + N pD = 1, we know that the estimated clutter rate is: ˆ k+1 λ

˚k+1|k = ˚ pD · N ∫ ˚k|k + (1 − pT (x′ )) · Dk|k (x′ )dx′ = ˚ pT N

(18.505) (18.506)

˚k|k + (1 − pT ) · Nk|k = ˚ pT N ˚k|k + ˚ = ˚ pT · N pT · Nk|k

(18.507)

˚k|k + Nk|k ) = ˚ = ˚ pT · (N pT · mk+1 .

(18.509)

(18.508)

3. The λ-PHD pseudofilter reduces to the λ-PHD filter if the intermixing motion model is disabled. The measurement-update equations for the λPHD filter and λ-PHD pseudofilter are already identical, so we need compare only the time-update equations. Let pT = ˚ pT = 1—that is, the intermixing model has been disabled, because targets can transition only to targets and generators only to generators. Then the pseudofilter time-update equations, (18.498) and (18.493), reduce to: ∫ fk+1|k (x|x′ ) · Dk|k (x′ )dx′ (18.510) Dk+1|k (x) = ˚k+1|k N

=

˚k|k . N

(18.511)

These are the same as the λ-PHD filter equations, (18.130) and (18.131), when there are no target appearances or disappearances. This means that the λ-PHD filter and λ-PHD pseudofilter will have approximately the same performance, when the target appearance rate and the target disappearance rate are small.

18.10

CPHD/PHD FILTERS WITH POISSON-MIXTURE CLUTTER

The Poisson-mixture model for unknown clutter was introduced in Section 16.3.3. This section provides the filtering equations for a CPHD filter for this Poissonmixture model, and its PHD filter special case. These equations were first proposed

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637

by Mahler in 2009 [155]. They are rederived in Section K.23 using Clark’s general chain rule, (3.91). It should also be noted that the derivation in Section K.23 can be easily generalized to clutter generators whose p.g.fl.’s have the form ( ) ∫ κ Gκk+1 [g|c, c] = Gk+1 1 − c + c g(z) · fk+1 (z|c)dz (18.512) where Gk+1 (z) is an arbitrary p.g.f. Thus the results of this section apply to clutter models more general than Poisson-mixture models. Since the measurement-update equations for the Poisson-mixture CPHD filter involve combinatorial sums, they are not computationally tractable as stands. They are presented here with the expectation that suitable approximation procedures may eventually be discovered. Recall that in the Poisson-mixture model, the measurement RFS has the form: Σk+1 = Tk+1 (x1 ) ∪ ... ∪ Tk+1 (xn ) ∪ Ck+1 (c1 , c1 ), ..., Ck+1 (cν , cν ) (18.513) where X = {x1 , ..., xn } with ˚ = {(c1 , c1 ), ..., (cν , cν )} with C where the p.g.fl.’s of Tk+1 (x) and Gk+1 [g|x]

=

Gκk+1 [g|c, c]

=

|X| = n is the set of target states and ˚ = ν is the set of clutter generators, and |C| Ck+1 (c, c) are, respectively, ∫ 1 − pD (x) + pD (x) g(z) · fk+1 (z|x)dz (18.514) ( ∫ ) κ exp c (g(z) − 1) · fk+1 (z|c)dz (18.515)

=

ec·fk+1 [g−1|c] .

κ

(18.516)

κ Here κk+1 (z|c, c) = c · fk+1 (z|c) is a family of elemental clutter intensity functions, parametrized by c and c where c > 0 is the unknown clutter rate and κ fk+1 (z|c) is the clutter spatial distribution. It follows that

˚ = Ck+1 (c1 , c1 ) ∪ ... ∪ Ck+1 (cν , cν ) Ck+1 (C)

(18.517)

is also Poisson, and its intensity function (PHD) is a mixture of the form ˚ = c1 · ck+1 (z|c1 ) + ... + cν · ck+1 (z|cν ). κk+1 (z|C)

(18.518)

Also, the p.g.fl. of the measurement RFS, for both targets and clutter generators, is ∏ κ ˚ = (1 − pD + pD · Lg )X Gk+1 [g|X ⊎ C] ec·fk+1 [g−1|c] . (18.519) ˚ (c,c)∈C

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18.10.1

Poisson-Mixture Clutter-Agnostic CPHD Filter

The Poisson clutter model was introduced in Section 16.3.3. The purpose of this section is to describe the CPHD and PHD filters associated with this model. The Poisson-mixture clutter CPHD filter has the form ... →

p¨k|k (¨ n)

... →

Dk|k (x)

... →

˚k|k (c, c) D

→ ↑ → ↑ →

p¨k+1|k (¨ n) ¨ k+1|k (x) D ˚k+1|k (c, c) D

→ ↑↓ → ↑↓ →

p¨k+1|k+1 (¨ n)

→ ...

Dk+1|k+1 (x)

→ ...

˚k+1|k+1 (c, c) D

→ ...

where the middle filter propagates PHDs for the targets; where the bottom filter propagates PHDs for the clutter generators; and where the top filter propagates the probability distribution on the number n ¨ = n +˚ n, where n is the number of targets and ˚ n is the number of clutter generators. The time-update equation for this filter is the same as that for the classical CPHD filter (see Section 8.5.2). Thus we need only specify the measurementupdate equations. Let ∫ Nk+1|k = Dk+1|k (x)dx (18.520) ∫ ∫ ∞ ˚k+1|k (c, c)dcdc ˚k+1|k = N D (18.521) 0 ( ∫ ) (1 − p (x)) · Dk+1|k (x)dx ∫ ∫ ∞ D−c ˚k+1|k (c, c)dcdc + e ·D 0 ϕ¨k = (18.522) ˚k+1|k Nk+1|k + N ∫ τW = pD (x) · LW (x) · Dk+1|k (x)dx (18.523) ∫ ∫ ∞ ˚k+1||k (c, c)dcdc (18.524) κW = e−c · c|W | · LκW (c) · D 0

and LW (x)

=

LκW (c)

=

{

Lz (x) if W = {z} 0 if otherwise ∏ κ fk+1 (z|c).

z∈W

(18.525) (18.526)

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Then: • Measurement update for the joint p.g.f.: ( ∑

(|P|)

P⊟Z ∏k+1

¨ ¨ x|P| · G k+1|k (x · ϕk )

)

W · W ∈P N τW +κ ˚ k+1|k +Nk+1|k ( ∑ ) ¨ (|Q|) (ϕ¨k ) G Q⊟Z k+1|k k+1 ∏ V · V ∈Q N τV +κ ˚ +N

¨ k+1|k+1 (x) = G

k+1|k

(18.527)

.

k+1|k

• Measurement update for the target PHD:

=

Dk+1|k+1 (x) Dk+1|k (x)  ∑ ωP 

P⊟Zk+1

(18.528) ¨k ) ¨ (|P|+1) (ϕ G 1−pD (x) k+1|k · (|P|) ˚ ¨ ¨ Nk+1|k +Nk+1|k Gk+1|k (ϕk ) ∑ W (x) + W ∈P pDτ(x)·L W +κW



.

• Measurement update for the clutter PHD: ˚k+1|k+1 (c, c) D ˚k+1|k (c, c) D =

e−c

(18.529) 

∑ P⊟Zk+1

 ωP 

1 ˚k+1|k Nk+1|k +N

+

∑

· c

W ∈P

¨k ) ¨ (|P|+1) (ϕ G k+1|k ¨k ) ¨ (|P|) (ϕ G k+1|k

|W |

·Lκ W (c) τW +κW

The summations are taken over all partitions P Zk+1 , and

 .

of the measurement set

τW +κW ¨ (|P|) (ϕ¨k ) · ∏ G ˚k+1|k W ∈P Nk+1|k +N ωP = ∑ ∏ (|Q|) τV +κV ¨ ¨ ˚ Q⊟Zk+1 Gk+1|k (ϕk ) · V ∈Q N +N k+1|k



.

(18.530)

k+1|k

• Estimate of the clutter intensity function: κ ˆ k+1 (z) =

∫ ∫

∞ κ ˚k+1|k (c, c)dcdc. c · fk+1 (z|c) · D 0

(18.531)

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Equation (18.531) follows from an analysis similar to that presented in Section 18.3.5 and in Section K.21. That is, from (K.436) of Section K.21, if ˚ is the p.g.fl. of the clutter RFS for a given set C ˚ of clutter generators, Gk+1 [g|C] then the predicted expected p.g.fl. of the clutter process is ∫ ¯ ˚ · f˚k+1|k (C|Z ˚ (k) )δ C. ˚ Gk+1 [g] = Gk+1 [g|C] (18.532) From (4.75) the predicted average intensity function is, therefore, κ ¯ k+1 (z)

¯ k+1 δG [1] ∫ δz δGk+1 ˚ ˚ (k) )δ C ˚ [1|C] · fk+1|k (C|Z δz ∫ ˚ · f˚k+1|k (C|Z ˚ (k) )δ C ˚ κk+1 (z|C)

= = =

(18.533) (18.534) (18.535)

˚ is given by (18.518). Thus where κk+1 (z|C)   ∫ ∑ κ ˚ (k) )δ C ˚ (18.536)  κ ¯ k+1 (z) = c · fk+1 (z|c) · f˚k+1|k (C|Z ˚ (c,c)∈C

∫ ∫

=

1 κ ˚k+1 (c, c)dcdc c · fk+1 (z|c) · D

(18.537)

0

where the last equation is due to Campbell’s theorem, (4.96). 18.10.2

Poisson-Mixture Clutter-Agnostic PHD Filter

The Poisson-mixture clutter (PMC) PHD filter arises when we assume that the joint predicted target-clutter RFS is Poisson, in which case ˚ ¨ k+1|k (¨ G x) = e(Nk+1|k +Nk+1|k )·(¨x−1) .

(18.538)

It has the form ... →

Dk|k (x)

→

¨ k+1|k (x) D

... →

˚k|k (c, c) D

→

˚k+1|k (c, c) D

→ ↑↓ →

Dk+1|k+1 (x)

→ ...

˚k+1|k+1 (c, c) D

→ ...

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where the top filter propagates PHDs for the targets and the bottom filter propagates PHDs for the augmented clutter generators. The time-update equation for this filter is the same as that for the classical PHD filter. The measurement-update equations are as follows ([155], Corollary 1): • Measurement update for target PHD: ∑

Dk+1|k+1 (x) = 1 − pD (x) + Dk+1|k (x)

ωP

P⊟Zk+1

∑ pD (x) · LW (x) . (18.539) τW + κ W W ∈P

• Measurement update for clutter PHD:  ˚k+1|k+1 (c, c) D = e−c 1 + ˚k+1|k (c, c) D

∑ P⊟Zk+1

ωP

 ∑ c|W | · LW (c)  . (18.540) τW + κ W

W ∈P

As before, the summations are taken over all partitions P of the measurement set Zk+1 . Also, ωP = ∑

18.11

∏

W ∈P (τW

Q⊟Zk+1

∏

+ κW ) . V ∈Q (τV + κV )

(18.541)

RELATED WORK

The following research efforts are related to that described in this chapter, in that they propose a PHD filter-like structure integrated with clutter estimation. They are as follows: 1. Section 18.11.1: The decoupled clutter-target PHD filter of Feng et al.—in which the clutter intensity function is estimated separately and then used in the classical PHD filter. 2. Section 18.11.2: The “dual PHD filter” of Jonsson et al.—an independently conceived version of the λ-PHD filter of Section 18.4.6. 3. Section 18.11.3: The “iFilter”—which is identical to the λ-PHD pseudofilter of Section 18.9.

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Decoupled Target-Clutter PHD Filter

This filter was proposed by Feng Lian, Chongzhao Han, and Weifeng Liu in 2010 [84]. Unlike the λ-CPHD and κ-CPHD filters, it is uncoupled: clutter estimation and target tracking are not inherently integrated as a single recursive procedure. Rather, the clutter intensity function is first estimated as a parametrized finitemixture model, and then employed in the classical PHD filter. As such, it can also be applied to the CPHD filter and to classical multitarget tracking algorithms such as MHT. The core of the approach is a particular method for estimating the clutter intensity function. The number of target-generated measurements is assumed to be much smaller than the average number of clutter measurements (that is, the clutter rate). It is also assumed that the unknown clutter intensity function does not vary with time. Finally, it is assumed that the unknown clutter RFS is Poisson. First the constant clutter rate is estimated as being approximately equal to the average number of measurements: k 1∑ λk ∼ |Zi | = k i=1

(18.542)

where Z1 , ..., Zk is the time sequence of measurement sets. Then the clutter spatial distribution is modeled as a Gaussian mixture: c(z|θ, µ) =

µ ∑

cj · NCj (z − cj ),

j=1

µ ∑

cj = 1

(18.543)

j=1

with unknown parameters θ = (c1 , C1 , c1 , ..., cµ , Cµ , cµ )

(18.544)

with cj ∈ RN and an unknown number µ of components. Assuming conditional independence of the measurements with respect to the parameters, the likelihood function with respect to measurement sets Z is ∏ Lθ (Z) = c(z|θ, µ). (18.545) z∈Z

The prior distribution f (θ) of θ is constructed by assuming that its parameter variables are distributed as follows: (c1 , ..., cM ) is Dirichlet-distributed ([83], p.

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62); each cj is normally distributed; and each Cj is Wishart-distributed ([83], p. 205). Given this, the posterior distribution f (θ|Z) is constructed and the MAP estimate computed using either expectation-maximization (EM) or Markov-chain Monte Carlo (MCMC) methods. Feng et al. implemented both EM and MCMC versions of their algorithm, and tested them in two-dimensional simulations. In these simulations, a linear-Gaussian sensor with pD = 0.95 observed up to five appearing and disappearing targets following curvilinear trajectories. The clutter process was Poisson with clutter rate λk+1 = 50, and the clutter spatial distribution ck+1 (z) was a three-component Gaussian mixture superimposed with a uniform spatial distribution. The authors compared the EM and MCMC versions of their algorithm with a conventional PHD filter—that is, one which was given the correct λk+1 and ck+1 (z). They reported that both versions greatly outperformed the conventional PHD filter in regard to both target-number estimation and target-localization accuracy. Their algorithms were also effective in estimating both λk+1 and ck+1 (z). 18.11.2

The “Dual PHD” Filter

In 2012 Jonsson, Degerman, Svensson, and Wintenby proposed this PHD filter as a means of detecting and tracking multiple targets in unknown background clutter [127].3 They also implemented it using Gaussian mixture techniques and applied it to targets moving in Doppler clutter. The measurement-update equations (the unnumbered equations in Section IV of [127]) for their filter correspond to the Bernoulli clutter model of Section 18.3.5, in which ˚ pκD (˚ c) = ˚ pκD (c). But since κ ˚ pD (c) must be assumed constant in a GM implementation, the “dual PHD filter” is identical to the GM implementation of the λ-PHD filter of Section 18.4.6. Jonsson et al. conducted simulations to test a Gaussian mixture (GM) implementation of their approach. The sensor is a high-flying airborne radar whose mission is to track low-flying objects. The radar also collects Doppler clutter measurements produced by ground road traffic. Exploiting the fact that ground and airborne objects have different dynamics, Jonsson et al. demonstrated that their PHD filter successfully detected and tracked six appearing and disappearing targets with traffic clutter on the order of 40 measurements per scan. 3

Though Jonsson et al. cited the paper [194] by Mahler et al., they apparently did not notice that their filter is a special case of the λ-CPHD filter described there.

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Advances in Statistical Multisource-Multitarget Information Fusion

The “iFilter ”

The “multitarget intensity filter” (MIF) was proposed by Streit and Stone in 2008 [281] and subsequently renamed the “iFilter.” Its measurement-update equation is identical to the measurement-update equation for the λ-PHD filter of Section 18.4.3. However, its time-update equations are

Dk+1|k (x)

=

˚k|k + ψk (x|ϕ) · N

∫

ψk (x|x′ ) · Dk|k (x′ )dx′ (18.546)

˚k+1|k N

=

˚k|k + ψk (ϕ|ϕ) · N

∫

ψk (ϕ|x′ ) · Dk|k (x′ )dx′ (18.547)

where, in the notation of Section 18.4.1, (a) ψk (ϕ|ϕ) = ˚ pT ; (b) ψk (x|ϕ) = ′ ′ ′ ′ (1 − ˚ pT ) · sB (x); (c) ψ (ϕ|x ) = 1 − p (x ); and (d) ψ k T k (x|x ) = pT (x ) · k+1|k fk+1|k (x|x′ ). The authors asserted that their claimed “Poisson point process” or “PPP” derivation of the MIF also demonstrated that “multitarget intensity filters [that is, the MIF and the PHD filter] can be understood in essentially elementary terms [using]. . . PPP’s at an elementary level” ([281] Section 1, 2nd paragraph). However, the claimed “PPP” derivations of both the MIF and the PHD filter have serious mathematical errors and seriously restrictive hidden assumptions (see Appendix A of [163]). A few of the mathematical errors were summarized in footnote four in Section 8.4.6.8. As for hidden assumptions: no target spawning; the distribution fk|k (X|Z (k) ) (and not just fk+1|k (X|Z (k) )) is Poisson; the intensity function of the target-birth RFS is constant; and the state space is bounded. In any case, and as can be seen from the discussion in Section 18.3, derivation of the MIF requires only straightforward algebra. Point process theory, elementary or otherwise, is completely unnecessary. Indeed, the MIF is identical to the λ-PHD pseudofilter of Section 18.9. As such, it exhibits the pathological behavior identified there (see Section 18.9.2). That is, it does not (as claimed) always estimate the clutter rate and does not (as claimed) always estimate the target-birth rate. It does not (as claimed) include the classical PHD filter as a special case. Finally, and as was noted in Section 18.9.2, if the intermixing motion model is approximately disabled (which occurs if the target appearance and target-disappearance rates are small) then the MIF will be approximately identical to the λ-PHD filter of Section 18.4.6.

Part IV

RFS Filters for Nonstandard Measurement Models

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Chapter 19 RFS Filters for Superpositional Sensors 19.1

INTRODUCTION

Many if not most sensors do not conform to the assumptions of the “standard” multitarget measurement model. Consider, for example, a mechanically rotating radar or a phased-array radar, either of which generates a continuous real-valued signature. The real-valued signature is the real part of a complex-valued signature, which in turn is a sum (“superposition”) of the complex-valued signatures generated by the individual targets and the background clutter. Those excursions of the realvalued signature that exceed a (fixed or adaptive) threshold, produce a finite set Z (a “scan” or “frame”) of “detections.” If the targets are sufficiently distant, then each target can be modeled as a mathematical point that generates at most a single detection at a time (the “small target” assumption). Detection-extraction schemes such as this tend to discard useful information, thus tending to reduce tracking performance. For example, when the targets are close together (relative to the angular and range resolution of the radar), amplitude detection approaches can merge detections, thereby failing to resolve adjoining targets. In principle, a tracking filter that exploits the full superpositional signal model could achieve better performance. The purpose of this chapter is to describe CPHD filters specifically designed to operate with superpositional models. These will be collectively described as Σ-CPHD filters.

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Examples of Superpositional Sensor Models

To emphasize the potential importance of this chapter, five different real-world applications involving superpositional sensors will be briefly described: 1. Section 19.1.1.1: Surveillance radar. 2. Section 19.1.1.2: Time-direction-of-arrival (TDOA) for sinusoidal signals. 3. Section 19.1.1.3: Multi-user detection (MUD) in communications networks. 4. Section 19.1.1.4: Radio-frequency tomography for surveillance of interior spaces. 5. Section 19.1.1.5: Thermopile arrays for heat-based localization. 19.1.1.1

Surveillance Radar

Suppose that targets are sufficiently distant that they can be regarded as point targets. Also assume that the radar is narrowband, that is, its radian center frequency ωc is much larger than its bandwidth. For a given azimuth and elevation α, θ, the transmitted signal has the form st = χt · eι·ωc t

(19.1)

√

where ι = −1 is the complex unit and where the complex envelope χt specifies the shape of the radar pulse. For example, if χt = 1[0,T ] (t), then the radar transmits a single rectangular pulse with pulsewidth T . If the transmitted signal impinges on a point target with state x, a reflected signal is received at the radar: ηt (x) = At (x) · χt−τt (x) · eι·(t−τt (x))·(ωc +ωt (x)) + Vt .

(19.2)

where Vt is sensor noise, τt (x) is the time delay, ωt (x) is the Doppler shift, and At (x) depends on factors such as radar cross section (RCS), atmospheric absorption, and so on In the event that there are multiple targets X = {x1 , ..., xn }, the noisy signal received at the radar is the superposition Zt =

∑ x∈X

ηt (x) + Vt .

(19.3)

RFS Filters for Superpositional Sensors

19.1.1.2

649

Time-Direction-of-Arrival (TDOA) for Sinusoidal Signals

Balakumar, Sinha, Kirubarajan, and Reilly [14] have considered the following TDOA problem. The sensor is a linear array of M identical receivers (for example, antennas or microphones) with sensor positions d0 , 2d0 , ..., M d0 . Let τ0 ≜ d0 /c where c is the signal propagation speed. Suppose that there are an unknown number of unknown sinusoidal sources (for example, radio-frequency transmitters or acoustic sources), which are sufficiently distant from the array that the signal from each source impinges on it as a plane wave. Each source is characterized by its signal amplitude α, its center frequency ω, its bandwidth β, and its angle of arrival ϕ at the array. Assume that all sources are narrowband—that is, that β ≪ ω—in which case β can be neglected. We wish to determine the number n and states xi = (αi , ωi , ϕi ) (19.4) of the sources for i = 1, ..., n. Towards this end, let X = {x1 , ..., xn } with |X| = n and ηj (x) η(x)

= =

η(X)

=

αi · e−ι·jτ0 ωi ·sin ϕi (η1 (x), ..., ηM (x))T n ∑ η(xi )

(19.5) (19.6) (19.7)

i=1

√ where ι = −1 denotes the imaginary unit. Then the complex-valued signal from the ith source, received at the jth receiver, is ([14], Eqs. (1,4)) Zj,i = ηj (xi ) + Wj,i

(19.8)

and Wj,i is a complex-valued zero-mean Gaussian noise vector (see Appendix H for a brief discussion of Gaussian distributions with complex-valued arguments). The total signal received at the jth receiver is the superposition of the signals generated by all of the sources Zj =

n ∑

ηj (xi ) + Wj ,

(19.9)

i=1

where Wi is also zero-mean. Thus the measurement received by the array is the random vector Z =η(X) + W (19.10)

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T

T

where Z = (Z1 , ..., ZM ) and W = (W1 , ..., WM ) . Equation (19.10) provides another example of a superpositional measurement model. To apply a PHD filter to this tracking problem, Balakumar et al. converted this superpositional measurement model to a detection measurement model. Any collected measurement z = (z1 , ..., zM )T is used to construct a set Z˜ = {˜ z1 , ..., ˜ zn˜ } of pseudo-measurements, which are then employed as inputs to the PHD filter. Let M 1 ∑ Fz (ω) = zj · e−ι·jωτ0 (19.11) M j=1 be the discrete Fourier transform (DFT) of z. The DFT will have n ˜ peaks at frequencies ω = ω ˜ 1 , ..., ω ˜ n˜ with respective amplitudes α1 , ..., αn˜ . Separate the peaks and interpret each as the DFT F˜zi (ω) of a pseudo-measurement ˜ zi generated by one of the sources. An approximate likelihood Li (x) is constructed for ˜ zi ([14], Eq. (11)) and used in the PHD measurement-update equation. An obvious question presents itself: Are there PHD or CPHD filters that can address the original superpositional data, rather than the detection data extracted from it? 19.1.1.3

Multi-User Detection (MUD) in Communications Networks

MUD problems occur in dynamic, mobile, multiple-access wireless digital communications networks in which the goal is to detect, track, and identify system users as they enter and exit. Current systems are typically based on the assumption that the number of active users is constant, known, and equal to the maximum number of registered users. In actuality, the actual number of users changes with time and is appreciably smaller than the maximum number. Also, the set of active users is typically unknown at the receiver. System efficiency and capacity could be increased if active users could be rapidly identified ([19], p. 54). Besides cellphone networks, applications include ad hoc networks (in which MUD facilitates optimal transmission strategies); and spatial multiplexing (in which MUD facilitates the proper allocation of system power). Biglieri, Lops, and Angelosante have written a series of publications applying finite-set statistics techniques to MUD and related applications [5], [6], [20], [7], [8], [21], including the monograph [19]. The purpose of this section is to sketch the basic elements of these applications. In MUD the time-varying state xt consists of various parameters that potentially allow the identification of a given user. One parameter is a label ℓ

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651

associated with the user; and another is the message symbol dt generated by the user (drawn from some source alphabet). Assume that there is a “reference user,” which has known state x0 and which is always active. The signal collected at the receiver at time t has the form ∑ Zt = ηt (x0 ) + ηt (xt ) + Vt (19.12) x∈Xt

where Xt is the set of unknown users and Vt is the receiver noise. Thus the likelihood function has the form ( ) ∑ ft (zt ) = fVt z − ηt (x0 ) − ηt (xt ) . (19.13) x∈Xt

Biglieri et al. have implemented a full multitarget recursive Bayes filter that directly incorporates this likelihood function. They also employ novel Bayesoptimal maximum a posteriori (MAP) multitarget state-estimators specifically designed for MUD. Nevertheless, it is doubtful that any approach based on the multitarget Bayes filter would be computationally tractable in a communications network of realistic size. Consequently, CPHD filters specifically designed for superpositional models are of potential interest. 19.1.1.4

Radio-Frequency Tomography

RF tomography [236] is an approach for determining the locations and velocities of unknown targets (such as people) in unknown, denied areas (such as closed buildings). A number ν of transmitter-receiver units are situated around the denied area. Each receiver measures the received signal strength (RSS) due to the signal generated by each transmitter. At any moment, this results in the collection of m = ν(ν − 1)/2 RSS measurements. Any object in the denied area, whether a target of interest or otherwise, attenuates the signal along any transmitter-to-receiver line-of-sight on which a target happens to be located. Prior to tracking, measurements are collected with the purpose of estimating the clutter and noise background. If ν is large enough, it is possible in principle to detect, locate, and track all moving objects in the denied area. Algorithmic approaches, involving techniques such as the EM algorithm and sequential Monte Carlo (SMC) filtering, have recently been devised for this purpose [329], [326]. Since RF tomography inherently involves superpositional sensors, it would be useful to develop PHD or CPHD filters applicable to RF tomography and related

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applications. This challenge was addressed by Thouin, Nannuru, and Coates in 2011 [290]. The model they employed is as follows. Let j = 1, ..., m be an index that specifies each transmitter-to-receiver link. Then the single-target measurement function for the RSS of each link at time tk+1 is ([290], Eq. (6)): ( ) λ2j (x) j ηk+1 (x) = ϕ · exp − (19.14) 2σλ2 where λj (x) is the perpendicular distance between a target located at x = (x, y) and the jth link; and where ϕ and σλ are empirically-determined constants. Thus the further a target at x is from the jth link, the larger the value of λj (x) and the smaller the RSS. Since the RSS due to multiple targets X = {x1 , ..., xn } is the superposition of the RSSs due to each target individually, the (noisy) total attenuation on the jth link is ([290], Eq. (7)) j ηk+1 (X) =

∑

j ηk+1 (x) + Vk+1

(19.15)

x∈X

with total likelihood function L(z1 ,...,zm ) (X) =

m ∏

j NRk+1 (zj − ηk+1 (X)).

(19.16)

j=1

19.1.1.5

Thermopile Arrays

A thermopile is a low-resolution heat detector that, within its field of view, measures the heat radiation of an object relative to ambient temperature. An array of thermopiles can, at least in principle, be used to detect, locate, and track moving objects (for example, humans) that are warm in comparison to their surrounding environment (for example, rooms). Hauschildt et al. have investigated the use of finite-set statistics techniques for this purpose [133], [104], [105]. Suppose that there are m thermopiles in the array, each regarded as a pixel in a heat image. Let x be the state of a single target. Then the measurement function for the jth pixel can be approximated as ([104], Eq. (57)): j ηk+1 (x) = aj · [arctan (bj (ϕj + ∆j ϕ + θj )) − arctan (b(ϕj − ∆j ϕ + θj ))] (19.17) where ϕ is the angle of arrival of the object at the pixel, ∆j ϕ is the angular size of the target, θj is the pixel orientation, and aj , bj are calibration constants. The

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653

measurement function for the entire image is, therefore, 1 m ηk+1 (x) = (ηk+1 (x), ..., ηk+1 (x))T .

(19.18)

Thus the noisy measurement model for multiple targets X = {x1 , ..., xn } is the superposition ∑ Zk+1 = ηk+1 (x) + Vk+1 . (19.19) x∈X

19.1.2

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter: • Superpositional sensors are ubiquitous, and multitarget tracking filters specifically designed for them could offer a significant improvement over conventional, detection-based tracking approaches (Section 19.1.1). • An exact formula for a CPHD filter for general superpositional sensors can be derived (Section 19.2). However, it is computationally intractable in general. • An exact, closed-form Gaussian mixture solution of the exact superpositional CPHD filter exists, but is only partially tractable (Section 19.3). • A computationally tractable approximate superpositional CPHD filter is possible, when implemented using particle techniques (Section 19.4). 19.1.3

Organization of the Chapter

The chapter is organized as follows: 1. Section 19.2: An exact (but computationally intractable) superpositional CPHD filter (“exact Σ-CPHD filter”). 2. Section 19.3: A semi-tractable closed-form Gaussian mixture superpositional CPHD filter, based on an approximation due to Hauschildt, and referred to here as the “Hauschildt Σ-CPHD filter.” 3. Section 19.4: A tractable approximate superpositional CPHD filter, based on a Campbell’s theorem approximation originally devised by Thouin, Nannuru, and Coates (“TNC Σ-CPHD filter”).

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19.2

EXACT SUPERPOSITIONAL CPHD FILTER

Mahler derived exact filtering equations for a general superpositional CPHD filter in 2009 [156]. It will hereafter be referred to as the exact superpositional CPHD filter, or exact Σ-CPHD filter for short. The purpose of this section is to summarize the filtering equations for this filter. Since the time-update equation is the same as that for the classical CPHD filter (Section 8.5.2), it is necessary to discuss only the measurement-update. Suppose that we have a single sensor with the superpositional measurement model with likelihood function fk+1 (z|X) = fVk+1 (z − ηk+1 (X)) where ηk+1 (X) =

{

∑

0 x∈X ηk+1 (x)

if if

X=∅ X ̸= ∅

(19.20)

(19.21)

where ηk+1 (x) is the (real- or complex-valued) signature-vector generated by a single target; and where Vk+1 is a zero-mean random (real or complex) noise vector. Suppose that the sensor delivers a new measurement zk+1 , after having collected a time-stream Z k : z1 , ..., zk of measurement vectors. Assume, as with the classical CPHD filter (Chapter 8), that the predicted multitarget distribution fk+1|k (X|Z k ) is approximately an i.i.d.c. RFS: fk+1|k (X|Z k ) ∼ = |X|! · pk+1|k (|X||Z (k) ) ·

∏

sk+1|k (x|Z (k) )

(19.22)

x∈X

where pk+1|k (n|Z k ) is the predicted cardinality distribution with probability generating function (p.g.f.) Gk+1|k (x|Z (k) ) =

∑

pk+1|k (n|Z k ) · xn

(19.23)

−1 sk+1|k (x|Z (k) ) = Nk+1|k · Dk+1|k (x|Z k )

(19.24)

n≥0

and where is the predicted target spatial distribution, and where ∫ Nk+1|k = Dk+1|k (x|Z k )dx.

(19.25)

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Abbreviate pn

=

pk+1|k (n|Z (k) ) (k)

s(x) f (z)

= =

sk+1|k (x|Z fVk+1 (z)

ηx

=

ηs (z)

=

ηk+1 (x) ∫ s(x) · δηx (z)dx

(19.26) (19.27) (19.28)

)

(19.29) (19.30)

where δηx (z) is the Dirac delta density function concentrated at ηk+1 (x). Also, let ∫ g1 (w) · g2 (w − z)dw

(g1 ⋆ g2 )(z) =

(19.31)

be the convolution of the functions g1 (z) and g2 (z). Then the convolutional power g ⋆j of g(z) is recursively defined by g ⋆0

=

δ0

⋆1

= =

g g ⋆ g ⋆(j−1)

g g ⋆j

(19.32) (19.33) (19.34)

(j ≥ 1) .

Given this, the measurement-update equations for the exact Σ-CPHD filter are ([156], Theorem 1): • Measurement update for the cardinality distribution: pk+1|k+1 (n) = ∑

(f ⋆ ηs⋆n )(zk+1 ) j≥0

pj · (f ⋆ ηs⋆j )(zk+1 )

· pn .

• Measurement update for the expected number of targets: ∑ ⋆n n≥1 n · pn · (f ⋆ ηs )(zk+1 ) . Nk+1|k+1 = ∑ ⋆j j≥0 pj · (f ⋆ ηs )(zk+1 )

(19.35)

(19.36)

• Measurement update for the spatial distribution: sk+1|k+1 (x) =

·

∑

1 Nk+1|k+1 ⋆(n−1)

n≥1

n · pn · (f ⋆ ηs )(zk+1 − ηx) · s(x). ∑ ⋆j j≥0 pj · (f ⋆ ηs )(zk+1 )

(19.37)

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n Substituting pn = e−Nk+1|k ·Nk+1|k /n! into (19.37) yields the measurementupdate for the exact Σ-PHD filter special case of the exact Σ-CPHD filter:

Dk+1|k+1 (x) =

∑

1 n≥0 n!

∑

1 j≥0 j!

where ηD (z) =

19.3

⋆n · (f ⋆ ηD )(zk+1 − ηx)

∫

⋆j · (f ⋆ ηD )(zk+1 )

· Dk+1|k (x)

Dk+1|k (x) · δηx (z)dx.

(19.38)

(19.39)

HAUSCHILDT’S APPROXIMATION

The measurement-update equations for the exact Σ-CPHD filter are computationally intractable in general. However, in 2011 Hauschildt proposed an approximation that results in a closed-form and at least partially tractable solution of the exact ΣCPHD filter [104]. He called it the “superpositional sensor (SPS) CPHD filter,” but it will be referred to here as the Hauschildt Σ-CPHD filter.1 The section is organized as follows: 1. Section 19.3.1: An overview of the Hauschildt approximation. 2. Section 19.3.2: Measurement models for the Hauschildt Σ-CPHD filter. 3. Section 19.3.3: Measurement update equations for the Hauschildt Σ-CPHD filter. 4. Section 19.3.4: Implementations of the Hauschildt Σ-CPHD filter. 19.3.1

Hauschildt Σ-CPHD Filter: Overview

The multitarget likelihood function for the superpositional model is fk+1 (z|X) = fVk+1 (zk+1 − ηk+1 (X)). 1

(19.40)

Note: The measurement-update equation in [104] is not quite correct, because of a minor algebra error in its derivation. The corrected filtering equations will be presented in this section.

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The Hauschildt approximation therefore begins with the observation that the measurement-updated cardinality distribution, PHD, and expected number of targets are given by the equations ∫ f (z − ηk+1 (X)) · fk+1|k (X)δX |X|=n Vk+1 k+1 pk+1|k+1 (n) = ∫ (19.41) fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY ( ∫ ) fVk+1 (zk+1 − ηk+1 (x) − ηk+1 (X)) ·fk+1|k (X ∪ {x})δX ∫ Dk+1|k+1 (x) = (19.42) fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY ∫ |X| · fVk+1 (zk+1 − ηk+1 (X)) · fk+1|k (X)δX ∫ Nk+1|k+1 = .(19.43) fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY Assume that the predicted multitarget distribution fk+1|k (X) is i.i.d.c. (as defined in Section 4.3.2): pk+1|k (|X|) fk+1|k (X) = |X|! ·

|X|

X · Dk+1|k .

(19.44)

Nk+1|k Let the predicted PHD be a Gaussian mixture as in Section 9.5: νk+1|k

Dk+1|k (x) =

∑

k+1|k

k+1|k

· NP k+1|k (x − xi

wi

)

(19.45)

i

i=1

and let the single-target likelihood function fk+1 (z−ηk+1 (x)) be linear-Gaussian: ηk+1 (x)

=

Hk+1 x

(19.46)

fVk+1 (z)

=

NRk+1 (z).

(19.47)

It follows that, for each X, the predicted multitarget distribution fk+1|k (X) is a (very complicated) Gaussian mixture. Consequently, repeated application of the fundamental Gaussian identity, (2.3), allows one to compute exact closedform formulas for the numerators and denominator of (19.41) through (19.43). Furthermore, the numerator of (19.42) turns out to be a Gaussian mixture. As with the GM-CPHD filter, it is assumed that the cardinality distribution vanishes for sufficiently large n. It then turns out that pk+1|k+1 (n) and Dk+1|k+1 (x) can both be written in exact closed form and that, in particular, Dk+1|k+1 (x) is a Gaussian mixture. Unscented Kalman filter (UKF) techniques can be used to extend the approach to moderately nonlinear measurement functions ηk+1 (x)—see [104], Section V-B.

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19.3.2

Hauschildt Σ-CPHD Filter: Models

The modeling assumptions underlying the Hauschildt Σ-CPHD filter are as follows: • Constant probability of target survival: pS,k+1|k (x′ ) = pS,k+1|k .

(19.48)

• Linear-Gaussian single-target motion model: fk+1|k (x|x′ ) = NQk (x − Fk x′ ).

(19.49)

• The single-target measurement function is linear: (19.50)

ηk+1 (x) = Hk+1 x where Hk+1 is the measurement matrix. Define { 0 (∑ ) if ∑ Hk+1 X = Hx = H x if x∈X x∈X

X=∅ . X ̸= ∅

(19.51)

• Measurement noise is Gaussian: (19.52)

fVk+1 (z) = NRk+1 (z).

The time-update equation for the Hauschildt Σ-CPHD filter is identical to that for the GM-CPHD filter (Section 9.5.5.2). Thus only the measurement-update equations need be described. 19.3.3

Hauschildt Σ-CPHD Filter: Measurement Update

Assume that the predicted PHD is a Gaussian mixture: νk+1|k

Dk+1|k (x) =

∑

k+1|k

wi

k+1|k

· NP k+1|k (x − xi

)

(19.53)

i

i=1

with νk+1|k

Nk+1|k =

∑ i=1

k+1|k

wi

.

(19.54)

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In what follows, let a multi-index with values in {1, ..., νk+1|k } be defined as follows: • For n ≥ 1, it is an n-tuple o = (o1 , ..., on ) with o1 , ..., on ∈ {1, ..., νk+1|k }. • For n = 0, the empty multi-index is denoted as o = (). Define |o| o1

=

{

=

{

0 n () o1

if if

o = () o = (o1 , ..., on )

if if

(19.55)

o = () o = (o1 , ..., on )

(19.56)

and wok+1|k

=

{

xk+1|k o

=

{

Pok+1|k

=

{

if if

w o1

1 k+1|k · · · w on

k+1|k x o1

0 k+1|k + ... + xon

k+1|k

k+1|k

P o1

0 k+1|k + ... + Pon

o = () o = (o1 , ..., on )

(19.57)

if if

(19.58)

if if

o = () o = (o1 , ..., on )

o = () . (19.59) o = (o1 , ..., on )

Given this, the measurement-update equations for the Hauschildt Σ-CPHD filter are: • Measurement updated cardinality distribution ([104], Eq. (43)): pk+1|k+1 (n)  

=

·NR

  

∑

(

k+1|k

zk+1 − Hk+1 xo

pk+1|k (|o′ |) 0≤|o′ |≤n

max

(19.60) 

k+1|k o:|o|=n wo

k+1|k T Hk+1 k+1 +Hk+1 Po

∑ ·NR

pk+1|k (n) n Nk+1|k

|o′ | Nk+1|k

k+1|k T Hk+1 k+1 +Hk+1 Po′

(

) 

k+1|k

· w o′

zk+1 −

k+1|k Hk+1 xo′



)  

where, by (19.55) through (19.59), if |o| = 0 then T wo · NRk+1 +Hk+1 Po Hk+1 (zk+1 − Hk+1 xo ) = NRk+1 (zk+1 ).

(19.61)

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• Measurement updated PHD ([104], Eqs. (30-42)): ∑ Dk+1|k+1 (x) = pk+1|k+1 (|o|) · |o|

(19.62)

1≤|o|≤nmax

·NP˜ k+1|k+1 (x − x ˜k+1|k+1 ) o o

where x ˜k+1|k+1 o

=

P˜ok+1|k+1

=

Ko

=

( ) xo1 + Ko zk+1 − Hk+1 xk+1|k o

(19.63)

(I − Ko Hk+1 )Pok+1|k (19.64) 1 −1 ( ) T T Pok+1|k Hk+1 Hk+1 Pok+1k Hk+1 + Rk+1 (.19.65) 1

• Measurement updated expected number of targets: ∑ Nk+1|k+1 = pk+1|k+1 (|o|) · |o|

(19.66)

1≤|o|≤nmax

=

n∑ max

n pk+1|k+1 (n) · n · νk+1|k .

(19.67)

n=1

These results are established in Section K.26. The PHD filter special case is given by the equation ∑ Dk+1|k+1 (x) = pk+1|k+1 (|o|) · |o| · NP˜ k+1|k+1 (x − x ˜o )

(19.68)

o

1≤|o|≤nmax

where (19.69)

pk+1|k+1 (n)  

=

·NR

 

1 n!

k+1|k k+1 +Hk+1 Po

∑ ·NR

∑

k+1|k w o:|o|=n ( o T Hk+1

0≤|o′ |≤n

max

k+1|k T Hk+1 k+1 +Hk+1 Po′



k+1|k

zk+1 − Hk+1 xo

1 |o′ |! (

)  .

k+1|k

· w o′ k+1|k

zk+1 − Hk+1 xo′

) 

The computational complexity of these filtering equations can be reduced by k+1|k k+1|k k+1|k noticing that wo , xo , Po are invariant with respect to permutations of the components of o.

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Remark 80 (Computational complexity) Hauschildt’s Σ-CPHD filter is computationally demanding, and thus probably applicable only when the number of targets is small. However, it should be pointed out that, in many applications, only a few targets are closely-spaced at any given time; and that superpositional models are necessary only when targets are closely spaced. (For, otherwise, conventional detection methods can be used to resolve them.) In such situations one could apply the Hauschildt Σ-CPHD filter to the closely-spaced targets, and CPHD or other nonsuperpositional filters to the remaining, less closely-spaced, targets. 19.3.4

Hauschildt Σ-CPHD Filter: Implementations

Hauschildt has applied an unscented Kalman filter (UKF) Gaussian-mixture implementation of his Σ-CPHD filter to the thermopile application discussed in Section 19.1.1.5 [104]. The thermopile was assumed to be a line array consisting of eight pixels, observing targets that move in a one-dimensional surveillance space at constant velocity. In the first simulation, three targets appear and disappear while moving in the positive direction. The second target is present for most of the scenario, and is crossed by the first target and later by the second target. The measurement noise in each pixel is small with variance σ 2 = (0.05)2 . The GM-Σ-CPHD filter was compared with a conventional GM-CPHD filter, with the measurements for the latter created by applying a detection threshold to the pixel data. While the GM-CPHD filter was unable to track the targets during the two target crossings, the Σ-CPHD filter successfully tracked them. In the second simulation, a fourth target was introduced at the same time as the third, and the noise variance was increased to a much more challenging value σ 2 = 0.25—approximately one-fourth of the maximum signal amplitude. In this case, the GM-CPHD filter could not be used since the thresholded data contained far too much clutter. Thus the UKF implementation was compared to an EKF implementation of the Σ-CPHD filter. It was observed that both implementations tracked the targets largely successfully, with the UKF implementation being somewhat better.

19.4

THOUIN-NANNURU-COATES (TNC) APPROXIMATION

This very clever approximation was devised by Thouin, Nannuru, and Coates in 2011. They employed it to derive a PHD filter for superpositional sensors, which they called the “additive likelihood moment (ALM)” filter” [290].

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The purpose of this section is to describe a generalization of the TNC approximation, and the approximate CPHD filter for superpositional sensors that results as a consequence. This filter, which will be called the TNC Σ-CPHD filter, was proposed by Nannuru, Coates, and Mahler in 2013 [222].2 The section is organized as follows: 1. Section 19.4.1: An overview of the generalized TNC approximation. 2. Section 19.4.2: Modeling assumptions for the TNC Σ-CPHD filter. 3. Section 19.4.3: Measurement update equations for the TNC Σ-CPHD filter. 4. Section 19.4.4: Implementations of the TNC Σ-CPHD filter. 19.4.1

Generalized TNC Approximation: Overview

The purpose of this section is to summarize the original TNC approximation (Section 19.4.1.1) and then its generalization (Section 19.4.1.2). 19.4.1.1

The Original TNC Approximation

As with the Hauschildt approximation, the TNC approximation begins with the measurement-update equations

=

Dk+1|k+1 (x)

=

= 2

pk+1|k+1 (n) (19.70) ∫ f (z − ηk+1 (X)) · fk+1|k (X)δX |X|=n Vk+1 k+1 ∫ fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY ( ∫ ) fVk+1 (zk+1 − ηk+1 (x) − ηk+1 (X)) ·fk+1|k (X ∪ {x})δX ∫ (19.71) fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY Nk+1|k+1 (19.72) ∫ |X| · fVk+1 (zk+1 − ηk+1 (X)) · fk+1|k (X)δX ∫ fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY

The ALM filter’s measurement-update equation ([290], Eq. (12)), is valid only for discrete state spaces. The TNC approximation itself remains valid, however. It was subsequently generalized by Mahler and used to derive an approximate CPHD filter for superpositional sensors [188]. The PHD filter special case of this CPHD filter is the correct form for the ALM filter for continuous spaces. Thouin, Nannuru, and Coates report that the original and the corrected ALM filters appear to have surprisingly similar performance, at least in an RF tomography application [223].

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where, since we are to derive a PHD filter, fk+1|k (X) is assumed to be Poisson. The approximation is based on the recognition that the denominator of these equations, ∫ fVk+1 (zk+1 − ηk+1 (X)) · fk+1|k (X)δX,

(19.73)

can be approximated using the following five-step process. 1. Step 1: Certain set integrals can be rewritten as ordinary integrals using the change of variables z = η(X), using the change of variables formula for set integrals of (3.46): ∫ ∫ T (η(X)) · f (X)δX = T (z) · P (z)dz, (19.74) where P (z) is a conventional probability density; and where, in our situation, T (z)

=

η(X)

=

fVk+1 (zk+1 − z) ∑ ηk+1 (X) = ηk+1 (x)

(19.75) (19.76)

x∈X

f (X)

=

fk+1|k (X).

(19.77)

2. Step 2: Given that fk+1|k (X) is Poisson and that η(X) has the superpositional form of (19.76), Campbell’s theorem (4.96) allows us to derive explicit formulas for the expected value ok+1|k and variance Ok+1|k of P (z): ∫ o = ηk+1 (x) · D(x)dx (19.78) ∫ O = ηk+1 (x)ηk+1 (x)T · D(x)dx (19.79) where D(x) is the PHD of f (X) and where (19.79) is true only if f (X) is Poisson. 3. Step 3: Approximate P (z) ∼ = NO (z − o).

(19.80)

4. Step 4: Assume that the noise is Gaussian, fVk+1 (z) = NRk+1 (z),

(19.81)

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in which case the right side of (19.74) can be solved, approximately, in closed form: ∫ ∫ T (z) · P (z)dz ∼ NRk+1 (zk+1 − z) · NO (z − o)dz (19.82) = =

NRk+1 +O (zk+1 − o)

(19.83)

and where the last equation results from the fundamental Gaussian identity, (2.3). 5. Step 5: Employ various algebraic stratagems to bring the numerators of (19.70) through (19.73) into a form that can be approximated using the previous four steps. 19.4.1.2

Generalized TNC Approximation

The generalized TNC approximation exploits the fact that the quadratic form of Campbell’s theorem, (4.102), allows us to derive an expression for O when f (X) is arbitrary. In general, (19.79) becomes ∫ O = ηk+1 (x)ηk+1 (x)T · D(x)dx (19.84) ∫ ∫ [ ] + ηk+1 (x1 )ηk+1 (x2 )T · D 2 (x1 , x2 ) − D(x1 ) · D(x2 ) dx1 dx2 where D(x) is the PHD of f (X) and where ∫ D 2 (x1 , x2 ) = f ({x1 , x2 } ∪ W )δW

(19.85)

is the second-order factorial-moment density of f (X) as defined (4.83). (Equations (19.78) and (19.84) are proved in Section K.27. Given this, we can reformulate (19.70) and (19.71) so that they have the form

pk+1|k+1 (n)

=

Dk+1|k+1 (x)

=

∫

∫

n

fVk+1 (zk+1 − ηk+1 (X)) · f k+1|k (X)δX

(19.86) fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY ( ∫ ) fVk+1 (zk+1 − ηk+1 (x) − ηk+1 (X)) x ·f k+1|k (X)δX ∫ (19.87) fVk+1 (zk+1 − ηk+1 (Y )) · fk+1|k (Y )δY

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x

n

for suitable definitions of f k+1|k (X) and f k+1|k (X). Applying the reasoning in Steps 1-4 and (19.78) and (19.84) to these formulas, we end up with measurementupdate equations for the TNC Σ-CPHD filter (as listed in Section 19.4.3). For the full proof, see [188]. Example 7 (A simple illustration) A simple example illustrates the correctness of (19.78) and (19.84). Let Ξ = {X1 , X2 , X3 } (19.88) where X1 , X2 , X3 are independent random state-vectors with respective linearGaussian probability distributions f1 (x) f2 (x)

= =

NP1 (x − x1 ) NP2 (x − x2 )

(19.89) (19.90)

f3 (x)

=

NP3 (x − x3 ).

(19.91)

Then the multitarget distribution of Ξ is 

 f1 (x1 ) · f2 (x2 ) · f3 (x3 ) + f1 (x3 ) · f2 (x1 ) · f3 (x2 ) f (X) = δ|X|,3 ·  +f1 (x2 ) · f2 (x3 ) · f3 (x1 ) + f1 (x1 ) · f2 (x3 ) · f3 (x2 )  +f1 (x2 ) · f2 (x1 ) · f3 (x3 ) + f1 (x3 ) · f2 (x2 ) · f3 (x1 ) (19.92) and its PHD and second factorial moment density are, respectively, D(x) D (x1 , x2 ) 2

= f1 (x) + f2 (x) + f3 (x) (19.93) = f1 (x1 ) · f2 (x2 ) + f1 (x2 ) · f2 (x1 ) + f2 (x1 ) · f3 (x2 ) (19.94) +f2 (x2 ) · f3 (x1 ) + f1 (x2 ) · f3 (x1 ) + f1 (x1 ) · f3 (x2 ).

Assume that ηk+1 (x) = x. Then the summed random vector is Z = ηk+1 (Ξ) = X1 + X2 + X3 .

(19.95)

Its probability distribution is the convolution fZ (x) = (f1 ⋆ f2 ⋆ f3 )(x) = NP1 +P2 +P3 (x − x1 − x2 − x3 ).

(19.96)

Thus the expected value and covariance of Z are o = x1 + x2 + x3 and O = P1 + P2 + P3 , respectively. Alternatively, the expected value can be computed

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using (4.96): o=

∫

x · D(x)dx = x1 + x2 + x3 .

(19.97)

Likewise, the covariance can be computed using (19.84). For, using (19.85),

=

D 2 (x1 , x2 ) − D(x1 ) · D(x2 ) −f1 (x1 ) · f1 (x2 ) − f2 (x1 ) · f2 (x2 ) − f3 (x1 ) · f3 (x2 ).

(19.98)

Then using (19.84) we get: O = P1 + P2 + P3 . 19.4.2

(19.99)

TNC Σ-CPHD Filter: Models

The TNC Σ-CPHD filter requires the following measurement model: • Gaussian superpositional likelihood function: fk+1 (z|X) = NRk+1 (z − ηk+1 (X))

(19.100)

where ηk+1 (X) was defined in (19.21). 19.4.3

TNC Σ-CPHD Filter: Measurement Update

The predictor step is the same as that for the usual CPHD filter (see Section 8.5.2). The corrector step is defined as follows. Suppose that we have a single sensor with superpositional likelihood function as in (19.100). Suppose that the sensor collects a measurement vector zk+1 , after having collected a time-stream Z k : z1 , ..., zk of measurements. Abbreviate the predicted cardinality distribution and PHD as pk+1|k (n) = pk+1|k (n|Z k ) and Dk+1|k (x) = Dk+1|k (x|Z k ), respectively. Define ∫ Nk+1|k = Dk+1|k (x|Z k )dx (19.101) sk+1|k (x)

=

Gk+1|k (x)

=

−1 Nk+1|k · Dk+1|k (x|Z k ) ∑ pk+1|k (n) · xn

(19.102) (19.103)

n≥0 n (n)

Gk+1|k (x)

=

d Gk+1|k (x). dxn

(19.104)

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(2)

667

(3)

2 Let σk+1|k , Gk+1|k (1), and Gk+1|k (1) be, respectively, the variance, second factorial moment, and third factorial moment of pk+1|k (n). Assume that:

• The likelihood function has the form fk+1 (z|X) = NRk+1 (z − ηk+1 (X)) = NRk+1

(

z−

∑

)

ηk+1 (x) .

x∈X

(19.105) • There exists an n0 ≥ 0 such that pk+1|k (n) < 1/n for all n > n0 (which will be true, for example, if pk+1|k (n) = 0 for all n > n0 ). Then the corrector equations for the TNC Σ-CPHD filter are ([188], Theorem 1): N

n

Rk+1 +O k

pk+1|k+1 (n)

∝

(zk+1 − nˆ ok )

NRk+1 +Ok (zk+1 − Nk+1|k ˆ ok )

· pk+1|k (n) (19.106)

NRk+1 +O ok ) ˚k (zk+1 − ηk+1 (x) −˚ Dk+1|k+1 (x)

=

(19.107)

NRk+1 +Ok (zk+1 − Nk+1|k ˆ ok ) ·Dk+1|k (x) where n

Ok Ok

= =

( ) ˆk − ˆ n· O ok ˆ oTk ˆk + Nk+1|k · O

2 (σk+1|k

(19.108) − Nk+1|k ) ·

ˆ ok ˆ oTk

(19.109)

(2) Gk+1|k (1)

˚ ok

= Nk+1|k

·ˆ ok

(19.110)

and where

˚k O

=

ok ˆ

=

ˆk O

=

  (3) (2) Gk+1|k (1) Gk+1|k (1)2 G(2) (1) ˆ ·ˆ · Ok +  − ok ˆ oTk (19.111) 2 Nk+1|k Nk+1|k Nk+1|k ∫ ηk+1 (x) · sk+1|k (x)dx (19.112) ∫ ηk+1 (x)ηk+1 (x)T · sk+1|k (x)dx. (19.113)

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The measurement-update for the corresponding PHD filter—that is, the corrected ALM filter—is Dk+1|k+1 (x)

(19.114)

NRk+1 +Nk+1|k Oˆ k (zk+1 − ηk+1 (x) − Nk+1|k ˆ ok ) = NRk+1 +Nk+1|k Oˆ k (zk+1 − Nk+1|k ˆ ok ) ·Dk+1|k+1 (x). (Note: This equation does not require the assumption that pk+1|k (n) < 1/n for all n > n0 .) Somewhat surprisingly, the performance of the corrected and original ALM filters do not appear to be significantly different [223]. 19.4.4

TNC Σ-CPHD Filter: Implementations

Because of the nonlinear, non-Gaussian nature of the measurement update formulas, both the TNC Σ-CPHD filter and the TNC Σ-PHD filter must be implemented using sequential Monte Carlo (SMC, also known as particle) techniques. In [222], Nannuru, Coates, and Mahler devised auxiliary particle filter implementations of both filters. The SMC implementation of the TNC Σ-CPHD filter was shown to be significantly faster than that of the TNC Σ-PHD filter: O(νM 2 +nmax M 3 +nmax ν) for the former versus O(νM 2 + M 3 + n2max ν 2 ) for the latter, where ν is the number of particles, nmax is the maximum possible number of targets, and M is the dimension of the measurement space. (For the passive-acoustic application, M is the number of passive-acoustic sensors. For the RF tomography application, M = 12 ν(ν − 1) where ν is the number of transmitter-receiver nodes.) As a baseline for comparison, a conventional Markov Chain Monte Carlo (MCMC) algorithm was implemented, based on an approach due to Septier, Pang, Carmi, and Godsill [268]. This algorithm was much more computationally intensive than the TNC Σ-CPHD filter. The authors applied three filters—MCMC, TNC Σ-CPHD, and TNC ΣPHD—to two applications: multitarget tracking using passive-acoustic sensors, and multitarget detection and tracking using RF tomography sensor arrays. The results for each are described in turn.

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669

TNC Σ-CPHD Filter: Application to Passive Acoustics

In these simulations, targets are observed by a field of 25 omnidirectional passive acoustic-amplitude sensors, which are located at the nodes of a rectangular grid. Four acoustically active targets follow trajectories that are somewhat separated and slightly curvilinear. The performance of the three filters was compared, for 100 Monte Carlo trials, using the OSPA metric (Section 6.2.2). The authors reported that the TNC Σ-CPHD filter outperformed both the MCMC algorithm (which had roughly two times more OSPA error) and the TNC Σ-PHD filter (roughly three times more OSPA error). The TNC Σ-CPHD filter was about 87 times faster than the MCMC filter and about 27 times faster than the TNC Σ-PHD filter. 19.4.4.2

TNC Σ-CPHD Filter: Application to Radio-Frequency Tomography

RF tomography was described in Section 19.1.1.4. In these simulations, 24 RF receiver/transmitter sensors are located on the perimeter of the monitoring region, resulting in a total of 276 unique bidirectional sensor-to-sensor links. The tomography array observes four targets moving along slightly curvilinear trajectories, with the first pair approaching and diverging, and the second pair approaching and diverging. As in the passive-acoustic application, the TNC Σ-CPHD filter outperformed both the MCMC algorithm (which had roughly two times more OSPA error) and the TNC Σ-PHD filter (roughly three times more OSPA error). However, in all cases the OSPA error was roughly half that of the passive-acoustic application. This is because the measurement dimension and the signal-to-noise ratio were much higher. The TNC Σ-CPHD filter was about 30 times faster than the MCMC filter and about 14 times faster than the TNC Σ-PHD filter.

Chapter 20 RFS Filters for Pixelized Images 20.1

INTRODUCTION

This chapter addresses the problem of detecting and tracking multiple targets that move within a time-series of pixelized images. Examples of sensors that produce such data include electro-optical (EO) cameras, infrared cameras, synthetic aperture radars (SARs), and inverse SARs (ISARs). Conventional algorithms for processing such data are typically based on a detection paradigm. For example, preprocessing algorithms extract “blobs” from the images using techniques such as thresholding (which identifies higher-intensity pixels) or edge detection (which identifies circumscribed regions in an image). Features, such as the centroids of the blobs, are then used as the input measurements to some kind of multitarget tracking algorithm. However, this approach can waste potentially valuable information. When signal-to-noise ratio (SNR) is small, feature-detection algorithms will often eliminate target pixels along with clutter pixels; and clutter pixels can be spuriously declared to be target pixels. When SNR is very low, targets will not even be identifiable in any given image, thus making feature extraction ineffective. Consequently, performance could be improved if target detection, and target tracking were performed jointly in a single, unified nonlinear algorithm, using entire images rather than the features extracted from them. The purpose of this chapter is to describe innovative RFS filters, proposed by B.-N. Vo and B.-T. Vo, that accomplish this task. The most notable of these filters is the image-observation multitarget multi-Bernoulli (IO-MeMBer) filter, which will be described in Section 20.5. It has been shown to outperform the previously best track-before-detect filters in this application; and has been successfully applied to

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real grayscale and color video recordings of complex, rapidly evolving multitarget scenarios such as hockey games (see Section 20.6). The discussion will be at a relatively high level. For greater detail, see the original papers cited in the chapter. 20.1.1

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter: • All of the RFS filters in this chapter are based on B.-N. Vo’s imageobservation (IO) measurement model (Section 20.2). It models targetilluminated regions of an image as adaptively parametrized, area-filling templates. • These filters can be applied to either grayscale or color images. • These RFS filters are based on the assumption that targets have physical extent in two dimensions. In particular, it is assumed that targets cannot overlap or pass over or through each other. • These RFS filters are exact closed-form solutions of the multitarget Bayes filter. They require no approximations other than (a) those in the modeling assumptions; and (b) the simplifying assumption that the predicted-target RFS has a specific form (multi-Bernoulli, i.i.d.c., or Poisson). • Because the IO measurement model is usually highly nonlinear, these RFS filters must typically be implemented using sequential Monte Carlo (SMC, also known as particle) techniques. • The IO-MeMBer filter has been shown to outperform the best previously known track-before-detect filter, the histogram-PMHT (Section 20.6.1). • Because of the difficulties associated with state estimation for multitarget SMC filters, the IO-PHD filter and IO-CPHD filter are computationally more demanding than the IO multi-Bernoulli filter. As a consequence, they are probably of lesser practical significance. 20.1.2

Organization of the Chapter

The chapter is organized as follows:

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1. Section 20.2: B.-N. Vo’s image-observation (IO) multitarget measurement model. 2. Section 20.3: An approximate multitarget motion model for the IO model. 3. Section 20.4: The IO-CPHD filter. 4. Section 20.5: The IO-MeMBer filter. 5. Section 20.6: Implementations of the IO-MeMBer filter.

20.2

THE IO MULTITARGET MEASUREMENT MODEL

The measurement model described in this section is due to B.-N. Vo [315], [316]. Suppose that an imaging sensor presents its measurement as an m1 × m2 array of pixels. The pixel measurement can be a real number representing a gray-scale intensity; or a three-dimensional vector (R, G, B)T , representing the intensities of the red, green, and blue color channels. Whatever their form, the pixel measurements are packaged into an image-vector z = (z1,1 , ..., zm1 ,m2 )T = (z 1 , ..., z M )T

(20.1)

of dimension M = m1 m2 . Assume: • Targets have a physical extent. The state-vector x of a target can include, besides position and velocity variables, variables that specify target shape, size, orientation, and identity-class. • Some pixels are “illuminated” by the target, whereas all other pixels are “background.” Let Jk+1 (x) be the set of indices of all pixels that are illuminated by a target with state x. • Targets move on a surface and thus, because they have physical extent, cannot overlap or pass over each other (see Figure 20.1). Thus if x1 and x2 are the states of two distinct targets then Jk+1 (x1 ) ∩ Jk+1 (x2 ) = ∅.

(20.2)

j • fk+1 (z|x) is the probability density of z in the jth pixel, given that it is target-illuminated; and it assumed to be known a priori.

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Figure 20.1 A schematic diagram of the image-observation (IO) measurement model. Shown are “blobs” consisting of pixels illuminated by targets or landmarks. It is assumed that targets—and therefore blobs—cannot physically overlap each other. j • fk+1 (z) is the probability density of z in the jth pixel, given that it is not target-illuminated; and it is assumed to be known a priori.

• Pixels are conditionally independent with respect to target state. Given this, the sensor likelihood function of a target-containing image is 

LZ (x) = fk+1 (z|x) = 

∏ j ∈J / k+1 (x)



j fk+1 (z j ) 

∏ j∈Jk+1 (x)



j fk+1 (z j |x) (20.3)

whereas the likelihood of an image with no targets is

ℓz =

m ∏ j=1

j fk+1 (z j ).

(20.4)

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Assume that a multitarget state X = {x1 , ..., xn } with |X| = n is “physically realizable” in the sense that, for all i, l = 1, ..., n with i ̸= l, (20.5)

Jk+1 (xi ) ∩ Jk+1 (xl ) = ∅. Then the multitarget likelihood function is ([316], Eq. (3)): fk+1 (z|X) = ℓz · χX z

(20.6)

where the power-functional notation χX was defined in (3.5) and where ∏

χz (x) =

j∈Jk+1 (x)

j fk+1 (z j |x) j fk+1 (z j )

(20.7)

.

If X = ∅ then fk+1 (z|X) = ℓz . Otherwise, let (20.8)

Jk+1 (X) = Jk+1 (x1 ) ⊎ ... ⊎ Jk+1 (xn )

be the set of indices of all target-illuminated pixels. Then because of (20.5),    ∏ ∏ j j j fk+1 (z|X) =  fk+1 (z j )  fk+1 (z j |x1 ) (20.9) j ∈J / k+1 (X)



···  =

=





∏ j∈Jk+1 (xn )

M ∏ j=1

ℓz ·

j∈Jk+1 (x1 )

j fk+1 (z j |xn )



j fk+1 (z j ) 

n ∏

n ∏

∏

i=1 j∈Jk+1 (xi )

χz (xi ) = ℓz · χX z .

j fk+1 (z j |xi ) j fk+1 (z j )



 (20.10) (20.11)

i=1

Remark 81 This measurement model is in some respects similar to that employed for SAR images in the DARPA MSTAR program of the 1990s [118]. The algorithms developed under MSTAR addressed a different problem than that considered here: automatic target recognition (ATR) of single, motionless ground targets. However, the MSTAR algorithms were also based on the presumptions that (1) the probability distribution of a target in any pixel is known (because of radar physics using CAD modeling of the targets), and (2) pixels are statistically independent.

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20.3

IO MOTION MODEL

Because targets cannot overlap, their motion is not statistically independent and thus the standard multitarget motion model of Section 7.4 is not strictly applicable. Assume, however, that targets are sufficiently small that they do not occupy large regions of any given image. Then the motion model for the conventional multiBernoulli filter (see Section 13.4.2) can be adopted as an approximation. Consequently, the time-update for the IO-CPHD filter is the same as that for the classical CPHD filter (Section 8.5.2). Thus we need only specify the measurement-update equations.

20.4

IO-CPHD FILTER

We are given the predicted PHD Dk+1|k (x) and the predicted cardinality distri∫ bution pk+1|k (n) or predicted p.g.f. Gk+1|k (x). Let Nk+1|k = Dk+1|k (x)dx. Then if the new measurement at time tk+1 is zk+1 , the exact closed-form measurement-updated versions of these are ([316], Eqs. (10,11)): • Measurement updates for the p.g.f. and cardinality distribution: Gk+1|k+1 (x)

=

pk+1|k+1 (n)

=

where

∫

1 ϕk = Nk+1|k

Gk+1|k (x · ϕk ) Gk+1|k (ϕk ) ∑

ϕnk · pk+1|k (n) l l≥0 ϕk · pk+1|k (l)

χzk+1 (x) · Dk+1|k (x)dx.

(20.12) (20.13)

(20.14)

• Measurement update for the expected number of targets: (1)

Gk+1|k (ϕk ) Nk+1|k+1 = Gk+1|k (ϕk )

· ϕk .

(20.15)

• Measurement update for the PHD: (1)

Gk+1|k (ϕk )

1 Dk+1|k+1 (x) = Nk+1|k

·

Gk+1|k (ϕk )

· χzk+1 (x) · Dk+1|k (x)

(20.16)

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The PHD filter special case is ([316], Corollary 1): Dk+1|k+1 (x) = χzk+1 (x) · Dk+1|k (x).

(20.17)

These equations are proved in Section K.28. Because the pseudolikelihood χzk+1 (x) is highly nonlinear in general, they must be implemented using sequential Monte Carlo (SMC) techniques.

20.5

IO-MEMBER FILTER

Despite its name, the IO-MeMBer filter is not a variant of the “classical” multiBernoulli filter of Chapter 13 (which is intended for use with the standard multitarget measurement model). Rather, it is a multi-Bernoulli filter intended for use with B.-N. Vo’s image-observation (IO) multitarget measurement model. Also, whereas the classical multi-Bernoulli filter requires additional approximations beyond those required in its modeling assumptions, the IO-MeMBer filter is exact closed-form, in the sense that no further approximations are required beyond those in the modeling assumptions. The time-update for the IO-MeMBer filter is the same as that for the classical multi-Bernoulli filter (Chapter 13). Thus we need only specify the measurementupdate equations. 20.5.1

IO-MeMBer Filter: Measurement Update

We are given the number νk+1|k of predicted tracks, the predicted target-existence k+1|k

k+1|k

probabilities qi , and the track distributions si (x). If the new measurement at time tk+1 is zk+1 then there are νk+1|k+1 = vk+1|k measurementupdated tracks, and the formulas for these are ([316], Eq. (13)): • Measurement update for target-existence probabilities: k+1|k

qi

k+1|k+1

qi

= 1−

where ϕk,i =

∫

k+1|k qi

· ϕk,i

(20.18)

k+1|k

+ qi

· ϕk,i

χzk+1 (x) · sik+1|k (x)dx.

(20.19)

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• Measurement update for track distributions: k+1|k k+1|k+1

si

(x) =

χzk+1 (x) · si ϕk,i

(x) (20.20)

.

These equations are proved in Section K.29. Once again, because the pseudolikelihood χzk+1 (x) is usually highly nonlinear, practical implementation usually requires SMC techniques. 20.5.2

IO-MeMBer Filter: Track Merging

It is possible that the state estimates corresponding to two tracks can be close enough together that the tracks overlap, thus violating the basic assumption of the IO model. When this occurs, such tracks must be merged. Suppose that the ith and lth tracks are to be merged. The merged existence probability and track distribution are k+1|k+1

qi,l

k+1|k+1

=

qi

=

∫

k+1|k+1

k+1|k+1

si

k+1|k+1

si,l

(x)

k+1|k+1

− qi

+ ql

k+1|k+1

· ql

(20.21)

k+1|k+1

(x) · sl

k+1|k+1 si (y)

·

(x) .

(20.22)

k+1|k+1 sl (y)dy

The first equation is due to the following reasoning. The probabilities that the ith k+1|k+1 k+1|k+1 and lth tracks do not exist are, respectively, 1 − qi and 1 − ql . k+1|k+1 k+1|k+1 Thus the probability that neither exists is (1 − qi )(1 − ql ). Thus the probability that at least one exists is given by (20.21). However, (20.22) is the Bayes parallel combination (see (22.139) or (10.71)) of the ith and lth track distributions, assuming an (improper) uniform prior distribution. 20.5.3

IO-MeMBer Filter: Multitarget State Estimation

Multitarget state estimation is accomplished in the same manner as with the CBMeMBer filter (Section 13.4.5). 20.5.4

IO-MeMBer Filter: Track Management

Track management is accomplished as described in Section 13.4.6 for the CBMeMBer filter.

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20.6

679

IMPLEMENTATIONS OF IO-MEMBER FILTERS

Since the IO-CPHD filter equations must be implemented using SMC methods, multitarget state estimation is—as with the classical CPHD filter—conceptually and computationally demanding. However—and as with the classical multi-Bernoulli filter—multitarget state estimation for the IO-MeMBer filter is easy. Consequently, to date the only implementations of IO-RFS filters have been of IO-MeMBer filters, all of them conducted by Vo, Vo and their associates. The purpose of this section is to briefly summarize these implementations. 20.6.1

Track-Before-Detect (TBD) in Image Data

A TBD algorithm is one that accepts “raw” image data—that is, the pixelized image itself without any prior preprocessing [58], [60]. Using the notation of Section 20.2, a common measurement model used in TBD studies is the following ([316], Eq. (24)): j j 2 fk+1 (z|x) = Nσk+1 (z − ηk+1 (x))

if

j ∈ Jk+1 (x)

(20.23)

and j 2 fk+1 (z) = Nσk+1 (z)

if

j∈ / Jk+1 (x)

(20.24)

j 2 where ηk+1 (x) is the measurement function in the jth pixel, and where σk+1 is the variance of the background noise (assumed to be the same in all pixels). Vo, Vo, Pham, and Suter compared the IO-MeMBer filter with the best current TBD filter (according to the then most recent study [58]), the “histogram probabilistic multihypothesis tracker” (H-PMHT).1 Images were assumed to be of a fixed square surveillance region with sides of length L. The measurement function was chosen to be the point-spread function defined by ([316], Eq. (25)), ( ) ∆x ∆y Ik+1 (∆x a − x)2 + (∆y b − y)2 j ηk+1 (x, y, vx , vy ) = · exp − , (20.25) 2πσh2 2σh2

where ∆x and ∆y are the horizontal and vertical sizes of the pixels (regarded as cells of the surveillance region), Ik+1 is the source intensity, and σh2 is the 1

Unlike RFS filters or conventional MHT, operation of the H-PMHT presumes that the number of targets is known a priori. Thus target number must first be heuristically estimated before H-PMHT can be applied. Given this, PMHT uses Dempster’s expectation-maximization (EM) algorithm to perform measurement-to-track association.

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blurring factor. The target template Jk+1 (x, y) was chosen to be the 4 × 4 pixel square region with center closest to (x, y). Two simulation scenarios were considered [316]. Since H-PMHT requires that target number be known, the first simulation involved four maneuvering targets of fixed and known numbers. Since the IO-MeMBer filter does not require this a priori knowledge, the second simulation involved four appearing and disappearing targets, with target at any time unknown to the trackers. In the simulations, the following parameters were assumed: L = 45m, ∆x = ∆y = 1m, Ik+1 = 30, 1 ≤ a, b ≤ 45, σ 2 = 1m2 , σh2 = 1. In the first (known target number) simulation, the IO-MeMBer filter presumed that no targets appear or disappear, but was otherwise ignorant of the initial target number. The IO-MeMBer filter used 1000 particles per track. Because the HPMHT was given the initial target number a priori, it was initially able to track the targets better than the IO-MeMBer filter. Thereafter, however, the H-PMHT performed considerably worse with respect to all measures. In particular, localization errors increased rapidly with time for the H-PMHT, whereas those for the IO-MeMBer filter were essentially constant. Moreover, the processing time for the H-PMHT algorithm was typically larger than that for the IO-MeMBer filter, and increased rapidly with improving image resolution (decreasing number of pixels per meter). Processing time for the IO-MeMBer was essentially flat with respect to improving resolution. In the second (unknown target number) simulation, only the IO-MeMBer filter was tested, using 1000 Monte Carlo trials. Vo et al. concluded that the IOMeMBer “is able to initiate, maintain, terminate and consequently estimate all target tracks satisfactorily, although there are occasional delays in the initiation and termination of tracks.” The authors also verified that the IO-MeMBer filter performs poorly when its primary modeling assumption—that targets have physical extent and thus cannot overlap—is violated. 20.6.2

Tracking in Color Videos

Hoseinnezhad, Vo, Vo, and Suter have applied the IO-MeMBer filter to real color video. Their first approach successfully performed in rapidly evolving multitarget scenarios such as hockey and football (soccer) games, but also required a priori information about the visual appearance of the targets [112]. Hoseinnezhad, Vo, and Vo subsequently removed this restriction, using kernel density estimation to learn a

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model of the background, which was then subtracted from the images to expose the foreground targets [115], [113]. It is this latter work that shall be described. The fundamental difficulty in applying the IO-MeMBer filter to real data is determining the form of the background factor ℓz and the target-related factor χz (x) in (20.5). Hoseinnezhad et al. designed an algorithm that gradually learns and updates the background function ℓz , which is then used to generate a grayscale foreground image function χz (x) for the targets in the scene. j j First, the RGB (red-green-blue) color vector (Rk+1 , Gjk+1 , Bk+1 )T is j j j converted to a chromaticity (rgI) color vector (rk+1 , gk+1 , Ik+1 )T with 0 ≤ j j j rk+1 , gk+1 , Ik+1 ≤ 1: j Ik+1

=

j rk+1

=

j gk+1

=

) 1 ( j j Rk+1 + Gjk+1 + Bk+1 256 j Rk+1 j 256Ik+1

Gjk+1

(20.26) (20.27)

(20.28)

j 256Ik+1

where the denominator applies to eight-bit color quantization. The reason for this conversion is the fact that the rgI representation is more robust with respect to ambient variations in light (for example, those due to shadows). The background is estimated over a sliding time-window (pushdown “stack”) of rgI vectors, which has fixed length N0 . The initial stack has the form 0, 2K0 , ...,

N0 − 1 K0 N0

(20.29)

where K0 determines the maximum length of time for which any target is expected to be stationary. At each time-step, the newest image is added to the top of the stack and the oldest image is deleted from the bottom. Second, the rgI vectors are subjected to a kernel estimator to produce grayscale pixel values, normalized to be in [0, 1], for the background ([115], Eq. (10)): j zk+1

( ) N0 −1 ∏ (ojk+1 − ojk−K0 l+1 )2 1 ∑ = exp − . 2 N0 2σo,k+1 l=1 o=r,g,I

(20.30)

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Here, the parameters 0 < σr,k+1 , σg,k+1 , σI,k+1 ≤ 1 are defined by j σo,k+1 = median |ojk+1 − ojk−K0 l+1 |

(20.31)

1≤l≤N0 −1

and are MAD (median absolute deviation) kernel-estimation bandwidths for the rgI colors. Also, the following notational convention is employed:

ojk+1

 j  rk+1 = gj  k+1 j Ik+1

if if if

o=r o=g . o=I

(20.32)

Third, let X = {x1 , ..., xn } with |X| = n be the targets in the scene and let z¯i,k+1 =

1 Mi

∑

j zk+1

(20.33)

j∈Tk+1 (xi )

be the averaged grayscale value for the ith target, where Mi = |Tk+1 (xi )|. Then Hoseinnezhad et al. show that ([115], Eq. (20))  M ∑ 1 zj  (20.34) ζB · exp  M · δB j=1 k+1 ( ) ( ) Mi · (1 − z¯i,k+1 ) z¯i,k+1 exp · ζF · exp − (20.35) M · δB δF 

ℓzk+1

=

χzk+1 (xi )

=

where δB , δF are control parameters and ζB , ζF are normalization factors. Hoseinnezhad et al. conducted experiments with moving people in three video sequences from a test sequence called the CAVIAR benchmark data set. Target state-vectors were assumed to have the form (x, y, lx , ly , vx , vy ) where x, y is the centroid, lx , ly are the width and height, and vx , vy the velocity components. The first video shows two persons who enter and then leave a laboratory. The second shows people walking in a shopping center and entering and leaving a shop. The third video shows four people entering and leaving a lobby. The authors report that “with a comparable computational cost, our method outperforms competitive and similar methods in terms of accuracy, especially for a relatively large number of targets” ([113], Section V).

RFS Filters for Pixelized Images

20.6.3

683

Tracking Road-Constrained Targets

Wong, Vo, and Vo report the application of the IO-MeMBer filter to the problem of tracking ground targets moving on road networks [325], using a trapezoidal road model described by Vo, See, Ma, and Ng in [303]. The algorithm was tested in a simulation involving two vertical and two horizontal roads intersecting at four points. Target motion was modeled using a road-constrained constant-turn model. Observations were 500 × 500 pixelized images, with each pixel corresponding to a square observation region 8m on a side. Four appearing and disappearing targets moved along the two vertical roads and top horizontal road. The authors reported acceptable performance, given the IO-MeMBer filters’s modeling assumptions (specifically, the fact that targets are assumed to not overlap).

Chapter 21 RFS Filters for Cluster-Type Targets 21.1

INTRODUCTION

Recall the multitarget measurement model defined in (5.21): all measurements

target-generated

target-generated

???? Σ

? ?? ? Υ(x1 )

? ?? ? Υ(xn )

=

∪... ∪

not target-generated

∪

???? C

.

(21.1)

There, extended targets and group targets were defined to be targets that (usually) generate multiple measurements. That is, the number |Υ(x)| of measurements generated by a (group or extended) target with state x can be arbitrary. More specifically: 1. Extended targets: Each measurement originates with a single physical target. However, this target can generate multiple measurements because the target is, for example, close enough to the sensor that multiple measurements are generated by scatterers on the target’s surface. (a) The state of an extended target—hereafter denoted as ˚ x—can include centroid, centroidal velocity, target type, and target-shape parameters. (b) Because extended targets have a contiguous physical shape, they cannot physically overlap and can occlude each other. (c) Their measurements Υ(˚ x) depend only on the state ˚ x of the extended target, and ˚ x can therefore be estimated directly from the measurements.

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2. Group targets. In this case, measurements are generated by the autonomous or semi-autonomous point targets belonging to a coordinated group of targets. Collectively, these targets constitute a tactically integrated “meta-target” such as an aircraft sortie or aircraft carrier group. They may also consist of many nested “force levels”—for example, a company consisting of several platoons, each of which consists of several squads. (a) The state of a group target—also denoted as ˚ x—can include centroid, centroidal velocity, formation type, number of targets in the group, and group-shape parameters. (b) Complete specification of a group target, however, also requires determination of the states of its constituent parts (including individual conventional targets). 3. Group targets versus extended targets [283], [274], [318]. Unlike extended targets, group targets can interpenetrate each other and do not occlude each other. Unlike extended targets, the measurements they generate do not depend directly on the state ˚ x. Rather, they depend on the states x ∈ Ξ˚ x of the conventional targets that comprise ˚ x. Estimation of ˚ x is therefore considerably more indirect and difficult than estimation of the state of an extended target. (a) For example, determination that ˚ x is a squad or a platoon largely depends on determination of the size |Ξ(˚ x)| of the group, and perhaps also its shape (spatial distribution). (b) Determination that ˚ x is a brigade or a regiment is more difficult. It additionally requires identification of more ambiguous features, such as the presence of particular target types (such as command and control vehicles); or the existence of coordinated action between subgroups of the group target. Shape (for example, chevron versus laager) can also be an important source of information, including information about tactical intent. 4. Unresolved targets. The standard multitarget measurement model is based on the “no unresolved targets” approximation mentioned in Section 5.5. That is, it is assumed that all targets are close enough to the sensor(s) that no measurement is generated by more than a single target. If targets are further removed, however, this approximation may no longer remain valid.

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687

(a) Consider, for example, a mechanically-scanned radar. For any given azimuth α and elevation θ, a pulse is transmitted and is reflected from the target. The time-delay between the transmitted and received pulse determines a range bin r. If a sufficiently “bright” target is present then a peak occurs in the signature, indicating that a target detection has occurred at the coordinates α, θ, r. (b) However, if many targets are sufficiently close together at that same range, the signature peak will actually be due to several targets rather than only one. Such targets are said to be unresolved. We will address the unresolved-target problem by modeling them as a very simple kind of group target: a point target-cluster. In addition to extended, group, and unresolved targets, a related kind of target will be considered in this chapter: 5. Cluster targets. A cluster target is best described as an extended target whose state ˚ x is not only unknown but unknowable, since the nature of the measurement-generation process is completely unknown. That is, a cluster is manifested only by the measurements it generates, in the form of a cluster of measurements. (a) The very existence of a cluster target must be inferred indirectly, through its manifestation as a “measurement cloud” that is dynamically persistent and coherent over time. (b) Detecting and tracking a cluster target is a dynamic generalization of the static problem of “clustering.” That is, it requires the analysis of a time sequence of measurement sets to determine if the measurements have a dynamically evolving structure, in the form of one or more clusters. The purpose of this chapter is to describe RFS detection and tracking filters for extended targets, cluster targets, and group targets.1 These will be considered each in turn. 1

Note: The IO-MeMBer filter, described in Chapter 20, is based on the presumption that targets have physical extent and thus illuminate multiple pixels in an image. It also can be regarded as an extended-target filter—but not, as in this chapter, for the standard multitarget measurement model.

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21.1.1

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter. First, for extended targets: • There exists a general PHD filter for tracking multiple extended targets— which, however, has combinatorial complexity (Section 21.4.1). • Extended targets can be modeled in three major ways: using an exact rigidbody (ERB) model (Section 21.2.2), an approximate rigid-body (ARB) model (Section 21.2.3), and an approximate Poisson-body (APB) model (Section 21.2.4). • The APB model leads to a PHD filter that has a simpler form than the alternatives, but also has combinatoric computational complexity (Section 21.4.3). • Various approximations—such as the Granstr¨om-Lundquist-Orguner (GLO) approximation—are being devised to render this PHD filter computationally tractable (Section 21.4.3). • Using the GLO approximation, the APB-PHD filter is being implemented using both Gaussian mixtures (Section 21.4.3.4) and Gaussian inverse-Wishart (GIW) mixtures (Section 21.4.3.8). Second, for cluster targets: • A dynamically evolving cluster target can be modeled as a Poisson cluster with unknown intensity function of the form x · θk+1 (z|x), where x is a convenient unknown parameter and where x > 0 is the unknown expected number of targets in the cluster. Consequently, the multiobject likelihood function for a measurement set consisting of multiple dynamically evolving cluster targets has the form: ˚ = e−(x1 +...+xn ) fk+1 (Z|X)

∏

(x1 · θk+1 (z|x1 ) + ... + xn · θk+1 (z|xn ))

z∈Z

(21.2) where ˚ = {(x1 , x1 ), ..., (xn , xn )} X ˚ = n (Section 21.6.1). with |X|

(21.3)

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689

• Using this model, it is possible to devise CPHD and PHD filters capable of detecting and tracking cluster targets in dynamically evolving measurement sets. However, these filters have combinatorial computational complexity (Sections 21.7.1 and 21.7.2). Third, for group targets: • A level-1 group target has two parts: a group state ˚ x that specifies the characteristics of the group target; and a finite set X = {x1 , ..., xn } that consists of the conventional targets that comprise the group target (Section 21.8). • The “natural” state representation of a system of multiple level-1 group targets has the form X = {(˚ x1 , X1 ), ..., (˚ xn , Xn )}

(21.4)

where Xi ̸= ∅ is the nonempty set of targets associated with the group target ˚ xi ; and where the ˚ x1 , ...,˚ xn are distinct (Section 21.8.2). • The natural state representation of group targets is mathematically difficult to work with, and thus must be replaced by a simplified state representation that has the form • X = {(˚ x1 , x1 ), ..., (˚ xν , xν )} (21.5) where the targets associated with the group target ˚ xi are those xj such that •

(˚ xi , xj ) ∈ X (Section 21.8.3.1). • Using this simplified representation, it is possible to devise computationally tractable PHD and CPHD filters for tracking multiple level-1 group targets (Sections 21.9.1 and 21.9.2). • Tracking a single (rather than multiple) group targets requires CPHD and PHD filters that have a special, two-filter form (Sections 21.9.4 and 21.9.3). • Like level-1 group targets, general level-ℓ group targets have a simplified state representation (Section 21.10.1). • This simplified state representation makes it possible, at least in principle, to construct computationally tractable CPHD and PHD filters (Section 21.11). Fourth, for unresolved targets:

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• It is possible to generalize the concept of target number so that it becomes continuous. • In this case, the corresponding multitarget likelihood function is continuous with respect to target number: lim fk+1 (Z|a, x)

=

fk+1 (Z|∅)

(21.6)

=

˚ fk+1 (Z|X)

(21.7)

a↘0

˚ ∪ (a, x)) lim fk+1 (Z|X

a↘0

where ‘lima↘0 ’ denotes the one-sided limit from above. • Given this, an unresolved target can be modeled as a point target-cluster (a, x)—that is, as a cluster of targets colocated at a whose average number is a. • An exact PHD filter for this measurement model can be derived (see Section 21.14). It is combinatoric, since it involves a summation over all partitions of the current measurement set. • Given that the point clusters are not too close together and that the clutter density is not too large, this PHD filter reduces to an approximate filter that has the same computational complexity as the classical PHD filter. 21.1.2

Organization of the Chapter

The chapter is organized as follows: Sections on extended targets: 1. Section 21.2: Measurement models for extended targets. 2. Section 21.3: Bernoulli filters for extended targets. 3. Section 21.4: PHD filters for extended targets. 4. Section 21.5: CPHD filters for extended targets. Sections on cluster targets: 1. Section 21.6: Measurement models for cluster targets. 2. Section 21.7: PHD filters for cluster targets. Sections on group targets:

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1. Section 21.8: Measurement models for level-1 group targets. 2. Section 21.9: PHD and CPHD filters for level-1 group targets. 3. Section 21.10: Measurement models for group targets with an arbitrary number ℓ of levels. 4. Section 21.11: PHD and CPHD filters for group targets with an arbitrary number ℓ of levels. Sections on unresolved targets: 1. Section 21.12: The continuous-cardinality measurement model for unresolved targets. 2. Section 21.13: Motion models for unresolved targets. 3. Section 21.14: The exact unresolved-targets PHD filter—that is, the PHD filter corresponding to the continuous-cardinality model. 4. Section 21.15: An approximation of the exact unresolved-targets PHD filter that has the same computational complexity as the classical PHD filter. 5. Section 21.16: Implementations of the unresolved-target CPHD and PHD filters.

21.2

EXTENDED-TARGET MEASUREMENT MODELS

The purpose of this section is to describe the statistical models necessary for devising RFS filters for extended targets. The section is organized as follows: 1. Section 21.2.1: The statistical representation of extended targets. 2. Section 21.2.2: The exact rigid-body (ERB) model of an extended target—as a finite ensemble of point-scatterers. 3. Section 21.2.3: The approximate rigid-body (ARB) model of an extended target—an approximation of the ERB model, assuming that scatterers are sufficiently well separated. 4. Section 21.2.4: The approximate Poisson-body (APB) model of an extended target—as a continuously distributed ensemble of scatterers.

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Advances in Statistical Multisource-Multitarget Information Fusion

The Statistics of Extended Targets

The modeling of an extended target is simpler than that of a group target. Like a point target, an extended target is characterized by a single state vector, say ˚ x. Unlike a point target, the measurement set Υ(˚ x) associated with ˚ x contains many measurements. There are three possibilities regarding state estimation for extended targets: 1. State-only: One is interested only in estimating the state ˚ x of the extended target—for example, state variables such as target centroid, centroidal velocity, target-shape and target-orientation parameters, and target identity. In this case, ˚ x is the complete specification of the extended target. 2. State and shape: In addition to ˚ x, one is also interested in determining the shape (spatial distribution) of the extended target. This typically amounts to determination of the PHD (intensity function) DΥ(˚ x) (z) of the random measurement set Υ(˚ x) of the extended target. 3. State and scatterers: Suppose that, in addition to ˚ x, one wishes to estimate the states of the scatterers on the extended target. In this case the complete state specification has the form (˚ x, X) where X is the set of scatterers for a particular target orientation and sensor pose. This possibility is more properly regarded as a type of group target detection, tracking, and state estimation, which will be addressed in Sections 21.8 and 21.9. 21.2.2

Exact Rigid-Body (ERB) Model

Reference [179], pp. 427-431, described a model of an extended target as a group of “scatterers” (measurement-generators) located at fixed points on a solid body, as seen from the line-of-sight extending from the sensor to the target. Like a point target, each scatterer generates at most a single measurement, and each measurement is generated by no more than a single scatterer. Thus, mathematically, a single extended target generates measurements as though it were a cluster of point targets. The state of an extended target has the general form ˚ x = (x, y, z, vx , vy , vz , θ, φ, ψ)

(21.8)

where x, y, z are the coordinates of its centroid c, vx , vy , vz are the coordinates of the velocity of c, θ, φ, ψ are body-frame coordinates, and c is the target type.

RFS Filters for Cluster-Type Targets

The known number L states

693

of point-scatterers on the extended target’s surface have x ˘1 + c

,...,

x ˘L + c.

(21.9)

That is, they are fixed offsets from the centroid c of the extended target and (implicitly) depend on target type c. ∗ For any given sensor state x, the sites that are visible to the sensor are specified by a visibility function: ℓ

∗

e (˚ x, x) =

{

1 0

if if

the site x ˘ℓ + c is visible to sensor . otherwise

(21.10)

As an illustration, the visibility function for a given sensor pose can be constructed from a CAD (computer-aided design) model of the oriented surface of a given target type. It follows that the probability of detection of the extended target at the site x ˘ℓ + x is ∗ ∗ ∗ pℓD (˚ x, x) = eℓ (˚ x, x) · pD (˘ xℓ + c, x) (21.11) ∗

where pD (x, x) is the usual probability of detection of x, given that the sensor ∗ has state x. Likewise, the likelihood function at the site x ˘ℓ + x is ∗

∗

ℓ fk+1 (z|˚ x, x) = fk+1 (z|˘ xℓ + c, x)

(21.12)

∗

where fk+1 (z|x, x) is the usual single-target likelihood function for the sensor. The measurement model for a single extended target is just the standard multitarget measurement model, with the scatterers x ˘ℓ + x treated like single point targets. Thus the likelihood function for a single extended target follows from (7.21): 0 fk+1 (Z|˚ x) = fk+1 (Z)

∏ ℓ:θ(ℓ)>0

(

ℓ pℓD (˚ x) · fk+1 (zθ(ℓ) |˚ x) ) ℓ 1 − pD (˚ x) · κk+1 (zθ(ℓ) )

(21.13)

where 0 fk+1 (Z)

κk+1 (Z)

= =

κk+1 (Z)

L ∏

e−λk+1 ·

ℓ=1 κZ k+1

(1 − pℓD (˚ x))

(21.14) (21.15)

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Advances in Statistical Multisource-Multitarget Information Fusion

∗

and where the dependence on the sensor state x has been suppressed for notational clarity. The summation is taken over all associations θ : {1, ..., L} → {0, 1, ..., m} where m = |Z| and where, recall, θ : {1, ..., L} → {0, 1, ..., m} is a function such that θ(ℓ) = θ(ℓ′ ) > 0 implies ℓ = ℓ′ . When there is no clutter, this reduces to the following variant of (7.24): f˜k+1 (Z|˚ x)

=

0 f˜k+1 (Z)

τ (j) τ (j) m ∑∏ pD (˚ x) · fk+1 (z|˚ x)

f˜0 (Z|∅)

=

L ∏

(21.16)

τ (j)

τ

1 − pD (˚ x)

j=1

(1 − pℓD (˚ x))

(21.17)

ℓ=1

where the summation is taken over all injective functions τ : {1, ..., m} → {1, ..., L}. The multitarget likelihood function for multiple ERB-model extended targets ˚ = {˚ ˚ = n (4.17): therefore is, for a state set X x1 , ...,˚ xn } with |X| ∑ ˚ = fk+1 (Z|X) κk+1 (W0 ) · f˜k+1 (W1 |˚ x1 ) · · · f˜k+1 (Wn |˚ xn ) W0 ⊎W1 ⊎...⊎Wm =Z

(21.18) where the summation is taken over all mutually disjoint subsets W0 , W1 , ..., Wm of Z (the null set included) such that W0 ∪ W1 ∪ ... ∪ Wm = Z. 21.2.3

Approximate Rigid-Body (ARB) Model

This is an approximation of the rigid-body scatterer model. The scatterers generate measurements as though they are point targets. Assume that these scatterers are not too close together. Then the approximate multitarget likelihood function of (7.50) applies, and under current assumptions takes the form: ) ( L˚ x ℓ ∏ ∑ pℓD (˚ x) · fk+1 (z|˚ x) ℓ ∼ fk+1 (Z|˚ x) = κk+1 (Z) (21.19) 1 − pD (˚ x) + κk+1 (z) ℓ=1

where κk+1 (Z) = e−λk+1

z∈Z

∏

κk+1 (z)

(21.20)

z∈Z

and where κk+1 (z) is the intensity function of the Poisson clutter RFS; and where L˚ x. x is the number of scatterers for the extended target with state ˚

RFS Filters for Cluster-Type Targets

21.2.4

695

Approximate Poisson-Body (APB) Model

This model is originally due to Gilholm, Godsill, Maskell, and Salmond [93] and was described in [179], pp. 431-432. It is based on the following assumption: • Any extended target is sufficiently far away from the sensor that its measurement set resembles a continuously distributed cluster rather than—as with the ERB and ARB models—a spatially structured point pattern. In this approximation, the measurements generated by an extended target with state ˚ x are assumed to be distributed according to a spatial distribution ϕz (˚ x) abbr. = sk+1 (z|˚ x)

(21.21)

and their number is distributed according to a Poisson distribution with parameter γ(˚ x) abbr. = γk+1 (˚ x) > 0.

(21.22)

Thus sk+1 (z|˚ x) can be used to approximately specify the shape, size, and orientation of the extended target. The PHD of the measurement RFS is: µ(z|˚ x) = γ(˚ x) · ϕz (˚ x). 21.2.4.1

(21.23)

APB Model for a Single Extended Target

Given this, the multitarget likelihood function for a single extended target is ∏ x) fk+1 (Z|˚ x) = e−γ(˚ · γ(˚ x)|Z| ϕz (˚ x). (21.24) z∈Z

Now suppose that the extended target is observed in Poisson clutter with clutter rate λk+1 and spatial distribution ck+1 (z) and let γ˜ (˚ x) µ ˜(z|˚ x)

= =

λk+1 + γ(˚ x) λk+1 ck+1 (z) + γ(˚ x) · ϕz (˚ x).

(21.25) (21.26)

Then the multitarget likelihood function for a single extended target in Poisson clutter is ([179], Eqs. (12.221-12.224)): ∏ x) f˜k+1 (Z|˚ x) = e−˜γ (˚ · µ ˜(z|˚ x). (21.27) z∈Z

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21.2.4.2

APB Model for Multiple Extended Targets

˚ = {˚ Suppose that we have multiple extended targets with state set X x1 , ...,˚ xn } ˚ with |X| = n. If the targets generate measurements independently, then the multitarget likelihood is given by ∏ ˚ ˚ = e−˜γ (X) ˚ fk+1 (Z|X) µ ˜(z|X) (21.28) z∈Z

where ˚ γ˜ (X)

=

λk+1 +

∑

(21.29)

γ(˚ x)

˚ ˚ x∈X

˚ µ ˜k+1 (z|X)

=

λk+1 ck+1 (z) +

∑

γ(˚ x) · ϕz (˚ x).

(21.30)

˚ ˚ x∈X

This follows from the fact that the p.g.fl. for multiple independent extended targets with independent clutter is ˚ Gk+1 [g|X]

= =

where

Gκk+1 [g] · Gk+1 [g|˚ x1 ] · · · Gk+1 [g|˚ xn ] (∫ ) ˚ exp (g(z) − 1) · µ ˜k+1 (z|X)dz (

(21.31) (21.32)

)

(21.33)

and where the p.g.fl. for Poisson clutter is ( ) ∫ Gκk+1 [g] = exp λk+1 (g(x) − 1) · ck+1 (z)dz .

(21.34)

Gk+1 [g|˚ x] = exp γ(˚ x)

21.3

∫

(g(z) − 1) · ϕz (˚ x)dz

EXTENDED-TARGET BERNOULLI FILTERS

Ristic and Sherrah [258] and Ristic, Vo, Vo, and Farina ([262], Section IX-C) have addressed the problem of detecting and tracking a single extended target, with unknown shape, using a Bernoulli filter.2 They address extended-target models that 2

The tracking of a single extended target with unknown spatial extent has been addressed by Hongyan Zhu, Chongzhao Han, and Chen Li [111], using PHD filter rather than Bernoulli filter techniques.

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are more general than the APB model of Section 21.2.4—namely, models that are i.i.d.c. of the form ∏ fk+1 [Z|˚ x] = |Z|! · p˚ fk+1 (z|˚ x) (21.35) x (|Z|) z∈Z Z |Z|! · p˚ x (|Z|) · f˚ x

=

(21.36)

abbr. where f˚ x) is the state-dependent spatial distribution of measurex (z) = fk+1 (z|˚ ments generated by the extended target; and where p˚ x (m) is the probability that m measurements will be generated by the extended target. If κk+1 (Z) is the probability distribution of the clutter process, it can be shown that the multitarget likelihood function for the extended target is ˚ = κk+1 (Z) if X ˚ = ∅ and, if X ˚ = {˚ fk+1 (Z|X) x},

fk+1 (Z|{˚ x})

=

κk+1 (Z) 

· p˚ x (0) +

(21.37) 

∑ ∅̸=W ⊆Z

κk+1 (Z − W ) W · |W |! · p˚ . x (|W |) · f˚ x κk+1 (Z)

This leads to measurement-update equations 1 − pk+1|k

1−pk+1|k+1 = 1 − pk+1|k + pk+1|k

∑

∅̸=W ⊆Z

κk+1 (Z−W ) κk+1 (Z)

· |W |! · pk+1 (|W |) · τW (21.38)

and (21.39)

fk+1|k+1 (˚ x) =

pk+1|k · fk+1|k (˚ x) pk+1|k+1 ∑ (Z−W ) W pk+1 (0) + ∅̸=W ⊆Z κk+1 · |W |! · pk+1 (|W |) · f˚ x κk+1 (Z) · ∑ (Z−W ) 1 − pk+1|k + pk+1|k ∅̸=W ⊆Z κk+1 · |W |! · pk+1 (|W |) · τW κk+1 (Z)

where τW =

∫

W f˚ x)d˚ x. x · fk+1|k (˚

(21.40)

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Advances in Statistical Multisource-Multitarget Information Fusion

Extended-Target Bernoulli Filters: Performance

Ristic and Sherrah [258] and Ristic, Vo, Vo, and Farina ([262], Section IX-C) have implemented the extended-target Bernoulli filter using particle techniques, calling it the “BPF-X filter” (“Bernoulli particle filter for extended targets”). They assume that the measurement cardinality distribution p˚ x (m) is a binomial distribution: m p˚ x) · (1 − pD (˚ x))Lk+1 −m x (m) = CLk+1 ,m · pD (˚

(21.41)

where pD (˚ x) abbr. = pD,k+1 (˚ x) is the probability of detection for scatterers on the surface of the extended target (assumed to be constant); where CL,m is the binomial coefficient (2.1); and where Lk+1 is the maximum number of measurements expected to be generated by the extended target at time tk+1 . The value of Lk+1 is estimated at each time-step, independently of the filter recursion ([258], Eq. (27)). They also assume that the extended-target likelihood function abbr. f˚ x) is elliptical in shape. x (z) = fk+1 (z|˚ Ristic et al. tested their filter in two-dimensional simulations, with pD = 0.6 and clutter rate λ = 5. The authors reported that the filter accurately determined target existence or nonexistence; accurately estimated the target trajectory when it does exist; and accurately estimated the size, shape, and orientation of the elliptical target-shape. They also reported that, with the exception of shape-estimation, performance is largely unaffected by errors in the estimation of Lk+1 . They concluded that, if shape-estimation is of no concern, a conventional Bernoulli filter performs as well as the extended-target Bernoulli filter. Ristic et al. also tested the filter using real video images [258]. The goal was to detect and track a moving car as it enters and exits the scene. To address large variations in the car’s velocity, shape, and orientation, the plant noise was assumed to be large. Given this, the authors reported that the extended-target Bernoulli filter accurately determined the presence or nonpresence of the car, and accurately estimated its trajectory (given the large magnitude of the plant noise).

21.4

EXTENDED-TARGET PHD/CPHD FILTERS

The purpose of this section is to describe various PHD and CPHD filters for detecting and tracking extended targets. It is organized as follows: 1. Section 21.4.1: A general PHD filter for extended targets.

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2. Section 21.4.2: Extended-target PHD filters for the exact rigid-body (ERB) model. 3. Section 21.4.3: Extended target PHD filters for the approximate Poissonbody (APB) model. 21.4.1

General Extended-Target PHD Filter

The material in this section is essentially the same as that reported by Swain and Clark [285]. A general PHD filter, valid for arbitrary clutter processes and arbitrary target-measurement processes, was described in Section 8.2. The latter condition includes extended targets, since the number of measurements generated by a target can be arbitrary. Let fk+1 (Z|˚ x) be the likelihood of collecting the measurement set Z, given that an extended target with state ˚ x is present. In particular, let (21.42)

˚ pD (˚ x) = 1 − fk+1 (∅|˚ x)

be the generalized probability of detection (that is, the probability of collecting at least one measurement from ˚ x). Assume that the clutter process is Poisson with clutter intensity function κk+1 (z). Then (8.29) through (8.32) reduce to the following general formula for an extended-target PHD filter with Poisson clutter: ˚k+1|k+1 (˚ D x) = 1−˚ pD (˚ x) + ˚k+1|k (˚ D x)

∑ P⊟Zk+1

ωP

∑

˚ LW (˚ x) κ W + τW

(21.43)

W ∈P

where τW

=

ωP

=

κW

=

∫

˚k+1|k (˚ fk+1 (W |˚ x) · D x)d˚ x ∏ W ∈P (κW + τW ) ∑ ∏ ′ Q ⊟Zk+1 V ∈Q (κV + τV ) { κk+1 (z) if W = {z} . 0 if otherwise

(21.44) (21.45) (21.46)

This follows immediately from (8.29) through (8.32) after noting that the p.g.fl. of the clutter process is Gκk+1 [g] = eκk+1 [g−1] .

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PHD Filter for Extended Targets: ERB Model

The exact rigid-body (ERB) model was described in Section 21.4.2. The likelihood functions for the single-target and multitarget ERB models were given in (21.13) and (21.18), respectively: f˜k+1 (Z|˚ x)

=

∏

0 f˜k+1 (Z)

ℓ:θ(ℓ)>0

˚ fk+1 (Z|X)

=

(

∑

ℓ pℓD (˚ x) · fk+1 (zθ(ℓ) |˚ x) ) ℓ 1 − pD (˚ x) · κk+1 (zθ(ℓ) )

(21.47)

κk+1 (W0 )

(21.48)

W0 ⊎W1 ⊎...⊎Wm =Z

·f˜k+1 (W1 |˚ x1 ) · · · f˜k+1 (Wn |˚ xn ). In principle, the ERB-PHD filter for these models follows directly from (21.43). A major difficulty in practice is, of course, the computationally challenging nature of (21.44). Similar remarks apply to the ERB models of (21.19). However, an ad hoc approximate approach is also possible. One can apply a standard PHD filter to the problem, thus implicitly regarding the scatterers in the ERB model as individual point targets. For a single extended target, the shape of the PHD can be regarded as an intensity map of the extended target; and parameters such as centroid and centroidal velocity can be inferred from it. For multiple extended targets, a clustering algorithm will be required to extract the distinct extended targets from the total PHD. 21.4.3

PHD Filter for Extended Targets: APB Model

The approximate Poisson-Body (APB) model was described in Section 21.2.4. The corresponding PHD filter was proposed by Mahler in 2009 [174]. This “APB-PHD filter” is the subject of this section, which is organized as follows: 1. Section 21.4.3.1: Time update for the APB-PHD filter. 2. Section 21.4.3.2: Measurement update for the APB-PHD filter, which has combinatorial computational complexity. 3. Section 21.4.3.3: Computational complexity reduction for the APB-PHD filter. 4. Section 21.4.3.4: Gaussian-mixture implementation of the APB-PHD filter.

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5. Section 21.4.3.5: Performance results for the APB-PHD filter. 21.4.3.1

APB-PHD Filter: Time Update

As long as the extended targets do not move so close together that they physically overlap, the usual PHD filter time-update equation can be used as an approximation: ˚k+1|k (˚ D x|Z (k) )

21.4.3.2

= ˚ bk+1|k (˚ x) (21.49) ∫ ˚k|k (˚ + ˚ pS (˚ x′ ) · fk+1|k (˚ x|˚ x′ ) · D x′ |Z (k) )d˚ x′ .

APB-PHD Filter: Measurement Update

For the measurement-update, the following models are required: • ϕz (˚ x) abbr. = µk+1 (z|˚ x): spatial distribution for the measurements generated by an extended target with state ˚ x. • ˚ pD (˚ x) abbr. = ˚ pD,k+1 (˚ x): probability of detection for an extended target with state ˚ x. • κ(z) abbr. = λk+1 ck+1 (z): intensity function of the Poisson clutter process. The quantity ˚ pD (˚ x) requires further discussion. From (21.24) we see that the probability of collecting no measurements at all from the extended target is x) fk+1 (∅|˚ x) = e−γ(˚ .

(21.50)

x) Thus the probability of collecting at least one measurement is 1 − e−γ(˚ . Since the APB model already has a built-in probability of detection, why do we also need x) ˚ pD (˚ x)? The reason is that 1 − e−γ(˚ > 0, and so it cannot be used to model complete occlusion of the extended target—whereas this can be accomplished using ˚ pD (˚ x). Thus the effective probability of detection for the extended target is x) ˚ peff. x) = ˚ pD (˚ x) · (1 − e−γ(˚ ), D (˚

(21.51)

x) which reduces to ˚ peff. x) = 1 − e−γ(˚ when ˚ pD (˚ x) = 1. D (˚ Assume that a new measurement set Zk+1 has been collected, and that ˚ (k) ) is Poisson. Then the predicted multi-extended target process fk+1|k (X|Z

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the measurement-update equations for the APB-PHD filter are as follows ([174], Theorem 1): ˚k+1|k+1 (˚ ˚k+1|k (˚ D x|Z (k+1) ) = ˚ LZk+1 (˚ x|Z (k) ) · D x|Z (k) )

(21.52)

where the PHD pseudolikelihood function is ˚ LZk+1 (˚ x|Z (k) )

=

1−˚ peff. x) (21.53) D (˚ x) |W | ∏ ∑ ∑ e−γ(˚ ·˚ pD (˚ x) · γ(˚ x) + ωP ϕz (˚ x) dW P⊟Zk+1

W ∈P

z∈W

where the summation vanishes if Zk+1 = ∅ and, if otherwise, is taken over all partitions P of the measurement set Zk+1 (see Appendix D for a discussion of partitions). Here ∏ W ∈P dW ∏ ωP = ∑ (21.54) Q⊟Zk+1 V ∈Q dV where

dW

=

δ|W |,1 · κ

W

[ ] ∏ −γ |W | ˚ + Dk+1|k e ˚ pD γ ϕz

(21.55)

z∈W

=

δ|W |,1 · κW ) ∫ ( x) e−γ(˚ ·˚ pD)(˚ x) · γ(˚ x)|W | ( ∏ + d˚ x. ϕz (˚ x) ˚k+1|k (˚ · ·D x|Z (k) ) z∈W κ(z)

(21.56)

The measurement-updated expected number of extended targets is ˚k+1|k [1 − ˚ Nk+1|k+1 = D peff. D ]+

∑

ωP · |P|

(21.57)

P⊟Zk+1

where the summation term is the weighted-average number of cells in the partitions of Zk+1 and where ˚k+1|k [1 − ˚ D peff. D ]=

∫

˚k+1|k (˚ (1 − ˚ peff. x)) · D x|Z (k) )d˚ x. D (˚

(21.58)

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Remark 82 Equations (21.52) through (21.55) are slightly different than, but equivalent to, those presented in [174]. Equations (21.52) through (21.55) become those in [174] when we substitute ϕz (˚ x) dW

?→

?→

ϕz (˚ x) κ(z)

(21.59)

[ ] ∏ ϕz −γ |W | ˚ δ|W |,1 + Dk+1|k e ˚ pD γ . κ(z)

(21.60)

z∈W

Example 8 (Well separated extended targets, no clutter) For conceptual clarity assume that ˚ pD (˚ x) = 1 and κ(z) = 0 and γ(˚ x) = γ0 , in which case (21.52) and (21.54) reduce to [ ] ∏ |W | ˚ −γ0 dW = e ·γ · Dk+1|k ϕz (21.61) 0

z∈W

and Dk+1|k+1 (˚ x|Z (k+1) ) ˚k+1|k (˚ D x|Z (k) )

e−γ0 · Dk+1|k (˚ x|Z (k) )

=

+e

−γ0

∑

ωP

(21.62)

∑ γ |W | 0 dW

(

W ∈P

P⊟Zk+1

∏

ϕz (˚ x)

)

z∈W

which, in turn, reduces to ˚k+1|k+1 (˚ D x|Z (k+1) ) ˚k+1|k (˚ D x|Z (k) )

=

e−γ0 ∑

+

P⊟Zk+1

where

(21.63)

ω ˜P

∑ W ∈P

∏

x) z∈W ϕz (˚ [∏ ] ˚ ′ Dk+1|k z′ ∈W ϕz

[ ] ˚k+1|k ∏ D z∈W ϕz ω ˜P = ∑ [∏ ]. ∏ ˚ ′ Q⊟Zk+1 V ∈Q Dk+1|k z′ ∈V ϕz ∏

W ∈P

(21.64)

Also, assume that the n extended targets are well separated. Then the measurements in Zk+1 will segregate into a “natural partition” ˆ 1 , ..., W ˆ n} Pˆ = {W

(21.65)

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ˆ i consists of those measurements generated by the ith extended where the cell W target. Because of the product [ ] ∏ ∏ ˚ Dk+1|k ϕz , W ∈P

z∈W

˚k+1|k+1 (˚ the weight ω ˜ Pˆ of this partition will dominate the value of D x|Z (k+1) ), resulting in ω ˜ Pˆ ∼ 1 and = ˚k+1|k+1 (˚ D x|Z (k+1) )

∼ =

˚k+1|k (˚ e−γ0 · D x|Z (k) ) ∏ n ∑ x) ˆ ϕz (˚ z∈W [∏i ]. + ˚ ′ ϕ ′ i=1 Dk+1|k ˆ z ∈Wi z

(21.66)

˚k+1|k+1 (˚ It follows that a partition will contribute more strongly to D x|Z (k+1) ) when ∏ its cells W consist of measurements that are near each other, in the sense that x) is large. Thus any partition that has a cell containing well separated z∈W ϕz (˚ measurements, is not one that accurately reflects the generation of measurements by extended targets. 21.4.3.3

APB-PHD Filter: The GLO Approximation

Because of the summation over partitions, the computational complexity of (21.52) is combinatorial. However, Granstr¨om, Lundquist, and Orguner have shown that— somewhat surprisingly—major reductions in computational tractability can be achieved [96], [97], [95]. The purpose of this section is to describe the basic elements of this Granstr¨om-Lundquist-Orguner (GLO) approximation ([95], pp. 32723275). The basic concept underlying the GLO approximation derives from Example 8 in the previous section. As noted there, a partition P of Zk+1 will most strongly contribute to the value of the PHD—that is, it will be most “informative”—if each of its cells corresponds to an actual cluster of target-generated measurements. In principle, therefore, we can neglect the terms in (21.54) that correspond to the noninformative partitions. Thus to reduce computational complexity, one should try to look for the nonoverlapping clusters in Zk+1 . Rather than using a conventional clustering algorithm, Granstr¨om et al. proposed what might be called an “n-degrees-of-separation” methodology. It consists of the following steps:

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705

1. Define the “natural” separation thresholds. Let d(z, z′ ) be a Mahalanobis distance on measurements and let di,j = d(zi , zj ) with i ̸= j. Order the di,j by increasing size, and denote this list as δ0 < δ1 < ... < δM where δ0 = 0 and M = 12 m(m − 1). 2. Select the most statistically appropriate thresholds. If z, z′ are measurements generated by the same target, then because d(zi , zj ) is a Mahalanobis distance, it is χ2 -distributed with multiple degrees of freedom. For a given probability P , let δ(P ) = IN V CHI(P ) be the threshold corresponding to P , where ‘IN V CHI’ indicates the inverse cumulative χ2 distribution. Retain only those δ0 , δ1 , ..., δM such that δ(PL ) < δi < δ(PU ) where PL ≤ 0.3 and PU ≥ 0.8 (and where these two numbers were determined empirically so as to achieve good tracking performance). Denote this new list of thresholds as δ˜1 > ... > δ˜M˜ . 3. For each di , construct the corresponding informative partition. Define δ z, z′ ∈ Zk+1 to be “δ-equivalent,” denoted as z ∼ z′ , if there is a sequence w1 , ..., wa ∈ Zk+1 for some a ≥ 1, such that w1 = z and wa = z′ δ and d(wi , wi+1 ) ≤ δ for all i = 1, ..., a − 1. Then ‘∼’ is clearly an equivalence relation on Zk+1 , and its equivalence classes W1 , ..., Wb form a partition of Zk+1 . It can be constructed as follows. Choose some v1 ∈ Zk+1 and let S1 = {v1 }. Augment S1 with all v ∈Zk+1 such that d(v1 , v) ≤ δ, to get S2 ⊇ S1 . If there are no such v, we stop and define Wδ (v1 ) = S1 = {v1 }. Otherwise, further augment S2 with all v that are within distance δ of some element of S2 (if there are any), thus getting S3 ⊇ S2 . Continue until we have constructed an Sc that cannot be further augmented, and denote Wδ (v1 ) = Sc . This is the cluster that contains v1 . Next, let v2 ∈ / Wδ (z1 ) and construct Wδ (v2 ). Continue until we end up with mutually disjoint Wδ (v1 ), Wδ (v2 ), ..., Wδ (vb ) with Wδ (v1 ) ∪ Wδ (v2 ) ∪ ... ∪ Wδ (vb ) = Zk+1 .3 That is, we end up with the informative partition corresponding to the threshold δ. 4. Eliminate the redundant partitions. Let P1 , ..., PM˜ be the informative partitions corresponding to the thresholds δ˜1 , ..., δ˜M˜ . If any of the P1 , ..., PM˜ are identical, eliminate the copies. These steps result in a drastic reduction in the number of partitions of Zk+1 that must be considered in (21.54). 3

Note that if δ = 0 then b = m and the partition is P = {{z1 }, ..., {zm }}. At the other extreme, if δ is larger than any ∆i,j then b = 1 and the partition is P = {Zk+1 }.

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Example 9 ([95], p. 3273). Suppose that there are four targets, each generating the same expected number γ = 20 of measurements. Let the clutter rate be λ = 50. Then m ¯ = 4 · 20 + 50 = 130 is the expected number of measurements in a frame. If |Zk+1 | = m ¯ then there are are approximately 10161 partitions of Zk+1 . Applying Steps 1 through 4, this number is reduced to a mere 27 partitions. Granstr¨om et al. compared this approach to a clustering algorithm called “k-means++” and found that it is much more effective in determining informative partitions. However, they also noted that it results in a PHD filter that tends to underestimate the number of extended targets when one or more of the extended targets are spatially close. The reason for this is as follows. Suppose that two measurement-clusters generated by two extended targets are sufficiently close that they share at least one measurement. Then when Steps 1-4 are applied, the measurements for these two clusters will be combined into a single cell, interpreted as having originated from a single extended target. As a partial remedy, Granstr¨om et al. proposed enhancing Steps 1-4 with a “subpartitioning” procedure that increases partition diversity. Assume that γ(˚ x) = γ. Then, on average, each expected target generates γ measurements and so n extended targets will generate an average of γn measurements. So, if a cell W contains γn measurements, we may conclude that it consists of measurements generated by approximately n targets. More formally, since the measurement processes of the extended targets are independent and Poisson, the probability that n targets will generate i measurements is p(i|n) = e−nγ ·

ni γ i . i!

(21.67)

This is a likelihood function for the event that there are i measurements, given the presence of n targets. Thus Granstr¨om et al. augment Steps 1-4 with the following additional step: 5. Subpartitioning. Assume that γ(˚ x) = γ is constant; and let P be a partition calculated using Steps 1-4; and let W ∈ P . Assume that the spatial extent of W is small enough that, approximately, W contains no clutter measurements. Then n ˆ W = arg max p(|W | |n)

(21.68)

n

is the maximum likelihood estimate of the number of extended targets that would be necessary to generate a measurement set of size |W |. If n ˆW > 1

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then W is too large and it should split into n ˆ W smaller cells. This is accomplished using the “k-means++” algorithm mentioned earlier. Granstr¨om et al. indicate that subpartitioning significantly improves cardinality estimation performance when targets are near each other, and especially when γ is larger: γ = 20 ([95], p. 3277). In a subsequent paper, Granstr¨om and Orguner proposed two additional approximate partitioning methods, which they termed “prediction partitioning” and “EM partitioning” ([101], p. 5664). The predictive approach employs the motion model (in [101], a constant velocity (CV) model) to construct partitions. The EM approach employs the expectation-maximization (EM) algorithm to construct the partitions. 21.4.3.4

APB-PHD Filter: Gaussian Mixture Implementation

Granstr¨om, Lundquist, and Orguner have proposed a Gaussian mixture implementation of the APB-PHD Filter [95]. In addition to the GLO approximation described in the previous subsection, they assume, as in (9.177) and (9.178), k+1|k

pD (˚ x) · NP k+1|k (˚ x −˚ xi

(21.69)

)

i

∼ =

k+1|k

k+1|k

) · NP k+1|k (˚ x −˚ xi

pD (˚ xi

)

i

(1 − pD (˚ x)) · ∼ =

k+1|k NP k+1|k (˚ x −˚ xi ) i

k+1|k

(1 − pD (˚ xi

(21.70)

k+1|k

)) · NP k+1|k (˚ x −˚ xi

)

i

and, more generally, that ([95], Eq. (9)) k+1|k

x) e−γ(˚ · γ(˚ x)n · ˚ pD (˚ x) · NP k+1|k (˚ x −˚ xi

(21.71)

)

i

∼ =

e

k+1|k −γ(˚ xi )

k+1|k n

· γ(˚ xi

k+1|k

) ·˚ pD (˚ xi

k+1|k

) · NP k+1|k (˚ x −˚ xi

).

i

They also assume that the extended-target likelihood function is linear-Gaussian: ϕz (˚ x) = NRk+1 (z − Hk+1˚ x). In this case it becomes possible to express the product ∏ ∏ ϕz (˚ x) = NRk+1 (z − Hk+1˚ x) z∈W

z∈W

(21.72)

(21.73)

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in (21.55) as a linear-Gaussian likelihood function on an expanded measurement space Z|W | . For, if W = {w1 , ..., w|W | } and if measurement vectors are represented as column vectors, then define zW HW

=

T T (w1T , ..., w|W |)

(21.74)

=

T T (Hk+1 , ..., Hk+1 )T

(21.75)

? RW

??

?

|W | times

blkdiag(Rk+1 , ..., Rk+1 ). ? ?? ?

=

(21.76)

|W | times

In this case we can write NRW (zW − HW ˚ x) =

∏

(21.77)

NRk+1 (z − Hk+1˚ x).

z∈W

Given this, (21.52,21.54) become ∑

∑

P⊟Zk+1

W ∈P

˚k+1|k+1 (˚ ˚ND D x|Z (k+1) ) = D x) + k+1|k+1 (˚

˚k+1|k (˚ D x, P, W ) (21.78)

where ˚ND D x) k+1|k+1 (˚

=

˚k+1|k (˚ D x, P, W )

=

˚k+1|k (x|Z (k) ) (1 − ˚ peff. x)) · D D (˚

(21.79)

x) e−γ(˚ ·˚ pD (˚ x) · γ(˚ x)|W | ωP · d ( ) W ∏ ˚k+1|k (˚ · ϕz (˚ x) · D x|Z (k) ).

(21.80)

z∈W

Assume that the predicted PHD for extended targets is a Gaussian mixture: νk+1|k

˚k+1|k (˚ D x|Z (k) ) =

∑

k+1|k

wi

k+1|k

· NP k+1|k (˚ x −˚ xi

(21.81)

).

i

i=1

Then: • Target-nondetection PHD: νk+1|k

˚ND D x) = k+1|k+1 (˚

∑ i=1

k+1|k

(1−˚ peff. x))·wi D (˚

k+1|k

·NP k+1|k (˚ x−˚ xi i

). (21.82)

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• Component PHDs of the target-detection PHD: νk+1|k

∑

˚k+1|k+1 (˚ D x, P, W ) =

k+1|k+1

k+1|k+1

· NP k+1|k+1 (˚ x −˚ xi

wi

) (21.83)

i

i=1

where (see Section K.30) 

k+1|k

ωP · wi k+1|k ) ·e−γ(xi k+1|k ·˚ pD (xi ) k+1|k |W | ·γ(xi ) ·NRW (zW − HW ˚ x) dW

      k+1|k+1

wi k+1|k+1 −1

=

(Pi

k+1|k+1 −1 k+1|k+1 ) ˚ xi

=

(Pi

(Pi (Pi

21.4.3.5

= )

k+1|k −1

)

       (21.84)

−1 T + HW RW HW (21.85)

k+1|k −1 k+1|k ) ˚ xi −1 T +HW RW zW .

(21.86)

Gaussian Mixture APB-PHD Filter: Performance Results

Granstr¨om, et al. implemented the APB-PHD filter using the Gaussian mixture and GLO approximations, and tested it with simulated and real data. In their baseline simulations, the probability of detection, clutter rate, and target-measurement rate were, respectively, pD = 0.99, λ = 10, and γ = 10. They considered three scenarios: • Two extended targets enter from opposing corners and then travel in parallel and close proximity for the remainder. • Two extended targets enter from opposing corners and then cross paths simultaneously at the midpoint. • Two extended targets enter from opposing corners and cross, and then one spawns a third extended target at the same time that a fourth one appears. Using the OSPA distance (Section 6.2.2), the APB-PHD filter was compared with a na¨ıve GM-PHD filter (that is, one that na¨ıvely processes each measurement as though it originated with clutter or a single target). As expected, the APB-PHD filter greatly outperformed the na¨ıve GM-PHD filter. It largely correctly estimated

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the correct number of extended targets, whereas the na¨ıve filter’s estimate was, as expected, wildly incorrect. The authors also conducted an “apples-with-apples” comparison, in which the extended targets generated, on average, only a single measurement each (γ = 1). In this case, the APB-PHD filter’s internal model for target-measurement generation (Poisson) is quite different than the actual measurement-generation process (Bernoulli). As would be expected, the APB-PHD filter was less effective than the standard GM-PHD filter ([95], p. 3278). Next, the authors considered the case when the true target-measurement rate γ is unknown. They empirically inferred that, if γˆ is the assumed rate, then filter performance (that is, cardinality estimation) is good provided that ([95], Eq. (32)) √ √ γˆ − γˆ ≤ γ < γˆ ≤ γˆ . (21.87) The authors additionally considered the case when the two targets generated different numbers of measurements (γ1 = 10 and γ2 = 20). It was observed that filter performance was better if the assumed rate was set to γˆ = 12 (γ1 + γ2 ) ([95], p. 3281). Finally, the APB-PHD filter was tested using measurements collected by a laser rangefinder from four moving human targets. Because the targets could move in front of each other and thus cause occlusions, a nonconstant pD (˚ x) was used. The APB-PHD filter tracking adequately through four out of six occlusion events ([95], p. 3282). 21.4.3.6

APB-PHD Filter: Gaussian Inverse-Wishart (GIW) Implementation

Granstr¨om and Orguner have proposed a generalization of the Gaussian-mixture implementation of the APB-PHD filter, in which target shape is elliptical (two dimensions) or ellipsoidal (three dimensions) but otherwise unknown [101], [147], [99]. Their approach is based on the “random matrix” model for single extended targets originally proposed by Koch [137], [86]. The basic idea is as follows. State of an Extended Target: The state of a single extended target is assumed to have the form ˚ x = (x, E) (21.88) where x is its kinematic state and E is its “extension state.” Here, x is an sd-dimensional vector of the form x = (pT , p˙ T , p ¨ T , ...)T ? ?? ? s

(21.89)

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711

where p denotes position in a d-dimensional Euclidean space, p˙ denotes velocity, and p ¨ denotes acceleration, and so on. Also, E is a d × d positive-definite symmetric matrix, which models the elliptical/ellipsoidal shape of the extended target. Dynamics of an Extended Target: The single-target Markov transition density is assumed to factor as ([137], Eq. (4)) fk+1|k (x, E|x′ , E ′ ) = fk+1|k (x|x′ , E ′ ) · fk+1|k (E|E ′ )

(21.90)

where ([137], Eqs. (24,30), [101], Eqs. (2,3a,3b)) fk+1|k (x|x′ , E ′ ) fk+1|k (E|E ′ )

= =

NQk ⊗E ′ (x − (Fk ⊗ Id )x) Wδk ,E ′ /δk (E)

(21.91) (21.92)

where Wδk ,E ′ /δk (E) is a Wishart distribution (see Appendix G); where Fk = {ϕi,j } is a s × s matrix; where Id is the d × d identity matrix; and where 

ϕ11 Id  .. Fk ⊗ Id =  . ϕs1 Id

··· ···

 ϕ1s Id  ..  . ϕss Id

(21.93)

is the sd × sd matrix defined by the Kronecker product 4 of Fk and Id : 5 According to Koch, this dynamics model has the effect of directing the acceleration of the extended target along the direction of the major axis of the ellipse ([137], p. 1045). 4

Let A = {ai,j } be a m × n matrix and B = {bi,j } a p × q matrix. Then the Kronecker product (also known as tensor product) A ⊗ B is the mp × nq matrix defined by   a11 B · · · a1n B   . . . .. .. A⊗B =   am1 B · · · amn B

5

For example, for s = 3 ([137], Eq. (18)):  1 tk − tk−1 1 Fk+1|k =  0 0 0

Qk+1|k

=

2

Σ · (1 − e

1 (t − tk−1 )2 2 k tk − tk−1 −(tk −tk−1 )/θ

e

−2(tk −tk−1 )/θ



0 )· 0 0

  0 0 0

 0 0 . 1

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Measurement Model for an APB-Model Extended Target: Measurements are assumed to be positions z = p. Presuming the APB measurement model, Granstr¨om and Orguner assume that the measurement spatial distribution ϕz (˚ x) = sk+1 (z|˚ x) of (21.21) has the form ([101], Eq. (11)) (21.94)

ϕz (x, E) = NE (z − (Hk+1 ⊗ Id )x)

where Hk+1 = (h1 , h2 , h3 )T = (1, 0, 0)T and, therefore, Hk+1 ⊗Id is the d×3d matrix defined by Hk+1 ⊗ Id = (h1 Id , h2 Id , h3 Id ) . (21.95) Equivalently, Hk+1 ⊗ Id is the projection operator defined by (Hk+1 ⊗ Id )(pT , p˙ T , p ¨ T , ...)T = p.

(21.96)

˚k|k (x, E|Z (k) ). It is no longer possible The PHD now must have the form D to employ a Gaussian mixture approximation as in (21.81). A Gaussian inverseWishart (GIW) mixture must be used instead: νk+1|k

˚k+1|k (x, E|Z (k) ) = D

∑

k+1|k

wi

k+1|k

· NP k+1|k (x − xi i

) · IWC k+1|k ,ν k+1|k (E). i

i

i=1

(21.97) GIW mixtures are employed because the inverse-Wishart distribution

IWC,ν (E) =

−1 1 det(C)ν/2 · (det E)−(ν+d+1)/2 · e− 2 tr(CE ) · Γd (ν/2)

(21.98)

2νd/2

is the conjugate prior of NE (z − (Hk+1 ⊗ Id )x) when E is assumed to be random.6 In addition to the previous assumptions, additional ones are made, analogous to those imposed in (21.69) through (21.71) ([101], pp. 5560-5661): • Target survival probability is constant: pS (x, E) = pS . • Clutter is Poisson with clutter rate λk+1 and spatial distribution ck+1 (z). • The target appearance PHD is a GIW mixture. 6

Here, ν > d − 1 is the number of degrees of)freedom; C is a positive-definite “scale matrix”; and ( ∏ Γd (ν/2) = π d(d−1)/4 · di=1 Γ ν−i+1 is the multivariate gamma distribution. If E is an 2 inverse-Wishart variate, then E −1 is a Wishart variate in the sense of (G.2) in Appendix G.

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713

• The probability of detection ˚ pD (x, E) abbr. = ˚ pD,k+1 (x, E) can be approximately separated by components: k+1|k

˚ pD (x, E) · NP k+1|k (x − xi

(21.99)

) · IWC k+1|k ,ν k+1|k (E)

i

i

i

k+1|k k+1|k k+1|k ∼ pD (xi , Ci ) · NP k+1|k (x − xi ) · IWC k+1|k ,ν k+1|k (E) = ˚ i

i

i

and k+1|k

(1 − ˚ pD (x, E)) · NP k+1|k (x − xi

(21.100)

)

i

∼ =

·IWC k+1|k ,ν k+1|k (E) i i ( ) k+1|k k+1|k k+1|k 1−˚ pD (xi , Ci ) · NP k+1|k (x − xi ) i

·IWC k+1|k ,ν k+1|k (E). i

i

• The product e−γ(x,E) · γ(x, E)n can be approximately separated by components: k+1|k

e−γ(x,E) · γ(x, E)n · NP k+1|k (x − xi

(21.101)

)

i

·IWC k+1|k ,ν k+1|k (E) i

k+1|k

∼ =

e

−γ(xi

i

k+1|k

,Ci

)

k+1|k

· γ(xi

k+1|k n

, Ci

k+1|k

) · NP k+1|k (x − xi

)

i

·IWC k+1|k ,ν k+1|k (E). i

i

Given this, the APB-PHD filter equations of Sections 21.4.3.1 and 21.4.3.2 can be implemented using the GIW approximation, together with the GLO approximation and its extensions. The equations for this filter are rather complex and are not presented here—see [101], pp. 5661-5663. 21.4.3.7

GIW-Mixture APB-PHD Filter: Performance Results

Using both simulated data and two extended targets, Granstr¨om et al. conducted simulations to assess the performance of the ET-GIW-PHD filter. Clutter was assumed to be uniformly distributed and Poisson with clutter rate λ = 10, with a linear-Gaussian sensor and linear-Gaussian target motion. The results were as follows:

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1. The targets have crossing trajectories: The ET-GIW-PHD filter successfully estimated the number of extended targets and their trajectories, as determined by the OSPA metric (Section 6.2.2). 2. The targets merge into parallel trajectories and then separate: The filter correctly estimated target number, but experienced some degradation in OSPA during the period when the targets moved in parallel. 3. The targets move in parallel and then separate: During the initial period when the targets move in parallel, the filter underestimated target number as 1; but correctly estimated it once the targets separated. The authors attributed this underestimation to the particular form of the time-update for the inverseWishart part of the GIW-mixture components. 4. The targets merge, move in parallel while taking a simultaneous right turn, then separate: The smaller the separation distance, or the faster the target speed during the turn, the more degraded the filter’s performance became. It correctly estimated target number for wider separations and slower speeds, but experienced greater difficulty otherwise. The authors speculated that an IMM version of the algorithm might be able to more effectively address the maneuver. Granstr¨om et al. compared the ET-GIW-PHD filter with the ET–GM-PHD filter of Section 21.4.3.4 using the same experiments as reported in [95]. In the first test, two human subjects move through the area, observed at waist-high level by a laser rangefinder. They repeatedly move toward and away from each other. The ETGIW-PHD filter slightly outperformed the ET-GM-PHD filter, correctly estimating target number at all times. In the second test, four subjects were involved. One was motionless for most of the experiment, the second walked behind the first, thus becoming occluded. The authors reported that the ET-GIW-PHD filter performed much better than the ET-GM-PHD filter, which underestimated target number on two occasions when subjects were spatially close, including the occlusion. The authors argued that this experiment proves that the ability to estimate the target extent also improves the ability to estimate target trajectory and number. Remark 83 At the time of writing two researchers, Zhang Yongquan and HongBing Ji, published a paper claiming to have significantly improved on the results of Granstr¨om et al. [334]. Unfortunately, these results appeared too late to be reported here.

RFS Filters for Cluster-Type Targets

21.4.3.8

715

APB-PHD Filter: Gamma Gaussian Inverse-Wishart (GGIW) Mixture Implementation

Granstr¨om and Orguner have proposed a generalization of the GIW-APB-PHD filter, using mixtures of gamma-Gaussian-inverse-Wishart (GGIW) distributions [100], [99]. In the process, they have also devised methods for modeling the merging and splitting of group targets. The purpose of this section is to briefly describe this work. In the GGIW-mixture approach, the state vector ˚ x = (x, E) is replaced by the state vector ˚ x = (γ, x, E) (21.102) where γ is the unknown measurement-rate of the APB measurement model in (21.22). The reason for the inclusion of γ > 0 as an unknown is, of course, due to the fact that the a priori measurement-rate γ(˚ x) in (21.22) is, in general, actually unknown. Thus if we incorrectly specify γ(˚ x), performance will likely suffer. The inclusion of γ as an additional state variable sidesteps this problem—but at the cost of forcing the PHD filter to estimate an additional state variable using the same information. x) Because of the addition of the new state variable γ, the factor e−γ(˚ in −γ (21.54) must be replaced by e . This suggests that (21.97)—the GIW-mixture approximation of a PHD—must be replaced by a mixture representation of the form ([100], Eq. (57a)) νk+1|k

˚k+1|k (γ, x, E|Z (k) ) D

=

∑

k+1|k

wi

(21.103)

· GAαk+1|k ,β k+1|k (γ) i

i

i=1 k+1|k

·NP k+1|k (x − xi i

) · IWC k+1|k ,ν k+1|k (E) i

i

where the gamma distribution GAα,β (γ) is defined by GAα,β (γ) =

βα · γ α−1 · e−βγ Γ(α)

(21.104)

and where α > 0 is the shape parameter; β > 0 is the inverse scale parameter; and Γ(x) is the gamma function. The gamma distribution is closed under multiplication since it satisfies the following identity:

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Advances in Statistical Multisource-Multitarget Information Fusion

GAα1 ,β1 (γ) · GAα2 ,β2 (γ)

=

β1α1 β2α2 · (21.105) (β1 + β2 )α1 +α2 −1 Γ(α1 + α2 − 1) · · GAα1 +α2 −1,β1 +β2 (γ). Γ(α1 ) · Γ(α2 )

Employing various approximations, Granstr¨om et al. show how to implement the APB-PHD filter using GGIW mixtures. Two additional innovations occur in the splitting or spawning model (prediction step) and the merging model (following the correction step). 21.4.3.9

GGIW-Mixture APB-PHD Filter: Performance Results

Granstr¨om et al. have conducted simulations to assess the performance of the ETGGIW-PHD filter, with emphasis on merging and splitting of extended targets. Clutter is assumed to be uniform and Poisson with clutter rate λ = 10. Two targets with linear-Gaussian motion are observed by a linear-Gaussian sensor. The following simulations were considered: 1. The targets merge: When the two extended targets were spatially close and moving in the same direction, they were successfully combined into a single, larger, extended target. 2. The targets split, or a single target spawns a new target: If the spawning model is included, a spawning or splitting event could be detected earlier (but at the cost of increased computational complexity). 3. One target occludes the other: The two targets were estimated as being a single target during the occlusion, and then as two following the occlusion.

21.5

EXTENDED-TARGET CPHD FILTER: APB MODEL

Two groups of researchers have simultaneously and independently claimed generalizations of the APB-PHD filter of Section 21.4.3 to CPHD filters: Orguner, Lundquist, and Granstr¨om [226], [227], [147]; and Feng Lian, Chongzhao Han, Weifeng Liu, Jing Liu, and Jian Sun [85]. These filters will here be called “APB-CPHD filters” to emphasize their dependence upon the APB model of an extended target. In [85], Feng et al.

RFS Filters for Cluster-Type Targets

717

implemented their algorithm assuming that the clutter rate is small and that the extended targets are separated. Orguner et al. have implemented their CPHD filter in exact closed form using two different approaches: Gaussian mixtures [226], [227] and gammaGaussian-inverse-Wishart (GGIW) mixtures [147]. They also employed the GLO approximation of Section 21.4.3.3. The purpose of this section is to briefly describe these implementations. It is organized as follows: 1. Section 21.5.1: Theory of the APB-CPHD filter for extended targets. 2. Section 21.5.2: Performance results for Orguner et al.’s Gaussian-mixture implementation of the APB-CPHD filter. 3. Section 21.5.3: Performance results for Orguner et al.’s gamma-Gaussian inverse–Wishart mixture implementation of the APB-CPHD filter. 4. Section 21.5.4: Performance results for Lian et al.’s particle implementation of the APB-CPHD filter. 21.5.1

APB-CPHD Filter: Theory

The measurement-update equations for the APB-CPHD filter are derived using the finite-set statistics theoretical approach described in Section 5.10.3 ([85]; [226], Section III; [227]). First, the p.g.fl. of the target-measurement process for an extended target is assumed to be i.i.d.c. (in particular, Poisson): Tg (˚ x) abbr. = Gk+1 [g|˚ x] = G˚ x (f˚ x [g])

(21.106)

abbr. where f˚ x) is the spatial distribution of the measurements, and x (z) = fk+1 (z|˚ abbr. G˚ x) is the p.g.f. of the probability distribution on the number x (z) = Gk+1 (z|˚ of target-generated measurements. Similarly, the clutter process is assumed to be i.i.d.c.: κg abbr. = Gκk+1 [g] = Gκk+1 (ck+1 [g]) (21.107)

where ck+1 (z) is the clutter spatial distribution and Gκk+1 (z) is the p.g.f. of the probability distribution on the number of clutter measurements. The p.g.fl. for the total measurement process is, therefore, ˚ ˚ = κg · TgX Gk+1 [g|X] .

(21.108)

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Advances in Statistical Multisource-Multitarget Information Fusion

Given this, the posterior p.g.fl. Gk+1|k+1 [˚ h] is determined from (5.58),

Gk+1|k+1 [˚ h] =

δF ˚ δZk+1 [0, h] δF δZk+1 [0, 1]

(21.109)

where F [g, ˚ h]

= =

∫

˚ ˚ ˚ · f˚k+1|k (X)δ ˚ X ˚ hX · Gk+1 [g|X] ( ) ˚k+1|k ˚ κg · G sk+1|k [˚ hTg ]

(21.110) (21.111)

˚ ˚ is i.i.d.c.: f˚k+1|k (X) ˚ = |X|! ˚ ·˚ ˚ ·˚ and where f˚k+1|k (X) pk+1|k (|X|) sX k+1|k . The formula for Gk+1|k+1 [˚ h] is then derived using the product rule and Clark’s general chain rule for functional derivatives, (3.70) and (3.91). This formula involves two combinatorial summations: one over all subsets of Zk+1 and one over all partitions of Zk+1 . The posterior cardinality distribution and posterior PHD are then derived using (4.66), (4.67), and (4.75). The derivation is essentially the same as that for the general PHD filter of Section 8.2.

21.5.2

Gaussian Mixture APB-CPHD Filter: Performance

Orguner et al. implemented the APB-CPHD filter using Gaussian mixture techniques and the GLO approximation of Section 21.4.3.3. They compared it to the APB-PHD filter using real laser range-finder data [226]. In their experiment, at most two human subjects were present, observed by a laser rangefinder whose beam was directed at waist level. The first subject entered the surveillance area, moved to its center, and then stood still. The second subject entered subsequently, moved behind the first subject (thereby becoming occluded), and finally left the surveillance area. Ground truth was not available, but the number of subjects could be determined from the measurements. The authors reported that both filters correctly estimated target number at all times, though for the APB-PHD filter, the expected number Nk|k of targets experienced some variation, especially during the occlusion. In a second experiment, the effective probability of detection was lowered from 0.99 to 0.7. In this case, the APB-CPHD filter still correctly estimated target number. The APB-PHD filter, however, estimated target number to be 3 rather than

RFS Filters for Cluster-Type Targets

719

2 during the occlusion. Its value of Nk|k also exhibited a notable upward bias after the second subject entered the scene. As the effective probability of detection was lowered further, the performance of the APB-CPHD filter began to deteriorate, though less so than that of the APB-PHD filter. 21.5.3

Gamma Gaussian Inverse-Wishart APB-CPHD Filter: Performance

As in the paper [226], Lundquist, Granstr¨om, and Orguner devised a version of the APB-CPHD filter that incorporates the GGIW mixture techniques described in Section 21.4.3.8. They conducted simulations comparing the performance of the GGIW-APB-CPHD with that of the GGIW-APB-PHD filter: 1. Four Targets Appearing and Disappearing at Different Times: The scenario was run for probabilities of detection pD = 0.80 and pD = 0.99 and for clutter rates λ = 5 and λ = 30. The GGIW-APB-CPHD filter correctly determined target number in all four possible scenarios, while the performance of the GGIW-APB-PHD filter became steadily worse as pD became smaller and λ larger. 2. Two Targets Converging, Moving in Parallel, Then Diverging: In this case pD = 0.99 and λ = 10. During the period of parallel motion, the performance of the GGIW-APB-CPHD was good, whereas that of the GGIWAPB-PHD filter steadily deteriorated with time. 21.5.4

APB-CPHD Filter of Lian et al.: Performance

Lian et al. implemented both the APB-PHD filter and the APB-CPHD filter using particle methods [85]. In their two-dimensional simulations, a range-bearing sensor observed four moving and appearing and disappearing extended targets, which followed curvilinear trajectories. The sensor probability of detection was 0.95, and clutter was uniformly distributed Poisson with clutter rate λ = 50. The authors reported that, in 300 Monte Carlo runs, both filters adequately estimated the number of extended targets, with the CPHD filter exhibiting significantly smaller variances. Similarly, when compared using the OSPA metric (Section 6.2.2), the CPHD filter exhibited significantly less OSPA error than the PHD filter, while requiring roughly four times more computational time.

720

21.6

Advances in Statistical Multisource-Multitarget Information Fusion

CLUSTER-TARGET MEASUREMENT MODEL

Cluster targets were defined in Section 21.1. The purpose of this and the next section is to describe the statistics and filtering of such targets. Suppose that a single sensor periodically collects measurement vectors z ∈ Z generated by an unknown, time-evolving random process. The output of the sensor is a time sequence Z (k) : Z1 , ...., Zk of measurement sets Z1 , ...., Zk . An unknown number of unknown stochastic entities—extended targets, for example— may be generating the measurements, resulting in a Time evolution of measurementclusters. Our goal is to recursively detect and characterize the shape of those clusters, despite the fact that the nature of the underlying measurement-generation process is completely unknown. In this sense, cluster-target detection and tracking can be regarded as a form of extended-target detection and tracking in which the underlying measurement model is completely unknown and must be inferred, on-the-fly, from the available measurements. If k = 1—that is, if we assume the static case—then this problem is more commonly known as “data clustering,” “data classification,” or “data categorization.” In the case of “hard clustering,” we are to partition Z1 into a disjoint union Z1 = C1 ⊎ ... ⊎ Cγ of an unknown number γ of unknown and mutually disjoint “classes” (“clusters,” “categories”) C1 , ..., Cγ according to some criterion of similarity or closeness. Alternatively, in “soft clustering” the classes C1 , ..., Cγ are regarded as having “fuzzy,” overlapping boundaries. When k = 1, innumerable approaches have been proposed, and they typically presume that the number of clusters in Z1 is known a priori. In 1989, however, Cheeseman proposed a Bayes-optimal approach to soft data clustering, one which estimated the number of clusters, along with estimates of their shapes and relative densities [33], [34]. In 2003, Mahler proposed an extension of Cheeseman’s approach to the dynamic case ([152], [179], pp. 452-458) and then, in 2009, PHD and CPHD filters for approximate data clustering. The purpose of this section is to describe this work. 21.6.1

Likelihood Function for Cluster Targets

Since the state variable of a cluster target is unknowable, we must make some assumptions about both its state and its likelihood function. Specifically, assume that the probability density function for measurement sets Z is a Poisson mixture

RFS Filters for Cluster-Type Targets

distribution: ˚ = e−(x1 +...+xn ) fk+1 (Z|X)

∏

˚ θk+1 (z|X)

721

(21.112)

z∈Z

where ˚ = x1 · θk+1 (z|x1 ) + ... + xn · θk+1 (z|xn ) θk+1 (z|X)

(21.113)

is the intensity function (PHD) of the Poisson process; where θk+1 (z|x) is a family of probability distributions parametrized by x in some parameter space X; and where ˚ = {(x1 , x1 ), ..., (xn , xn )} X (21.114) ˚ = n is a set of pairs ˚ with |X| x = (x, x) with x > 0. The p.g.fl. for this likelihood function is ([179], Eq. (12.371)): ˚ Gk+1 [g|X]

˚

= =

eθk+1 [g−1|X] ex1 ·θk+1 [g−1|x1 ] · · · exn ·θk+1 [g−1|xn ]

(21.115) (21.116)

where ˚ θk+1 [g − 1|X] θk+1 [g − 1|x]

=

∫

˚ (g(z) − 1) · θk+1 (z|X)dz

(21.117)

=

∫

(g(z) − 1) · θk+1 (z|x)dz.

(21.118)

In addition, assume that there is a Markov transition density for the parameters, of the form fk+1|k (x, x|x′ , x′ ). (21.119) Given this, a Bayes-optimal solution to the dynamic data clustering problem is the multiobject Bayes filter ... →

21.6.2

˚ (k) ) f˚k|k (X|Z

→

˚ (k) ) f˚k+1|k (X|Z

→

˚ (k+1) ) f˚k+1|k+1 (X|Z

→ ...

Estimation of Soft Clusters

Cheeseman proposed the following approach for estimating the form of soft clus˚ (k) ) to get ters. Apply a Bayes-optimal multiobject state estimator to f˚k|k (X|Z

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˚k|k = {(ˆ X x1 , x ˆ1 ), ..., (ˆ xnˆ , x ˆnˆ )}

(21.120)

˚k|k | = n with |X ˆ . For j = 1, ..., n ˆ , let ϕj (z) be the degree to which the jth data class contributed to the generation of z. This is given by the fuzzy membership function ([179], Eq. (12.365)) ϕj (z) =

x ˆj · θk+1 (z|ˆ xj ) . x ˆ1 · θk+1 (z|ˆ x1 ) + ... + x ˆnˆ · θk+1 (z|ˆ xnˆ )

(21.121)

That is, the jth data class is specified as a fuzzy subset of the measurement space Z. If ϕj (z) is small, then z is unlikely to be part of the jth cluster; whereas if ϕj (z) is large, then z is very likely part of the jth cluster.

21.7

CLUSTER-TARGET PHD AND CPHD FILTERS

Since the cluster-target multiobject Bayes filter is computationally intractable in general, principled approximations are required. In this section, two approximations of this filter are described: the cluster-target CPHD filter (Section 21.7.1) and its special case, the cluster-target PHD filter (Section 21.7.2). 21.7.1

Cluster-Target CPHD Filter

˚ (k) ) is i.i.d.c.: Assume that the predicted multiobject process f˚k+1|k (X|Z ˚ (k) ) = |X|! ˚ ·˚ ˚ f˚k+1|k (X|Z pk+1|k (|X|)

∏

˚ sk+1|k (x, x)

(21.122)

˚ (x,x)∈X

˚k+1|k (x|Z (k) ) and assume that a new measurement set Zk+1 is collected. Let G (k) and ˚ sk+1|k (x, x|Z ) be the predicted p.g.f. and spatial distribution, respectively. Then the measurement-update equations for the cluster-target CPHD filter are ([154], Theorem 1): • Measurement update for p.g.f.: ˚k+1|k+1 (x|Z G

(k+1)

)=

∑

˚(|P|) (x · ϕk ) · ∏ x|P| · G W ∈P τW k+1|k ∑ ∏ ′ |) (|P ˚ P ′ ⊟Zk+1 Gk+1|k (ϕk ) · W ′ ∈P ′ τW ′ (21.123)

P⊟Zk+1

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723

where the summation is taken over all partitions P of Zk+1 and where ∫ ∫ ∞ ϕk = e−x · x · ˚ sk+1|k (x, x)dxdx (21.124) 0 ( ) ∫ ∫ ∞ ∏ −x |W | τW = e ·x · θk+1 (z|x) (21.125) 0

z∈W

·˚ sk+1|k (x, x)dxdx. • Measurement update for spatial distribution: ˚ sk+1|k+1 (x, x|Z (k+1) ) e−x · Nk+1|k = (21.126) (k) Nk+1|k+1 ˚ sk+1|k (x, x|Z )   ˚(|P|+1) (ϕk ) G k+1|k ∑   ˚(|P|) (ϕk ) G k+1|k · ωP   |W | ∏ ∑ x · z∈W θk+1 (z|x) P⊟Zk+1 + W ∈P τW where ˚(|P|) (ϕk ) · ∏ G W ∈P τW k+1|k ωP = ∑ ∏ ′ |) (|P ˚ ′ P ′ ⊟Zk+1 Gk+1|k (ϕk ) · W ′ ∈P ′ τW

(21.127)

and

=

(21.128)

Nk+1|k+1 ∫ ∫ Nk+1|k

∞

e−x 0

·



∑ P⊟Zk+1

ωP 

˚(|P|+1) (ϕk ) G k+1|k + ˚(|P|) (ϕk ) G k+1|k

∑ x|W | · W ∈P

∏

z∈W

τW

θk+1 (z|x)

 

·˚ sk+1|k (x, x|Z (k) )dxdx. Remark 84 Although these equations were derived in [154], they are a consequence of Clark’s general chain rule, (3.91), with F [h] T˚ h [g](x, x)

˚k+1|k (˚ G sk+1|k [h]) (21.129) ( ∫ ) = ˚ h(x, x) · exp x (g(z) − 1) · θk+1 (z|x)dz (21.130)

=

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21.7.2

Cluster-Target PHD Filter

This PHD filter results if we assume that the predicted p.g.f. is Poisson: Gk+1|k (x) = eNk+1|k (x−1) . In this case, the measurement-update for the clutter-target PHD filter is ([154], Corollary 1): ( ) ∏ ˚k+1|k+1 (x, x|Z (k+1) ) ∑ ∑ x|W | D z∈W θk+1 (z|x) −x =e ωP 1 + ˚k+1|k (x, x|Z (k) ) τ˜W D W ∈P

P⊟Zk+1

(21.131) where τ˜W

=

∫ ∫

∞

e−x · x|W | ·

(

0

ωP

=

21.8

θk+1 (z|x)

)

(21.132)

z∈W

˚k+1|k (x, x)dxdx ·D ∏ ˜W W ∈P τ ∑ ∏ P ′ ⊟Zk+1

∏

W ′ ∈P ′

.

(21.133)

τ˜W ′

MEASUREMENT MODELS FOR LEVEL-1 GROUP TARGETS

Group targets were defined in Section 21.1. The purpose of this and Sections 21.9 through 21.11 is to describe the statistics and filtering of group targets. Group targets are often distinguished by the fact that they consist of multiply nested target-subgroups. A typical example of such subgroups are the force levels (force structure) corresponding to the U.S. Army chain of command: fire team, squad, platoon, company, battalion, brigade/regiment, division, corps, and various levels of “army.” A level-1 group target is one that has only a single force level: a squad consisting of multiple fire teams; a platoon consisting of multiple squads; a company consisting of multiple platoons; and so on. Group targets with multiple force layers are discussed in Section 21.10. The conventional multitarget detection and tracking problem has two “layers”: • A hidden target layer—the space X of target states x. • A visible observation layer—the space Z of measurements z generated by the targets and the background.

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Consider now the simplest multigroup detection and tracking scenario, in which each group target ˚ x (for example, a squad of dismounted troops) is itself an ensemble X of conventional point targets (that is, the individuals in a particular squad). In what follows, this will be referred to as the single-level group target problem. This detection and tracking problem has three layers: • A doubly-hidden group target layer—the space ˚ X of group states ˚ x. • A singly-hidden layer X of conventional targets. • A visible observation layer Z of measurements. More formally, a level-1 group target is characterized by two items: • A state vector ˚ x ∈ ˚ X, which can include group parameters such as group centroid, centroidal velocity, number of targets in the group, group-shape parameters, and group identity (for example, “squad,” “platoon,” “company,” “battalion,” “aircraft carrier group,” and so on). • A nonempty set X ⊆ X of the states of the individual targets that constitute the group. The section is organized as follows: 1. Section 21.8.1: The “natural” state representation of single level-1 group targets. 2. Section 21.8.2: The “natural” state representation of multiple level-1 group targets. 3. Section 21.8.3: The simplified state representation of multiple level-1 group targets. 4. Section 21.8.4: Level-1 group targets with the standard multitarget measurement model. 21.8.1

“Natural” State Representation of Single Level-1 Group Targets

The complete state representation of a single-level group target is a pair of the form (˚ x, X) with X ̸= ∅. A function whose argument is such a state has the form f (˚ x, X) where f (˚ x, ∅) = 0. (21.134)

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The corresponding integral has the form ∫

f (˚ x, X)d˚ xδX =

∑ 1 ∫ f (˚ x, {x1 , ..., xn })d˚ xdx1 · · · dxn n!

(21.135)

n≥1

where the summation is taken over all n ≥ 1 because f (˚ x, ∅) = 0. The optimal solution for detection and tracking of single level-1 group targets is a Bayes filter of the form ... → fk|k (˚ x, X|Z (k) ) → fk+1|k (˚ x, X|Z (k) ) → fk+1|k+1 (˚ x, X|Z (k+1) ) → ... where fk+1|k (˚ x, X|Z (k) )

=

fk+1|k+1 (˚ x, X|Z (k+1) )

=

fk+1 (Z|Z (k) )

=

∫

fk+1|k (˚ x, X|˚ x′ , X ′ )

·fk|k (˚ x′ , X ′ |Z (k) )d˚ x′ δX ′ ( ) fk+1 (Zk+1 |˚ x, X) ·fk+1|k (˚ x, X|Z (k) ) fk+1 (Zk+1 |Z (k) ) ∫ fk+1 (Z|˚ x, X)

(21.136)

(21.137) (21.138)

·fk+1|k (˚ x, X|Z (k) )d˚ xδX and where fk+1|k (˚ x, X|˚ x′ , X ′ ) is the Markov density and fk+1 (Zk+1 |˚ x, X) is the likelihood function. For any k, the JoM-type estimate (5.9) will be

(˚ xk|k , Xk|k ) = arg sup ˚ x,X

c|X| · fk|k (˚ x, X|Z (k) ) . |X|!

(21.139)

That is, ˚ xk|k is the optimal estimate of the state of the group target and Xk|k is the optimal estimate of the targets of which it is constituted. 21.8.2

“Natural” State Representation of Multiple Level-1 Group Targets

As Swain and Clark have noted [286], a system of multiple group targets is a cluster RFS, as was introduced in Section 4.4.2. That is, let Ξ˚ x ⊆ X be the RFS of targets

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associated with the group target with state ˚ x, and let ˚ Ξ ⊆ X be the RFS of group targets. Then the RFS of all conventional targets is ∪ Ξ= Ξ˚ (21.140) x. ˚ x∈˚ Ξ

A particular instantiation of the joint process consists of a instantiation {˚ x1 , ...,˚ xn } of ˚ Ξ and instantiations Xi of each Ξ˚ . Thus the complete state representation xi of a system consisting of multiple level-1 group targets must have the form [158], [159] X = {(˚ x1 , X1 ), ..., (˚ xn , Xn )} (21.141) where Xi ̸= ∅ is the set of targets associated with the group target ˚ xi . The ˚ x1 , ...,˚ xn must be distinct. This is because a pair (˚ x, X1 ), (˚ x, X2 ) with X1 ̸= X2 is physically unrealizable. It would indicate that the same group target ˚ x is constituted by, simultaneously, the target group X1 and the different target group X2 . It follows that a function for this state representation must satisfy fˇ({(˚ x1 , X1 ), ..., (˚ xn , Xn )}) = 0

(21.142)

if any of the following situations occur: Xi = ∅ for some i; or ˚ xi = ˚ xj for some i ̸= j. The corresponding integral has the form ∫ ∑ 1 ∫ ˇ f (X)δX = fˇ({(˚ x1 , X1 ), ..., (˚ xn , Xn )})d˚ x1 · · · d˚ xn δX1 · · · δXn . n! n≥0

(21.143) The optimal solution for detection and tracking of multiple level-1 group targets is a Bayes filter of the form ... →

fˇk|k (X|Z (k) )

→

fˇk+1|k (X|Z (k) )

→

fˇk+1|k+1 (X|Z (k+1) )

→ ...

where fˇk+1|k (X|Z (k) )

=

∫

fˇ(X|X′ ) · fˇk|k (X′ |Z (k) )δX′ (21.144) k+1|k

fˇk+1|k+1 (X|Z (k+1) )

=

fk+1 (Z|Z (k) )

=

fk+1 (Zk+1 |X) · fˇk+1|k (X|Z (k) ) (21.145) fk+1 (Zk+1 |Z (k) ) ∫ fk+1 (Z|X) · fˇk+1|k (X|Z (k) )δX (21.146)

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and where fˇk+1|k (X|X) is the Markov transition density and fk+1 (Z|X) is the likelihood function. For any k, the multigroup target analog of the JoM estimator (5.9) is (21.147)

Xk|k =

k|k k|k k|k {(˚ x1 , X1 ), ..., (˚ xk|k nk|k , Xnk|k )}

( =

arg sup n,˚ x1 ,X1 ,...,˚ xn ,Xn

˚ cn · c|X1 |+...+|Xn | ˇ ·fk|k ({(˚ x1 , X1 ), ..., (˚ xn , Xn )}) n! · |X1 |! · · · |Xn |!

) (21.148)

where ˚ c is a constant whose units of measurement are the same as those of ˚ x, and where c is a constant whose units of measurement are the same as those of x. k|k k|k Here, ˚ x1 , ...,˚ xnk|k are the estimates of the states of the group targets, and the k|k

k|k

X1 , ..., Xnk|k are the estimates of their respective constituent target groups. 21.8.3

Simplified State Representation of Multiple Level-1 Group Targets

The representation of multiple level-1 group targets in (21.141) is “natural.” It is also mathematically complex and implementationally problematic. For example, the p.g.fl. of a multiple level-1 group target must have the form ˇ = ˇ h] G[

∫

ˇ X · fˇ(X)δX h

(21.149)

ˇ x, X). The space of finite sets of where the test function must have the form h(˚ pairs of the form (˚ x, X), where X is itself a finite set, is very abstract and its theoretical treatment is too mathematically involved for practical purposes. For example, the PHD for the natural representation must have the form ˇ x, X) = D(˚

∫

fˇ({(˚ x, X)} ∪ Y)δY =

ˇ δG [1]. δ(˚ x, X)

(21.150)

This definition of a PHD is unusable in practice. Any PHD filter defined in terms of it would require computationally intractable set integrals. Consequently, a more useful state representation would be desirable. Such a representation exists, and the purpose of this section is to describe it. The section is organized as follows:

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1. Section 21.8.3.1 Simplified-representation states, set integrals, and the Bayes filter for the multiple level-1 group target problem. 2. Section 21.8.3.2: The statistical relationship between the simplified and natural state representations. 21.8.3.1

Simplified State Representation: States and Set Integrals

A system {(˚ x1 , X1 ), ..., (˚ xn , Xn )} of level-1 group targets can be equivalently represented as an arbitrary finite set of pairs of the form •

X = {(˚ x1 , x1 ), ..., (˚ xν , xν )}

(21.151)

where ν = |X1 | + ... + |Xn | and where ˚ x1 , ...,˚ xν will be referred to as the • • “ordinates” of X and x1 , ..., xν will be referred to as the “abscissas” of X. • To see why, first let us be given a simplified representation X and let us construct its corresponding natural representation X: •

• Let ˚ x1 , ...,˚ xn be the distinct ordinates of X and let Xi consist of all x • such that (˚ xi , x) ∈ X. Then X = {(˚ x1 , X1 ), ..., (˚ xn , Xn )} has the correct mathematical form for a group target in natural representation. Conversely, let us be given a natural representation X and let us construct its corresponding simplified representation: •

• For any X let X be the set of all pairs (˚ x, x) such that x is a member of the group target ˚ x—that is, such that x ∈ X for some (˚ x, X) ∈ X. Then • X is a simplified representation of X. These two procedures are obviously inverse to each other, and so (21.151) is an equivalent representation of (21.141). As an example, note that {(˚ x, x1 ), ..., (˚ x, xn )} is an equivalent representation of (˚ x, {x1 , ..., xn }). It follows that in a Bayesian formulation, the unknown state of a level-1 • multigroup system is a random finite subset Ξ of the space of all pairs •

• x = (˚ x, x) ∈ X = ˚ X × X.

(21.152)

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The definition of a set integral must be modified to accommodate this new state representation. It has the form ∫ ∑ 1 ∫ • • • • f (X)δ X = f ({(˚ x1 , x1 ), ..., (˚ xν , xν )})d˚ x1 · · · d˚ xν dx1 · · · dxν . ν! ν≥0

(21.153) Similarly, the p.g.fl. and PHD have the forms ∫ • • • • • • • G[h] = hX · f (X)δ X ∫

•

D(˚ x, x)

=

(21.154) •

•

•

δG •

•

f ({(˚ x, x)} ∪ Y )δ Y =

Ξ

(21.155)

[1]

δ(˚ x, x)

•

where the test function now has the form h(˚ x, x). • The quantity D(˚ x, x) is the probability (density) that there is a group target with state ˚ x and that x is one of its constituent targets. It is clearly more ˇ x, X) corresponding to the natural implementation-friendly than the PHD D(˚ representation. For the simplified representation, the optimal multigroup Bayes filter will have the form •

... →

•

•

f k|k (X|Z (k) )

•

•

f k+1|k (X|Z (k) )

→

→

•

f k+1|k+1 (X|Z (k+1) ) → ...

where •

•

f k+1|k (X|Z (k) ) •

•

f k+1|k+1 (X|Z

(k+1)

)

fk+1 (Z|Z (k) ) •

•

=

= =

∫

•

•

•

•

•

•

•

•

f k+1|k (X|X ′ ) · f k|k (X ′ |Z (k) )δ X ′ (21.156) •

fk+1 (Zk+1 |X) · f k+1|k (X|Z (k) ) ∫

(21.157)

fk+1 (Zk+1 |Z (k) ) •

•

•

•

fk+1 (Z|X) · f k+1|k (X|Z (k) )δ X (21.158)

•

•

and where f k+1|k (X|X ′ ) is the Markov transition density and fk+1 (Zk+1 |X) is the likelihood function. 21.8.3.2

Mathematical Relationships Between Representations

For the sake of completeness, the basic mathematical relationships between the natural and the simplified state representations of level-1 group targets are presented in

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˜ for the natural state representation, ˜ h] Appendix I. In summary, given the p.g.fl. G[ define the p.g.fl. for the simplified representation as •

•

ˇ Tˇ• ] G[h] = G[

(21.159)

h •

where the functional transformation h ?→ Tˇ• is defined as h

∏

Tˇ• (˚ x, X) = h

•

h(˚ x, x).

(21.160)

x∈X

It follows that •

•

G[h] =

∫

•

hX · fˇ(X)δX

(21.161)

•

where hX = 1 if X = ∅ and, if otherwise, ∏ ∏ • • hX = h(˚ x, x).

(21.162)

(˚ x,X)∈X x∈X

Given this, the multigroup distribution for the simplified representation can be defined in terms of the natural representation as [ ] • • δ ˇ ˇ f (X) = G[ T ] . (21.163) • • h • δX h=0 Similarly, the PHD for the simplified representation can be defined in terms of the natural representation as • •

D(˚ x, x) =

δG [1] = δ(˚ x, x)

∫ ∫

fˇ(Y ∪ {(˚ x, {x} ∪ Y )})δYδY.

(21.164)

˚⊆ ˚ Also, let S ⊆ X and S X be measurable subsets. Then it can be shown that ∫ ∫ • D(˚ x, x)dxd˚ x (21.165) ˚ S S

is the expected number of conventional targets that are in S, given that their ˚ Thus, in particular, the integral respective group targets are in S. ∫ • • N = D(˚ x, x)d˚ xdx (21.166) is the expected number of all (conventional) targets in all of the group targets.

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21.8.4

Multiple Level-1 Group Targets with the Standard Measurement Model

The simplified state representation for level-1 group targets, together with suitable independence assumptions, immediately leads to a simple measurement model for level-1 group targets. Assume that the constituent targets of all group targets in the scene generate measurements independently. Then the p.g.fl. ∫ • • Gk+1 [g|X] = g Z · fk+1 (Z|X)δZ (21.167) of the multiobject likelihood function (simplified representation) factors as follows: ∏ • Gk+1 [g|X] = Gk+1 [g|˚ x, x]. (21.168) •

(˚ x,x)∈X

For the standard measurement model (Section 7.2), the factor Gk+1 [g|˚ x, x] must have the form •

•

•

Gk+1 [g|˚ x, x] = 1 − pD (˚ x, x) + pD (˚ x, x) · Lg (˚ x, x) where •

Lg (˚ x, x) =

∫

g(z) · fk+1 (z|˚ x, x)dz

(21.169)

(21.170)

and where: • Probability of detection: •

•

pD (˚ x, x) abbr. = pD,k+1 (˚ x, x).

(21.171)

This is the probability that a measurement will be generated, if a target with state x is present, and also belongs to a level-1 group target with state ˚ x. • Likelihood function: •

Lg (˚ x, x) abbr. = fk+1 (z|˚ x, x).

(21.172)

If a measurement is generated, this is the likelihood that z will be generated, if a target with state x is present and also belongs to a level-1 group target with state ˚ x.

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•

Note that pD (˚ x, x) can be written in the form •

pD (˚ x, x) = pD (x) · ˚ p(˚ x|x)

(21.173)

Here, pD (x) is the usual probability of detection for a target with state x. In addition, the “relative probability of detection” ˚ p(˚ x|x) can be interpreted as the probability that a group target with state ˚ x will contain a target with state x (see Remark 85). The probability ˚ p(˚ x|x) can be used to specify the constituent targets of a group target. Suppose, for example, that a target x can never be found in a group target ˚ x. Then ˚ p(˚ x|x) = 0. Or, suppose that x is rarely a constituent of ˚ x. Then ˚ p(˚ x|x) is small. Similarly, the likelihood fk+1 (z|˚ x, x) can be used to specify how targets generate different measurements, depending on the group target to which they belong. For example, a target with state x may generate a particular measurement if it belongs to group target ˚ x1 , but an entirely different one if it belongs to group target ˚ x2 . One example might be a SIGINT sensor, in which case a command and control (C&C) vehicle might use different transmitters, depending on the tactical nature of the particular group target of which it is a part. Remark 85 (Relative probabilities of detection) The quantity ˚ p(˚ x|x) is the probability that the group target ˚ x is detectable, given that the constituent target x is detectable. To see why, note that probabilities of detection can be written in the form pD (x) = Pr(x ∈ Θ) = E[1Θ (x)] where Θ is a random closed subset of X. That is, Θ is a “random field of view,” with each instantiation Θ = S corresponding to a “cookie cutter” field of view pD (x) = 1S (x). It follows that, ˚⊆˚ if Θ X is a random field of view for group targets, then •

pD (˚ x, x)

= =

˚ x ∈ Θ) Pr(˚ x ∈ Θ, ˚ x ∈ Θ) Pr(˚ x ∈ Θ,

Pr(x ∈ Θ) = ˚ p(˚ x|x) · pD (x).

(21.174) · Pr(x ∈ Θ)

(21.175) (21.176)

In addition to the standard measurement model, assume also the standard multitarget motion model. Then the following are required: • Probability of survival—the probability that a target with state x′ , belonging to a group target with state ˚ x′ , will survive from time tk to time tk+1 : •

pS (˚ x′ , x′ ).

(21.177)

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• Markov transition density—the probability (density) that, at time tk , a target with state x and belonging to a group target with state ˚ x′ , will transition to a target with state x that belongs to a group target with state ˚ x: •

f k+1|k (˚ x, x|˚ x′ , x′ ).

(21.178)

If one insists that a target always belongs to a single group target and, therefore, cannot switch from one group target to another, then •

f k+1|k (˚ x, x|˚ x′ , x′ ) = fk+1|k (x|˚ x′ , x′ ) · δ˚ x). x′ (˚

21.9

(21.179)

PHD/CPHD FILTERS FOR LEVEL-1 GROUP TARGETS

The purpose of this section is to address the problem of devising PHD and CPHD filters for detecting and tracking level-1 group targets. Since detecting and tracking single level-1 group targets is a problem of independent interest, it will be treated separately. The section is organized as follows: 1. Section 21.9.1: A PHD filter for multiple level-1 group targets, assuming the standard multitarget measurement model for conventional targets. 2. Section 21.9.2: A CPHD filter for multiple level-1 group targets, assuming the standard multitarget measurement model for conventional targets. 3. Section 21.9.3: A PHD filter for single level-1 group targets, assuming the standard multitarget measurement model for conventional targets. 4. Section 21.9.4: A CPHD filter for single level-1 group targets, assuming the standard multitarget measurement model for conventional targets. 21.9.1

PHD Filter for Level-1 Group Targets: Standard Model

The standard multitarget measurement model for level-1 group targets was defined in Section 21.8.4. From the discussion there, it is clear that the corresponding PHD and CPHD filters for level-1 group targets will have the usual forms described in Section 8.4 and Section 8.5 —but using p(˚ x|x) · pD (x) in place of pD (x) and fk+1 (z|˚ x, x) = ˚ Lz (˚ x, x) in place of fk+1 (z|x). Also, one has to specify a Markov transition density •

f k+1|k (˚ x, x|˚ x′ , x′ ) = f˚k+1|k (˚ x|˚ x′ ) · fk+1|k (x|˚ x′ , x′ ).

(21.180)

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21.9.1.1

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PHD Filter for Multiple Level-1 Group Targets: Predictor and Corrector Equations

The time-update and measurement-update equations for the PHD filter are •

•

D k+1|k (˚ x, x)

=

bk+1|k (˚ x, x) ∫ • • + pS (˚ x′ , x′ ) · f k+1|k (˚ x, x|˚ x′ , x′ )

(21.181) (21.182)

•

·D k|k (˚ x, x)d˚ x′ dx′ •

D k+1|k+1 (˚ x, x) =

•

(21.183)

1−˚ p(˚ x|x) · pD (x)

D k+1|k (˚ x, x) +

∫

•

˚ p(˚ x|x) · pD (x) · Lz (˚ x, x) κk+1 (z) + τk+1 (z)

where τk+1 (z) =

∫

•

•

˚ p(˚ x|x) · pD (x) · Lz (˚ x, x) · D k+1|k (˚ x, x)d˚ xdx.

(21.184)

As usual, (21.183) is based on the assumption that the predicted RFS is approximately Poisson: • • • • • f k+1|k (X|Z (k) ) ∼ (21.185) = e−N k+1|k · D X k+1|k . 21.9.1.2

PHD Filter for Multiple Level-1 Group Targets: State Estimation

State estimation can be accomplished as follows. Construct the total expected number of targets ∫ • Nk+1|k+1 = D k+1|k+1 (˚ x, x)d˚ xdx (21.186) and round it off to the nearest integer ν. Then find the states (˚ x1 , x1 ), ..., (˚ xν , xν ) • corresponding to the ν largest peaks of D k+1|k+1 (˚ x, x). Then partition the ˚1 , ..., C ˚n based on proximity. That is, the elements of ˚ x1 , ..., ˚ xν into clusters C ˚i ⊆ ˚ C X are all close to each other. Finally, define Xi to consist of those xj ˚i . corresponding to the elements of C

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21.9.2

CPHD Filter for Level-1 Group Targets: Standard Model

The corresponding CPHD filter can be defined in the same manner, by making suitable substitutions into the equations in Section 8.5. Since this is straightforward, the details are left to the reader. 21.9.3

PHD Filter for Single Level-1 Group Targets: Standard Measurement Model

The tracking of a single group target is of interest in and of itself. The optimal Bayes filter for this problem was presented in (21.136) through (21.138). However, the PHD filter considered in Section 21.9.1 is inapplicable. This is because the Poisson assumption required for the PHD filter measurement-update is inconsistent with the a priori assumption that only a single group target is present. The purpose of this section is to describe a PHD filter for this problem, which requires the “mixed” or “factored” multitarget filter described in Section 5.9. It is based on the following application of Bayes’ rule: fk|k (˚ x, X|Z (k) ) = f˚k|k (˚ x|Z (k) ) · fk|k (X|˚ x, Z (k) ),

(21.187)

where f˚k|k (˚ x|Z (k) ) is the probability distribution of the group state ˚ x, and (k) fk|k (X|˚ x, Z ) is a probability distribution on the multitarget state X, given that X is the set of targets corresponding to the group target ˚ x. The Markov transition density fk+1|k (˚ x, X|˚ x′ , X ′ ) satisfies the following identity: fk+1|k (˚ x, X|˚ x′ , X ′ ) = f˚k+1|k (˚ x|˚ x′ , X ′ ) · fk+1|k (X|˚ x,˚ x′ , X ′ ).

(21.188)

Assume that f˚k+1|k (˚ x|˚ x′ , X ′ ) ′

′

fk+1|k (X|˚ x,˚ x ,X )

=

f˚k+1|k (˚ x|˚ x′ )

=

′

(21.189) ′

fk+1|k (X|˚ x , X ).

(21.190)

The first equation is true if the future state ˚ x of a group target is determined only by its previous state ˚ x′ and not by the targets X ′ that constituted that earlier group target. The second equation is true if the future target set X of a group target does not depend on the current state ˚ x but, rather, came about as a transition from the previous target set X ′ , given that this was the target set of the earlier group target ˚ x′ . Thus we may write fk+1|k (˚ x, X|˚ x′ , X ′ ) = f˚k+1|k (˚ x|˚ x′ ) · fk+1|k (X|˚ x′ , X ′ ).

(21.191)

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Given this, it is shown in Section K.31 that the optimal Bayes filter for single level-1 group targets can be cast into the form ... → f˚k|k (˚ x|Z (k) ) → ... → fk|k (X|˚ x, Z (k) ) →

f˚k+1|k (˚ x|Z (k) ) ↑↓ fk+1|k (X|˚ x, Z (k) )

→ f˚k+1|k+1 (˚ x|Z (k+1) ) → ... ↑↓ → fk+1|k+1 (X|˚ x, Z (k+1) ) → ...

Here the top row is a conventional Bayes filter for the group state ˚ x, and the bottom row is a conventional multitarget Bayes filter on the target set X associated with ˚ x. The filtering equations for the time-update are fk+1|k (˚ x|Z (k) )

=

fk+1|k (X|˚ x, Z (k) )

=

∫

fk+1|k (˚ x|˚ x′ ) · fk|k (˚ x′ |Z (k) )d˚ x′ (21.192) ( ∫ ) f˚k+1|k (˚ x|˚ x′ ) · fk|k (˚ x′ |Z (k) ) ·f˜k+1|k (X|˚ x′ , Z (k) )d˚ x′ (21.193) f˚k+1|k (˚ x|Z (k) )

where, for fixed ˚ x′ , f˜k+1|k (X|˚ x′ , Z (k) ) =

∫

fk+1|k (X|˚ x′ , X ′ ) · fk|k (X ′ |˚ x′ , Z (k) )δX ′

(21.194)

is a conventional multitarget prediction integral. The filtering equations for the measurement-update are ( f˚k+1|k+1 (˚ x|Z (k+1) )

=

fk+1|k+1 (X|˚ x, Z (k+1) )

=

) f˚k+1|k (˚ x|Z (k) ) ·fk+1 (Zk+1 |˚ x, Z (k) ) fk+1 (Zk+1 |Z (k) ) ( ) fk+1 (Zk+1 |˚ x, X) ·fk+1|k (X|˚ x, Z (k) ) fk+1 (Zk+1 |˚ x, Z (k) )

(21.195)

(21.196)

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where fk+1 (Zk+1 |Z (k) )

fk+1 (Zk+1 |˚ x, Z (k) )

=

∫

=

·f (Z |˚ x, Z (k) )d˚ x ∫ k+1 k+1 fk+1 (Zk+1 |˚ x, X)

fk+1|k (˚ x|Z (k) )

(21.197)

(21.198)

·fk+1|k (X|˚ x, Z (k) )δX and where, for fixed ˚ x, the last equation is a conventional multitarget Bayes factor. If we assume that fk+1|k (X|˚ x, Z (k) ) is (approximately) Poisson, then the PHD filter for the bottom row can be constructed, and the corresponding PHD filter then has the form ... → f˚k|k (˚ x|Z (k) ) ... → Dk|k (x|˚ x, Z (k) )

→ f˚k+1|k (˚ x|Z (k) ) → ↑↓ → Dk+1|k (x|˚ x, Z (k) ) →

f˚k+1|k+1 (˚ x|Z (k+1) ) → ... ↑↓ Dk+1|k+1 (x|˚ x, Z (k+1) ) → ...

The purpose of the following subsections is to describe this filter. 21.9.3.1

PHD Filter for Single Level-1 Group Targets: Models

The following models are required for this PHD filter: • Probability of target survival: •

′ abbr. pS,˚ x′ , x′ ). x′ (x ) = pS,k+1 (˚

• Markov transition density for the targets surviving at time group target ˚ x′ at time tk : fk+1|k (x|˚ x′ , x′ ).

(21.199) tk+1 from a

(21.200)

• Markov transition density for group targets: f˚k+1|k (˚ x|˚ x′ ).

(21.201)

• PHD of the process for targets appearing at time tk+1 in a group target ˚ x′ at time tk : bk+1|k (x|˚ x′ ). (21.202)

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• Probability of detection: •

abbr. pD,˚ x, x). x (x) = pD,k+1 (˚

(21.203)

abbr. Lz,˚ x, x). x (x) = fk+1 (z|˚

(21.204)

• Likelihood function:

21.9.3.2

PHD Filter for Single Level-1 Group Targets: Time and Measurement Updates

The following equations are derived in Section K.32: • Time update equations: ∫

f˚k+1|k (˚ x|˚ x′ ) · f˚k|k (˚ x′ |Z (k) )d˚ x′ (21.205) ∫   f˚k+1|k (˚ x|  ˚  x′ ) · f˚k|k (˚ x′ |Z (k) ) ′ (k) ′ ˜ ·Dk+1|k (x|˚ x , Z )d˚ x Dk+1|k (x|˚ x, Z (k) ) = (21.206) (k) ˚ fk+1|k (˚ x|Z ) f˚k+1|k (˚ x|Z (k) ) =

where ˜ k+1|k (x|˚ D x′ , Z (k) )

=

bk+1|k (x|˚ x′ ) (21.207) ∫ ′ + pS,˚ x′ , x′ ) x′ (x ) · fk+1|k (x|˚ ·Dk|k (x′ |˚ x′ , Z (k) )dx′

is a conventional PHD filter time-update (with spawning neglected for the sake of conceptual clarity).

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Advances in Statistical Multisource-Multitarget Information Fusion

• Measurement update equations: (

f˚k+1|k+1 (˚ x|Z (k+1) )

=

) fk+1 (Zk+1 |˚ x) ·f˚k+1|k (˚ x|Z (k) ) ( ∫ ) fk+1 (Zk+1 |˚ y) ·f˚k+1|k (˚ y|Z (k) )d˚ y

Dk+1|k+1 (x|˚ x, Z (k+1) ) Dk+1|k (x|˚ x, Z (k) )

=

1 − pD,˚ x (x) +

(21.208)

(21.209)

∑ pD,˚ x, x) x (x) · fk+1 (z|˚ κk+1 (z) + τ˚ x (z) z∈Zk+1

where, for fixed ˚ x, (21.209) is a conventional PHD corrector equation; and where Z

fk+1 (Z|˚ x)

=

τ˚ x (z)

=

e−λk+1 −D˚x[pD,˚x] · (κk+1 + τ˚ x) ∫ pD,˚ x (x) · Lz,˚ x (x)

=

·Dk+1|k (x|˚ x, Z (k) )dx ∫ pD,˚ x, Z (k) )dx. x (x) · Dk+1|k (x|˚

D˚ x [pD,˚ x]

(21.210) (21.211)

(21.212)

Also, the power-functional notation g Z was defined in (3.5). Remark 86 The multiobject density fk+1 (Z|˚ x) of (21.210) is a multiobject likelihood function since, as is easily verified (see (K.790)) in Section K.32, ∫ 21.9.3.3

fk+1 (Z|˚ x)δZ = 1.

(21.213)

PHD Filter for Single Level-1 Group Targets: State Estimation

The state of the group target can be estimated using a standard Bayes-optimal state estimator—for example, the MAP estimator: ˚ xk|k = arg sup f˚k|k (˚ x|Z (k) ). ˚ x

(21.214)

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Given this, the targets in ˚ xk|k can be estimated using the usual procedure for PHD filters (Section 8.4.4). That is, compute Nk|k =

∫

Dk|k (x|˚ xk|k , Z (k) )dx

(21.215)

and round it off to the nearest integer ν. Determine the states x1 , ..., xν corresponding to the ν largest suprema of Dk|k (x|˚ xk|k , Z (k) ). Then X = {x1 , ..., xν } is the state estimate of the targets in the group target. 21.9.3.4

PHD Filter for Single Level-1 Group Targets: Implementation

One obstacle to implementation of this PHD filter is the denominator f˚k+1|k (˚ x|Z (k) ) in (21.206): ∫

˜ k+1|k (x|˚ f˚k+1|k (˚ x|˚ x′ ) · f˚k|k (˚ x′ |Z (k) ) · D x′ , Z (k) )d˚ x′ . f˚k+1|k (˚ x|Z (k) ) (21.216) Because the likelihood function Dk+1|k (x|˚ x, Z

(k)

)=

˚

fk+1 (Z|˚ x) = e−λk+1 −Dk+1|k [pD,˚x] · (κk+1 + τ˚ x)

Z

(21.217)

in (21.210) is highly nonlinear in ˚ x, the distributions fk+1|k (˚ x|Z (k) ) must be approximated using particle (SMC) techniques: νk+1|k

f˚k+1|k (˚ x|Z (k) ) ∼ =

∑

k+1|k

wi

· δ˚ x). k+1|k (˚ x

(21.218)

i

i=1

In this case, fk+1|k (˚ x|Z (k) ) cannot be employed in the denominator of (21.216). (k) If fk+1|k (˚ x|Z ) is approximately unimodal, one possible way around this would be to approximate the particle representation of fk+1|k (˚ x|Z (k) ) as a Gaussian: νk+1|k

∑ i=1

k+1|k

wi

k+1|k

· θ(˚ xi

)∼ =

∫

NP˚k+1|k (˚ x −˚ xk+1|k ) · θ(˚ x)d˚ x

(21.219)

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where νk+1|k

˚ xk+1|k

=

∑

k+1|k

k+1|k

(21.220)

·˚ xi

wi

i=1 νk+1|k

˚k+1|k P

=

∑

k+1|k

wi

k+1|k

· (˚ xi

k+1|k

−˚ xk+1|k )T (˚ xi

−˚ x k+1|k) (.21.221)

i=1

Given this, Dk+1|k (x|˚ x, Z (k) ) can be approximated using either Gaussian mixture or particle methods. 21.9.4

CPHD Filter for Single Level-1 Group Targets: Standard Model

The PHD filter of Section 21.9.3 can be extended to derive a CPHD filter for detecting and tracking single level-1 group targets. The filtering equations are: • Time update equations: f˚k+1|k (˚ x|Z (k) )

=

Dk+1|k (x|˚ x, Z (k) )

=

Gk+1|k (x|˚ x)

=

˜ k+1|k (x|˚ D x′ , Z (k) )

=

∫

f˚k+1|k (˚ x|˚ x′ ) · f˚k|k (˚ x′ |Z (k) )d˚ x′ (21.222) ∫   f˚k+1|k (˚ x|˚ x′ )   ·f˚k|k (˚ x′ |Z (k) ) ′ (k) ′ ˜ k+1|k (x|˚ ·D x , Z )d˚ x (21.223) (k) ˚ fk+1|k (˚ x|Z )  ∫  fk+1|k (˚ x|˚ x′ )  ·fk|k (˚ x′ |Z (k) )  ˜ k+1|k (x|˚ ·G x′ )d˚ x′ (21.224) f˚k+1|k (˚ x|Z (k) )

where bk+1|k (x|˚ x′ ) (21.225) ∫ ′ + pS,˚ x′ , x′ ) x′ (x ) · fk+1|k (x|˚ ·Dk|k (x′ |˚ x′ , Z (k) )dx′ ˜ k+1|k (x|˚ G x′ )

=

usual p.g.f. predictor

(21.226)

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743

and where, in the usual p.g.f. predictor equation, we employ the models specified in Section 21.9.3.1. • Measurement update equations:

=

fk+1|k+1 (˚ x|Z (k+1) ) fk+1 (Zk+1 |˚ x) · f˚k+1|k (˚ x|Z (k) ) ∫ fk+1 (Zk+1 |˚ y) · f˚k+1|k (˚ y|Z (k) )d˚ y

(21.227)

and Dk+1|k+1 (x|˚ x, Z (k+1) ) Gk+1|k+1 (x|˚ x)

= =

usual PHD corrector usual p.g.f. corrector

(21.228) (21.229)

where fk+1 (Z|˚ x)

=

cZ k+1

m ∑

(m − j)! · pκk+1 (m − j)

(21.230)

j=1 (j)

·Gk+1|k (sk+1|k [1 − pD,˚ x) · σj (Zk+1 |˚ x) x ]|˚ ( ) ˚ sk+1|k [pD,˚ x L˚ x,z ] 1

σi (Zk+1 |˚ x)

=

σm,i

, ...,

ck+1 (z1 ) ˚ sk+1|k [pD,˚ x L˚ x,zm ] ck+1 (zm )

(21.231)

and where in the usual PHD and p.g.f. corrector equations we employ the models specified in Section 21.9.3.1. The derivation of these equations is similar to that of Section K.32, except that [176], Eq. (126) is used in place of (8.56).

21.10

MEASUREMENT MODELS FOR GENERAL GROUP TARGETS

The standard measurement models for both the “natural” and “simplified” state representations for multiple level-1 group targets were discussed in Section 21.8. The purpose of this section is to generalize the simplified state representation to group targets with an arbitrary number ℓ of levels. We will begin with the simplified state representation in Section 21.10.1, and then discuss the standard multitarget measurement model for this representation in Section 21.10.2.

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21.10.1

Simplified State Representation of Level-ℓ Group Targets ◦2

Consider the two-level multigroup scenario. A level-2 group target x (for example, a platoon consisting of several squads) is an ensemble of level-1 group ◦1 targets x = ˚ x (for example, of the individual squads), each of which is, in turn, an ◦0 ensemble of conventional targets x = x (the individual troops in the squad). The level-two multigroup detection and tracking problem thus has four layers: ◦2

• A triply-hidden level-two group target layer—the space X of second-level ◦2 group states x. • The doubly-hidden level-1 group target layer—the space ˚ X of first-level group states ˚ x. • The singly-hidden layer X of conventional targets. • The visible observation layer Z of measurements. In the general, or level-ℓ, multigroup detection and tracking scenario, a level-ℓ ◦ℓ group target x is an ensemble of level-(ℓ − 1) group targets, each of which is an ensemble of level-(ℓ − 2) group targets, and so on. ◦2 Consider a single level-2 group target x. Its complete representation consists ◦2

◦1

of a pair ( x, X), where ◦1

X = {(˚ x1 , X1 ), ..., (˚ xn , Xn )}

(21.232)

and where Xi is the set of conventional point targets that constitute the level-1 group target ˚ xi . Thus the detailed “natural” state representation of multiple level-2 group targets has the rather complicated form ◦2

◦1

◦2

◦1

{( x 1 , X 1 ), ..., ( x ν , X ν )} =

(21.233)

◦2

{( x 1 , {(˚ x1,1 , X1,1 ), ..., (˚ x1,n1 , X1,n1 )}) ◦2

, ..., ( x ν , {(˚ xν,1 , X1,1 ), ..., (˚ x1,nν , X1,nν )})}. Clearly, it would be preferable to have a less complex state representation of level-2 ◦2 ◦2 group targets. Towards that end, note that x 1 , ..., x ν must be distinct. Otherwise, it would be possible for the same level-2 group target to consist of two different ensembles of level-1 group targets. The ˚ x1,1 , ...,˚ x1,n1 , ...,˚ xν,1 , ...,˚ x1,nν must be distinct for similar reasons. Thus, applying the same reasoning as in Section 21.8.3,

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745

multiple level-2 group targets can be equivalently represented as a set •2

◦2

◦2

(21.234)

X = {( x 1 ,˚ x1 , x1 ), ..., ( x ν ,˚ xν , xν )} of triples of the form •2

•2

◦2

◦2 x = ( x,˚ x, x) ∈ X = X × ˚ X × X.

(21.235)

In general, multiple level-ℓ group targets can be represented as a set •ℓ

◦ℓ

◦ℓ

(21.236)

X = {( x 1 , ...,˚ x1 , x1 ), ..., ( x ν , ...,˚ xν , xν )} of (ℓ + 1)-tuples of the form •ℓ

•ℓ

◦ℓ

◦2

◦ℓ ◦2 x = ( x, ..., x,˚ x, x) ∈ X = X × ... × X × ˚ X × X.

(21.237)

The corresponding set integrals have the form ∫ •ℓ •ℓ f (X)δ X ∑ 1 ∫ ◦ℓ ◦ℓ = f ({( x 1 , ...,˚ x1 , x1 ), ..., ( x ν , ...,˚ xν , xν )}) ν!

(21.238)

ν≥0 ◦ℓ

◦ℓ

·d x 1 · · · d x ν · · · d˚ x1 · · · d˚ xν · dx1 · · · dxν . The optimal solution for the level-ℓ group target problem will be the Bayes filter •ℓ

... →

•ℓ

•ℓ

f k|k (X|Z (k) )

•ℓ

•ℓ

f k+1|k (X|Z (k) )

→

→

•ℓ

f k+1|k+1 (X|Z (k+1) )

→ ...

where •ℓ

•ℓ

f k+1|k (X|Z (k) ) =

∫

•ℓ

•ℓ

•ℓ

•ℓ

•ℓ

•ℓ

•ℓ

f k+1|k+1 (X|Z

(k+1)

fk+1 (Z|Z (k) ) = •ℓ

•ℓ

•ℓ

∫

•ℓ

•ℓ

f k+1|k (X|X ′ ) · f k|k (X ′ |Z (k) )δ X ′ (21.239)

)=

•ℓ

•ℓ

fk+1 (Zk+1 |X) · f k+1|k (X|Z (k) )

(21.240)

fk+1 (Zk+1 |Z (k) ) •ℓ

•ℓ

•ℓ

•ℓ

fk+1 (Z|X) · f k+1|k (X|Z (k) )δ X

(21.241) •ℓ

and where f k+1|k (X|X ′ ) is the Markov transition density and fk+1 (Zk+1 |X) is the likelihood function.

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21.10.2

Standard Measurement Model for Level-ℓ Group Targets

This model is the obvious extension of the model for level-1 group targets described in Section 21.8.4. The fundamental models of interest are: • Probability of detection: •ℓ

◦ℓ

◦ℓ

•ℓ

p D ( x, ...,˚ x, x) abbr. = p D,k+1 ( x, ...,˚ x, x).

(21.242)

This is the probability that a measurement will be generated, if a target with state x is present, and also belongs to a level-1 group target with state ˚ x; ◦2 which in turn belongs to a level-2 group target with state x; which in turn ◦3 belongs to a level-3 group target with state x; and so on. • Likelihood function: •ℓ

◦ℓ

◦ℓ

Lg ( x, ...,˚ x, x) abbr. = fk+1 (z| x, ...,˚ x, x).

(21.243)

If a measurement is generated, this is the likelihood that the particular measurement z will be generated. The corresponding motion-related models are: • Probability of survival (the probability that a target with state x′ , belonging to a group target with state ˚ x′ , will survive from time tk to time tk+1 ): •ℓ

◦ℓ

p S ( x ′ , ...,˚ x′ , x′ ).

• Intensity function for target appearances: •ℓ

◦ℓ

b k+1|k ( x, ...,˚ x, x). • Markov transition density (the probability (density) that, at time tk , a target with state x and belonging to a group target with state ˚ x′ , will transition to a target with state x that belongs to a group target with state ˚ x): •ℓ

◦ℓ

◦ℓ

f k+1|k ( x, ...,˚ x, x| x ′ , ...,˚ x′ , x′ ).

If one insists that group targets cannot switch from one level to another, then •ℓ

◦ℓ

◦ℓ

◦ℓ

◦ℓ

f k+1|k ( x, ...,˚ x, x| x ′ , ...,˚ x′ , x′ ) = fk+1|k (x| x ′ , ...,˚ x′ , x′ ) · δ◦ℓ ( x) · · · δ˚ x). x′ (˚ x′ (21.244)

RFS Filters for Cluster-Type Targets

21.11

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PHD/CPHD FILTERS FOR LEVEL-ℓ GROUP TARGETS

The PHD filter for multiple level-ℓ is given by the equations •ℓ

◦ℓ

(21.245)

D k+1|k ( x, ...,˚ x, x) •ℓ

=

◦ℓ

b k+1|k ( x, ...,˚ x, x) ∫ • ◦ℓ ◦ℓ ◦ℓ •ℓ + p S ( x ′ , ...,˚ x′ , x′ ) · f k+1|k ( x, ...,˚ x, x| x ′ , ...,˚ x′ , x′ ) •ℓ

◦ℓ

◦ℓ

·D k|k ( x ′ , ...,˚ x′ , x′ )d x ′ · · · d˚ x′ dx′ and •ℓ

◦ℓ

D k+1|k+1 ( x, ...,˚ x, x) •ℓ

•ℓ

◦ℓ

= 1 − p D ( x, ...,˚ x, x)

◦ℓ

(21.246)

D k+1|k ( x, ...,˚ x, x) •ℓ

∑

+

•ℓ

◦ℓ

◦ℓ

p D ( x, ...,˚ x, x) · Lz ( x, ...,˚ x, x) κk+1 (z) + τk+1 (z)

z∈Zk+1

where τk+1 (z)

=

∫ •ℓ

•ℓ

•ℓ

◦ℓ

◦ℓ

p D ( x, ...,˚ x, x) · Lz ( x, ...,˚ x, x) ◦ℓ

(21.247)

◦ℓ

·D k+1|k ( x, ...,˚ x, x)d x · · · d˚ xdx. Equation (21.183) is based on the assumption that the predicted process is approximately Poisson: •ℓ •ℓ •ℓ •ℓ •ℓ ∼ e−N k+1|k · D X f (X|Z (k) ) = . (21.248) k+1|k

k+1|k

State estimation can be accomplished using a generalization of the approach described in Section 21.9.1.2. Specifically, construct the total expected number of targets ∫ •

Nk+1|k+1 =

◦ℓ

◦ℓ

D k+1|k+1 ( x, ...,˚ x, x)d x · · · d˚ xdx

and round it off to the nearest integer ν. Then find the states ◦ℓ

◦ℓ

( x 1 , ...,˚ x1 , x1 ), ..., ( x ν , ...,˚ xν , xν )

(21.249)

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•

◦ℓ

corresponding to the ν largest peaks of D k+1|k+1 ( x, ...,˚ x, x). Then partition the ˚ x1 , ..., ˚ xν into clusters based on their closeness of proximity. Next, cluster the ◦2 ◦2 ◦ℓ ◦ℓ x 1 , ..., x ν into clusters based on proximity. Repeat until the x 1 , ..., x ν have been clustered on the basis of proximity.

21.12

A MODEL FOR UNRESOLVED TARGETS

The purpose of this and the remaining sections of the chapter is to describe PHD filters that can detect and track multiple targets, even when some or all of them may be unresolved. The multitarget measurement model for the approach was first described in [179], pp. 432-444. The corresponding PHD filter was introduced in 2009 [175]. A generalization to CPHD filters was proposed in 2012 in [85]. The measurement model is based on the state representation of a group of unresolved targets as a point target-cluster. Whereas a point target has a conventional state representation x, a point target-cluster has the augmented form (21.250)

˚ x = (ν, x)

where ν > 0 is an integer indicating the number of targets that are in the cluster. The measurement set Z generated by a point cluster is the usual measurement set generated by a multitarget state X = {x1 , ..., xν } where x1 → x, ..., xν → x. That is, the measurements are those generated by ordinary targets, all of which are colocated at x. It follows that the likelihood function for a point target has the form fk+1 (Z|˚ x) = fk+1 (Z|ν, x) =

lim x1 →x,...,xν →x

fk+1 (Z|{x1 , ..., xν }).

(21.251)

Given this, one would like to define multitarget filters defined on targets whose states have the augmented form (ν, x). This is not possible, however, since ν is discrete. Thus we must find a way to generalize (ν, x) to the form ˚ x = (a, x)

(21.252)

where a > 0 is an arbitrary real number. That is, we must generalize it to a model in which target number (target cardinality) is allowed to be continuous. Similarly, we must find a way to generalize (21.251) to arrive at likelihoods of the form fk+1 (Z|˚ x) = fk+1 (Z|a, x).

(21.253)

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749

In this case, the number a is interpreted to be the expected number of targets in the point cluster. Then, given multiple unresolved targets with state set ˚ = {˚ X x1 , ...,˚ xn } with conditionally independent measurements, the multitarget likelihood function for all unresolved targets, in clutter, is given by the fundamental convolution theorem (4.17): ˚ fk+1 (Z|X)

=

∑

κk+1 (W0 )

(21.254)

W0 ⊎W1 ⊎...⊎Wn =Z

·fk+1 (W1 |˚ x1 ) · · · fk+1 (Wn |˚ xn ). This is the continuous-cardinality model for unresolved targets, and is defined in greater detail as follows. First it is noted that if a = ν is an integer, then the multitarget likelihood function for the standard model for ν colocated targets is ([179], Eqs. (12.231,12.232)) { Bν,pD (x) (0) if Z = ∅ ∏ fk+1 (Z|ν, x) = (21.255) |Z|! · Bν,pD (x) (|Z|) z∈Z f (z|x) if Z ̸= ∅ where Bν,p (m) = Cν,m · pm (1 − p)ν−m

(21.256)

is the binomial distribution and Cν,m is the binomial coefficient (2.1). By analogy, in the general case we define ([179], Eq. (12.263)): f (Z|a, x) =

{

Ba,pD (x) (0) ∏ |Z|! · Ba,pD (x) (|Z|) · z∈Z f (z|x)

if if

Z=∅ Z ̸= ∅

(21.257)

where Ba,p (m) is a generalized binomial distribution. It is defined as the probability distribution 1 dGa,p Ba,p (m) = (0) (21.258) m! dz corresponding to the p.g.f. ([179], Eq. (12.247)):

Ga,p (z) =

∞ ∏

(1 − σi (a) · p + σi (a) · p · z) .

(21.259)

i=0

Here, σi (a) = σ(a − i) where σ(a) is a sigmoidal function—that is, an infinitely differentiable function such that (1) σ(a) = 0 for a ≤ 0, (2) σ(a) ∼ = a for

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0 < a < 1, and (3) σ(a) ∼ = 1 for a ≥ 1. Because of these properties, (21.259) is actually a finite product and is thus well defined. Consider the following examples ([179], Eqs. (12.248-12.257)). For 0 < a < 1, Ga,p (z) ∼ (21.260) = 1 − a · p + a · p · z. For 1 < a < 2, Ga,p (z) ∼ = (1 − p + p · z) · {1 − (a − 1) · p + (a − 1) · p · z} .

(21.261)

For 2 < a < 3, 2

Ga,p (z) ∼ = (1 − p + p · z) · {1 − (a − 2) · p + (a − 2) · p · z} .

(21.262)

In general, if a ˘ is the largest integer smaller than a, then for n < a < n + 1 ([179], Eq. (12.258)) we have a ˘ = n and so a ˘

Ga,p (z) ∼ ˘) · p + (a − a ˘) · p · z} (21.263) = (1 − p + p · z) · {1 − (a − a =

Ga˘,p (z) · {1 − (a − a ˘) · p + (a − a ˘) · p · z} .

(21.264)

That is, an unresolved target with a targets can be interpreted as: • a ˘ ordinary targets. • A single partially existing target whose probability of existence is a−˘ a ≤ 1. Remark 87 (Continuity properties) For notational clarity, assume that there is no clutter. Then it follows that for m = |Z|, lim Ga,p (z) = 1

(21.265)

a↘0

and so lim Ba,pD (x) (0) = 1,

lim Ba,pD (x) (m) = 0

a↘0

a↘0

(21.266)

and thus lim fk+1 (Z|a, x)

=

a↘0

=

fk+1 (Z|∅) { 1 if Z = ∅ . 0 if |Z| ≥ 1

(21.267) (21.268)

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˚ = {˚ ˚ = n then from (21.254) Similarly, if X x1 , ...,˚ xn } with |X| ˚ lim fk+1 (Z|(a, x) ∪ X) ∑ lim fk+1 (W0 |a, x)

(21.269)

a↘0

=

a↘0

W0 ⊎W1 ⊎...⊎Wn =Z

=

·fk+1 (W1 |˚ x1 ) · · · fk+1 (Wn |˚ xn ) ∑ fk+1 (W0 |∅)

(21.270)

W0 ⊎W1 ⊎...⊎Wn =Z

·fk+1 (W1 |˚ x1 ) · · · fk+1 (Wn |˚ xn ) and so ˚ lim fk+1 (Z|(a, x) ∪ X)

∑

=

a↘0

fk+1 (W1 |˚ x1 )

(21.271)

W1 ⊎...⊎Wn =Z

=

· · · fk+1 (Wn |˚ xn ) ˚ fk+1 (Z|X).

(21.272)

Thus the multicluster likelihood function is continuous with respect to target number. A word of caution, however: it is not also continuous with respect to target state: lim fk+1 (Z|{(a, x), (a′ , x′ )}) ̸= fk+1 (Z|{(a + a′ , x)}).

(21.273)

x′ →x

Given these preliminaries, let us be given the following modeling assumptions: • Single-target probability of detection: pD (x). • Single-target likelihood function: Lz (x) = fk+1 (z|x). • Likelihood function for point target-clusters: ˚|Z| (a, x) LZ (a, x) = fk+1 (Z|a, x) = |Z|! · β

∏

Lz (x)

(21.274)

z∈Z

where ˚m (a, x) def. β = Ba,pD (x) (m)

(21.275)

with associated p.g.f. ˚z (a, x) = G

∑ m≥0

˚m (a, x) · z m . β

(21.276)

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21.13

MOTION MODEL FOR UNRESOLVED TARGETS

As noted in [179], pp. 475-477, the motion of point target-clusters can be very complex. A cluster can resolve into several clusters with smaller numbers of targets; it can eventually resolve into individual targets; or multiple targets can deresolve into one or more point clusters. Here as in [179] a simplified model is considered, one in which clusters transition to other clusters using a Markov transition of the form f˚k+1|k (a, x|a′ , x′ ) = fk+1|k (a|a′ , x′ ) · fk+1|k (x|x′ ) (21.277) where fk+1|k (x|x′ ) is a conventional Markov density for point targets, and where fk+1|k (a|a′ , x′ ) is a Markov transition for target number. Since a, a > 0, it follows that fk+1|k (a|a′ , x′ ) cannot have a simple linear-Gaussian form.

21.14

THE UNRESOLVED-TARGET PHD FILTER

The general solution for unresolved targets would be a multitarget Bayes filter of the form ... →

˚ (k) ) f˚k|k (X|Z

→

˚ (k) ) f˚k+1|k (X|Z

→

˚ (k+1) ) f˚k+1|k+1 (X|Z

→ ...

Since this will be computationally intractable in general, we must devise principled approximations. The purpose of this section is to describe one such approximate, the unresolved-target PHD filter. ˚k+1|k (˚ ˚k+1|k (a, x) for unresolved We are given the predicted PHD D x) = D targets with ∫ ∫ ∞ ˚k+1|k (a, x)dadx. Nk+1|k = D (21.278) 0

˚ (k) ) is assumed to be Poisson: The predicted multitarget process f˚k+1|k (X|Z ˚k+1|k ˚X ˚ ˚ ˚ (k) ) = e−N f˚k+1|k (X|Z · Dk+1|k = e−Nk+1|k

∏

˚k+1|k (˚ D x).

(21.279)

˚ ˚ x∈X

Assume that a new measurement set Zk+1 has been collected. Then the corrector equation for the exact unresolved-target PHD filter is ([175], Eqs. (1-5)): ˚k+1|k+1 (a, x) = ˚ ˚k+1|k (a, x) D LZk+1 (a, x) · D

(21.280)

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where the PHD pseudolikelihood is ˚0 (a, x) + ˚ LZk+1 (a, x) = β

∑

ωP

P⊟Zk+1

∑ G ˚(|W |) (a, x) · ˚ LW 0 ; W δ · κk+1 + τW W ⊆P 1,|W |

(21.281)

where the summation is taken over all partitions P of the measurement set Zk+1 ; where δi,j is the Kronecker delta; and where κW k+1

=

∏

κk+1 (z) =

z∈W

τW

=

∏

λk+1 ck+1 (z)

(21.282)

z∈W

∫ ∫

∞

˚(|W |) (a, x) · ˚ G LW (a, x) 0

(21.283)

0

˚ LW (a, x) ˚(i) (a, x) G 0 Also,

=

=

˚k+1|k (a, x)dadx ·D ∏ ˚ Lz (x) z∈W [ i

d ˚ Gz (a, x) dz i

]

(21.284)

(21.285)

. z=0

) Zk+1 δ1,|W | · κk+1 + τW ( ). ωP = ∑ ∏ Zk+1 Q⊟Zk+1 V ∈Q δ1,|V | · κk+1 + τV ∏

W ∈P

(

(21.286)

˚(i) (a, x) can be approximated as ([175], Note that, because of (21.264), G 0 Eq. (13))   ) i (  ∏ 1 (i) ˚(i) (a, x) ∼ G 1+ = Gn,pD (x) (0) · 1 − da + da · (1 − pD (x)) 0  n−i+j  j=1

(21.287)

where da = a − a ˘

(21.288)

where a ˘ is the largest integer smaller than a; and where i n−i G(i) n,q (x) = i! · Cn,i · q · (1 − q + qx)

is the ith derivative of the p.g.f. of the binomial distribution.

(21.289)

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APPROXIMATE UNRESOLVED-TARGET PHD FILTER

Suppose that, relative to sensor resolution, the unresolved target groups are not too close together and the false alarm density is not too large. Then (19.32) reduces to a form that has the same computational complexity as the measurement-update for the classical PHD filter ([175], Eq. (9)): ˚0 (a, x) + ˚ LZk+1 (a, x) ∼ =β

∑

˚1 (a, x) · Lz (x) β κk+1 (z) + τk+1 (z)

(21.290)

z∈Zk+1

where τk+1 (z)

= =

˚1 · ˚ ˚k+1|k [β D Lz ] (21.291) ∫ ∫ ∞ ˚1 (a, x) · ˚ ˚k+1|k (a, x)dadx. (21.292) β Lz (a, x) · D 0

21.16

APPROXIMATE UNRESOLVED-TARGET CPHD FILTER

In 2012, Feng Lian, Chongzhao Han, Weifeng Liu, Jing Liu, and Jian Sun proposed the first (and to date) only generalization of the unresolved-targets PHD filter to an unresolved-targets CPHD filter [85]. Their theoretical methodology was the same as that described in Section 21.5 for extended targets. Indeed, they simultaneously derived formulas for the APB-CPHD filter and the extended-targets CPHD filter. The formulas for the unresolved-targets CPHD filter measurement-update are complicated—they involve combinatorial sums over both the subsets and the partitions of the new measurement set Zk+2 . For that reason they will not be listed here. The purpose of this section is, rather, to report the authors’ implementation and simulation results. Feng et al. implemented both the unresolved-targets PHD filter and the unresolved-targets CPHD filter, using particle methods. In their two-dimensional simulations, a range-bearing sensor observes four moving and appearing and disappearing targets, which follow curvilinear trajectories. The sensor probability of detection is 0.95, and clutter is uniformly distributed Poisson with clutter rate λ = 50.

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The authors reported that, in 300 Monte Carlo runs, both filters adequately estimated the number of unresolved targets, with the CPHD filter exhibiting significantly smaller variances. Similarly, when compared using the OSPA metric (Section 6.2.2), the CPHD filter exhibited significantly less OSPA error, while requiring roughly four times more computational time.

Chapter 22 RFS Filters for Ambiguous Measurements 22.1

INTRODUCTION

The concept of a nontraditional measurement was introduced in Chapter 3 of [179], and described in Sections 1.1.5 and 1.2.8. Such measurements include: • Quantized measurements—as, for example, used for data compression in communications networks. • Attributes—as, for example, extracted by a human operator from a camera image containing a target. • Features—as, for example, extracted from a sensor signature by a digital signal processing (DSP) algorithm. • Natural-language statements—as for example, provided by a human scout or a textual document. • Inference rules—as, for example, drawn from a knowledge-base. The purpose of this chapter is to show how Bayesian processing of conventional measurements can be rigorously extended to nontraditional measurements. For the most familiar measurement types, such as radar detections, relatively little ambiguity adheres to the mathematical representation of a measurement: typically, the measurement is represented as a vector z. Thus the only real uncertainty is that associated with the randomness of the generation of measurements by targets. This uncertainty is most commonly characterized by a nonlinear-additive measurement model Z = η(x) + V (22.1)

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and a corresponding likelihood function f (z|x) = fV (z − η(x)).

(22.2)

Thus it is conventional to think of the z in f (z|x) as the actual “measurement” and of f (z|x) as the full encapsulation of its uncertainty model. But in actuality, z is nothing more than a mathematical model zζ of some real-world measurement ζ. The likelihood thus actually has the form f (ζ|x) = f (zζ |x).

(22.3)

Stated differently: • Actual real-world measurements ζ are mediated by intervening mathematical representations of them. This observation becomes most obvious when one considers nontraditional measurement types. To motivate what follows in this chapter, this section begins with the simplest and most familiar example of a nontraditional measurement type: a quantized measurement. 22.1.1

Motivation: Quantized Measurements

Quantized measurements provide a method for reducing bandwidth usage in communications networks [102]. Let z be a measurement vector in some measurement space Z. Let Z be partitioned (“quantized”) into cells T1 , ..., Tm that are mutually disjoint, and which span Z: T1 ⊎...⊎Tm = Z. Given this, any specific instantiation Z = z of the random measurement Z can be replaced by (“compressed into”) the unique index i such that z ∈ Ti . The index i is sometimes referred to as the “quantized measurement” or “quantization” of z. However, this is a misnomer. We could alternatively employ in place of i some specific choice zi ∈ Ti (such as the centroid of Ti ) to represent Ti . In this case zi is also often described as the “quantized measurement” corresponding to z. This is once again a misnomer. The quantized measurement is neither the abbreviation i nor the abbreviation zi . Rather it is: • The entire subset Ti [54]. In returning Ti as a measurement, we are stating that we are unable to specify the actual measurement z any more precisely than containment within Ti : z ∈ Ti .

(22.4)

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Stated differently, • The quantized measurement Ti is imprecise—it is an imprecise measurement. The ambiguity in a quantized measurement is due not only to the fact that it is imprecise, but also to the fact that it is random. Different quantums Ti will be selected depending on the value of the random measurement Z—that is, on which Ti contains z. Consequently, a random quantized measurement is actually a discrete random subset Ω of Z, with instantiations Ω = T1 , ..., Tm and which is defined by: Ω = Ti if and only if Z ∈ Ti . (22.5) That is, • The quantum Ti is a constraint on the possible values of the underlying random measurement Z. It therefore follows that the probability distribution of the random quantum Ω is defined by pΥ (Ti ) = Pr(Ω = Ti ) = Pr(Z ∈ Ti ). (22.6) Given a nonlinear-additive measurement model, Pr(Z ∈ Ti ) can be explicitly computed: ∫ ∫ Pr(η(x) + V ∈ Ti ) = fV (z)dz = fV (z − η(x))dz (22.7) Ti −η(x) Ti ∫ = f (z|x)dz (22.8) Ti

where Ti − η(x) def. = {z − η(x)| z ∈ Ti }. 22.1.2

(22.9)

Generalized Measurements, Measurement Models, and Likelihoods

The purpose of this chapter is to directly generalize this reasoning to nontraditional measurements of arbitrary form. The following points will be made: 1. Any random closed subset Θ of Z is a generalized measurement. 2. Certain kinds of nontraditional measurements—attributes, features, naturallanguage statements, inference rules—can be mathematically represented as generalized measurements.

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3. Let Θ be a generalized measurement and let Z = η(x) + V a nonlinearadditive measurement model. Then (22.10)

Z∈Θ is a generalized measurement model.

4. Thus a generalized measurement is a “random constraint” of some kind on the possible values of an underlying random measurement process Z. 5. The function ρ(Θ|x) def. = Pr(Z ∈ Θ|x) = Pr(η(x) + V ∈ Θ)

(22.11)

will be called a generalized likelihood function (GLF). 6. Similarly, if Θ1 , ..., Θm are generalized measurements and V1 , ..., Vm are i.i.d. copies of V, then ρ(Θ1 , ..., Θm |x) def. = Pr(η(x, V1 ) ∈ Θ1 , ..., η(x, Vm ) ∈ Θm )

(22.12)

is the joint generalized likelihood function (joint GLF) of Θ1 , ..., Θm . The following question then presents itself: • Is the definition of a GLF as in (22.11) mathematically rigorous from a strict Bayesian point of view? At first glance and generally speaking, this would not appear to be the case. Consider, for example, quantized measurements. In this case, the generalized measurements are Ti with strict Bayesian likelihoods Pr(Ω = Ti |x). The instantiations Ω = T1 , ..., Tm are the only generalized measurements that can be collected; and, according to (22.8) their likelihoods sum to unity: m ∑ i=1

Pr(Ω = Ti |x) =

m ∑ i=1

Pr(η(x) + V ∈ Ti ) =

m ∫ ∑ i=1

f (z|x)dz = 1. (22.13) Ti

In general, however, an arbitrary imprecise measurement T may not necessarily be a quantum in some quantization scheme Ω. In this case, how is it possible to claim that ρ(T |x) is a legitimate likelihood function for all T ? For one thing, ρ(T |x) cannot be a density function because:

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• It never has units of measurement. • Even if we defined a measure-theoretic integral of the form ∫ ρ(T |x)dT,

(22.14)

this integral would most probably be infinite—as opposed to the value 1 that is required for a conventional likelihood function. These two facts will be even more true for nonconstant generalized measurements Θ and their generalized likelihoods ρ(Θ|x). Nevertheless, as has been shown in [161] and as will be explained in Section 22.3.4, • The concept of a GLF ρ(Θ|x) as defined in (22.11) is indeed rigorous from a strict Bayesian viewpoint. Specifically, ρ(Θ|x) = Pr(Z ∈ Θ|X = x).

(22.15)

That is, the GLF is the conditional probability, rigorously defined, that the event Z ∈ Θ will occur given that the event X = x is true. Note that a conventional likelihood f (z|x) is the conditional probability (density) that the event Z = z will occur, given that X = x is true. Thus: • The fact that the likelihood f (z|x) has units and integrates to unity can be regarded as a peculiarity of precise measurements z. This is because – Since Z = z is a zero-probability event, f (z|x) must be a density and thus have units. – Since the z exhaust Z and happen to be mutually exclusive, f (z|x) must integrate to unity. Similar reasoning applies to quantized measurements. In this case, the measurement space is {T1 , ..., Tm } and not Z itself. Thus the fact that ∑m ρ(T |x) = 1 is a peculiarity of quantized measurements. i i=1 22.1.3

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter:

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• Nontraditional measurements—attributes, features, natural language statements, and rules—can be represented as generalized measurements, that is, random closed subsets Θ of some measurement space Z (Section 22.2). • Nontraditional measurements are interpreted as (possibly random) constraints on an underlying nonlinear-additive measurement process Z = η(x) + V,

(22.16)

which in turn means that they obey a generalized measurement model of the form η(x) + V ∈ Θ. (22.17) • Nontraditional measurements can be processed using generalized likelihood functions (GLFs) defined by ρ(Θ|x) = Pr(η(x) + V ∈ Θ)

(22.18)

(Sections 22.3.3 and 22.5.5). • If targets are precisely characterized—that is, if the value of the measurement function η(x) is precisely specified for every x—then it can be shown that the posterior distribution f (x|Z ∈ Θ) conditioned on the event Z ∈ Θ is given by: ρ(Θ|x) · f0 (x) f (x|Z ∈ Θ) = ∫ . (22.19) ρ(Θ|y) · f0 (y)dy That is, suppose that we employ a GLF in Bayes’ rule, just as though it were a conventional likelihood. Then this is a provably Bayes-optimal procedure (Section 22.3.4). The reason for this is that GLFs are actually conditional probabilities of the form ρ(Θ|x) = Pr(Z ∈ Θ|X = x)

(22.20)

and that, consequently, Pr(Z ∈ Θ) =

∫

ρ(Θ|x) · f0 (x)dx.

(22.21)

• Furthermore, the GLF approach appears to be the only existing method for processing nontraditional measurements for which the claim of Bayes optimality can be made.

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• As a consequence of the GLF approach, certain aspects of expert-systems theory—Bayesian inference, fuzzy logic, the Dempster-Shafer theory of evidence, and rule-based inference—can be rigorously unified under a single Bayesian paradigm (Section 22.4.1). • As a consequence, it becomes possible to convert from one uncertainty representation to another, in a provably Bayes-optimal manner (Section 22.4.3). • The GLF approach can be extended to a methodology for addressing targets that are imperfectly characterized—that is, targets for which η(x) is not precisely known for all x. In this case η(x) = Σx is random set-valued (Section 22.5), and the GLF assumes the form ρ(Θ|x) = Pr(Σx ∩ Θ ̸= ∅).

(22.22)

• At the current time it is not known if these more general GLFs are theoretically justified from a strict Bayesian point of view. • In addition, the GLF approach (in the sense of both (22.18) and (22.22)) can be extended to an approach for addressing targets that are completely uncharacterized—that is, are not even in the target model base (Section 22.5). • The GLF approach (in the sense of both (22.18) and (22.22)) leads to a heuristic approach for modeling unknown correlations between information sources (Section 22.7). • It also leads to a rigorous approach for taking into account the unreliability of information sources (Section 22.8). • The GLF approach can be extended to multitarget detection and tracking using RFS filters such as Bernoulli filters, PHD filters, CPHD filters, and the CBMeMBer filter (Section 22.10). • When η(x) is perfectly characterized, this extension is provably Bayesoptimal (even if the RFS filter is itself not provably Bayes-optimal). • A single-target Bernoulli filter implementation due to Bishop and Ristic, using only rather vague natural-language statements as data, is particularly interesting (Section 22.10.5.3). • The GLF approach can be further extended to conventional measurement-totrack association (MTA) techniques (Section 22.11).

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22.1.4

Organization of the Chapter

The remainder of the chapter is organized as follows: 1. Section 22.2: Random set representations of nontraditional measurements. 2. Section 22.3: Generalized likelihood functions (GLFs) for nontraditional measurements, including Bayes optimality of the random set approach GLF approach. 3. Section 22.4: Bayes-optimal unification of fuzzy logic, the Dempster-Shafer theory of evidence, and rule-based inference. 4. Section 22.5: GLFs for incompletely characterized target types. 5. Section 22.6: GLFs for completely uncharacterized (“none-of-the-above” or NOTA) target types. 6. Section 22.7: GLFs for information sources with unknown correlations. 7. Section 22.8: GLFs for unreliable information sources. 8. Section 22.9: Using GLFs in multitarget detection and tracking filters. 9. Section 22.10: Using GLFs in RFS multitarget detection and tracking filters. 10. Section 22.11: Using GLFs in conventional multitarget detection and tracking filters.

22.2

RANDOM SET MODELS OF AMBIGUOUS MEASUREMENTS

Because random sets (generalized measurements) are very abstract entities, one might ask: • How does one construct the random set representation of specific real-world nontraditional measurements, such as attributes, features, natural-language statements, and inference rules? This is accomplished by exploiting the intuitive tools of conventional expertsystem uncertainty representations, as described in the following subsections: 1. Section 22.2.1: Imprecise measurements. 2. Section 22.2.2: Vague (also known as fuzzy) measurements.

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3. Section 22.2.3: Uncertain (also known as Dempster-Shafer or fuzzy DempsterShafer) measurements. 4. Section 22.2.4: Contingent measurements (also known as inference rules on fuzzy measurements). 22.2.1

Imprecise Measurements

The simplest example of a generalized measurement occurs when Θ = T0 is a constant (that is, nonrandom) subset of Z. In this case, T0 is said to be an imprecise measurement. Its semantic meaning is as follows: • A conventional point measurement z0 ∈ Z has been collected, but the only thing known about it is that it is constrained to membership in T0 . That is, its semantic meaning is the relationship z0 ∈ T0 . As already noted, quantized measurements are the most commonly encountered examples of imprecise measurements in practical application. 22.2.2

Vague Measurements

Let g(z) be a fuzzy membership function on Z—that is, a real-valued function with argument z such that 0 ≤ g(z) ≤ 1 for all z ∈ Z. Then g(z) is a vague (also known as fuzzy) measurement. The value g(z) is interpreted as being the degree of membership of the element z in the fuzzy set defined by g. The Zadeh fuzzy logic is defined by: (g ∧ g ′ )(z) (g ∨ g ′ )(z) g c (z)

= = =

min{g(z), g ′ (z)} max{g(z), g ′ (z)} 1 − g(z).

(22.23) (22.24) (22.25)

Another commonly employed logic is the prodsum fuzzy logic defined by: •

(g ∧ g ′ )(z) •

′

(g ∨ g )(z)

= =

g(z) · g ′ (z)

(22.26) ′

1 − (1 − g(z)) · (1 − g (z)).

(22.27)

These are two examples of “copula” fuzzy logics—that is, fuzzy logics that are probabilistic in origin (see [179], pp. 129-131). In general, a fuzzy conjunction operator ‘∧’ on fuzzy membership functions is a copula fuzzy conjunction if it can be written in the form ([179], p. 129, Eq.

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(4.43)) (g ∧A,A′ g ′ )(z) = Pr(A ≤ g(z), A′ ≤ g ′ (z))

(22.28)

where A, A′ are uniformly distributed random numbers on [0, 1]. Its corresponding disjunction operator is given by g ∨A,A′ g ′ = 1 − (1 − g) ∧A,A′ (1 − g ′ ).

(22.29)

Copulas model the statistical dependence between random numbers [224]. Copula conjunctions are central to the unification of fuzzy and Bayesian logic to be described in (22.115). How can we determine whether or not a fuzzy conjunction is also a copula conjunction? The following inequality is satisfied if and only if a fuzzy conjunction ‘∧’ is also a copula ([179], p. 130): a ∧ b + a ′ ∧ b′ ≥ a ∧ b′ + a ′ ∧ b

(22.30)

for all a, a′ , b, b′ ∈ [0, 1] such that a ≤ a′ , b ≤ b′ . Note that the concept of a vague measurement generalizes the concept of an imprecise measurement, since g(z) = 1T (z) is a fuzzy membership function, where 1T (z) is the set indicator function of the quantum T . A second interesting special case occurs when g(z) has only a finite number of distinct values ℓ1 < ... < ℓM . Let Tℓ = {z| ℓ ≤ g(z)}

(22.31)

be the level set of g(z) corresponding to ℓ, in which case T0 = Z for ℓ = 0 and TM +1 = ∅ for ℓ = M + 1. Let Ti abbr. = Tℓi . Then TM ⊂ ... ⊂ T1 and g(z) can be written as ([179], pp. 126-129): g(z)

(22.32)

=

ℓM · 1TM −TM −1 (z) + ℓM −1 · 1TM −1 −TM −2 (z) +... + ℓ2 · 1T2 −T1 (z) + ℓ1 · 1T0 −T1 (z)

=

(ℓM − ℓM −1 ) · 1TM (z) + ... + (ℓ2 − ℓ1 ) · 1T2 (z) + ℓ1 . (22.33)

The semantic meaning of a finite-level fuzzy measurement g(z) is as follows: 1. A conventional point measurement z0 ∈ Z has been collected.

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2. We guess, with degree of confidence ℓM , that z0 is constrained to within membership in TM —that is, by the relationship z0 ∈ TM . 3. As a second and less constraining guess, z0 ∈ TM −1 confidence ℓM −1 .

with degree of

4. Continue in this fashion. The following example provides a more detailed illustration of this concept. 22.2.2.1

Example of a Vague Measurement

Consider the following natural-language report supplied by a human observer ([179], pp. 102-103): ζ = “Gustav is near the tower.” Suppose that Gustav’s state x consists of his position x, y, velocity vx , vy , and identity c—that is, it has the form x = (x, y, vx , vy , c)T . Then the statement ζ provides information about Gustav’s state. Specifically, it tells us that his identity is c = “Gustav” and that his position ηk+1 (x) = (x, y)T is not arbitrary. Rather, it is constrained in regard to his nearness to a particular “anchor” (that is, reference landmark), namely the tower. Two forms of ambiguity impede the modeling of this “measurement:” • Ambiguity due to randomness: Observers will make unknown random errors in their assessments of what they are seeing. • Ambiguity due to ignorance—that is, the ambiguities associated with constructing a mathematical model of ζ. How do we model “fuzzy” and context-dependent concepts such as “near”? As a start, we could model ζ as a solid closed disk T1 ⊆ Z with the tower at its center. Anything within the disk is regarded as “near” and anything outside of it as “far.” Stated differently: The observer has made an assessment of what has been observed—what measurement z = ηk+1 (x) the observer thinks has been perceived—but could do no better than surmise that ηk+1 (x) ∈ T1 . The actual “measurement” that has been collected—and the only information that we have about Gustav’s state—is the constraint T1 on the possible values of ηk+1 (x). However, T1 is just a guess as to what the observer meant by the word “near.” So, we can hedge against our ignorance by specifying a nested sequence T1 ⊆ ... ⊆ Te of successively less restrictive constraints, with the constraint Ti

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assigned a belief τi > 0 that it is the correct one, with τ1 + ... + τe = 1. That is, we guess that ηk+1 (x) ∈ T1 (most restrictive interpretation of “near”) with weight τ1 ; or we guess that ηk+1 (x) ∈ T2 with weight τ2 (a less restrictive interpretation of “near”); and so on. If Te = R2 then this, the “null hypothesis,” is a statement that the observer was completely uncertain about what was perceived, with weight τe . The nested constraints T1 ⊆ ... ⊆ Te , taken together with their associated weights, constitutes the “measurement” that models the natural-language statement ζ. It can be represented as a random closed subset Θζ of measurement space, by defining Pr(Θζ = Ti ) = τi (22.34) for i = 1, ..., e. The nested constraints ηk+1 (x) ∈ Ti on ηk+1 (z) can therefore be equivalently restated as a random constraint ηk+1 (x) ∈ Θζ . This example is actually a specific instance of a fuzzy measurement. Let g(z) be a fuzzy membership function on measurements z, with level sets Ta = {z| a ≤ g(z)}

(22.35)

for 0 ≤ a ≤ 1. The Ta are nested, since Ta ⊆ Ta′ if a ≥ a′ . If the Ta are all centered at the location of the tower, then g is an interpretation of the statement ζ using an infinite number of nested constraints. Thus the fuzzy measurement g(z) is the measurement that has actually been collected. Now define the random subset ([179], p. 123, Eq. (4.21)): Θζ = {z| A ≤ g(z)}

(22.36)

where A is a uniformly distributed random number in [0, 1]. Then Θζ is a random set representation of g—which is, therefore, also a random set representation of the natural-language statement ζ. However, this is not the only such representation. 22.2.2.2

Random Set Representation of Vague Measurements

Every random closed subset Θ of Z defines a fuzzy membership function ([179], p. 123, Eq. (4.20)): µΘ (z) = Pr(z ∈ Θ), (22.37) known as Goodman’s “one-point covering function” of Θ.

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Conversely, given a fuzzy membership function g(z), it is possible to define a family of random closed subsets whose one-point covering functions are g(z). Specifically, Let αz be a uniform random scalar field on Z. That is, for each z ∈ Z, αz is a uniformly distributed random number in [0, 1]. Then the following generalization of (22.36) defines a random closed subset of Z ([179], p. 132, Eq. (4.59): Σα (g) ≜ {z| αz ≤ g(z)}. (22.38) It follows that the one-point covering function of Σα (g) is µΣα (g) (z) = Pr(z ∈ Σα (g)) = Pr(αz ≤ g(z)) = g(z)

(22.39)

where the final equality is due to the fact that αz is uniformly distributed on [0, 1] for every z. When αz = A is constant, (22.38) reduces to the form of (22.36): ΣA (g) = {z| A ≤ g(z)}.

(22.40)

In this case, it is clear that the instantiations Σa (g) for a ∈ [0, 1], are linearly ordered with respect to set inclusion. That is, for any a ̸= a′ , either Σa (g) ⊆ Σa′ (g) or Σa′ (g) ⊆ Σa (g) or both. This is consistent with the “Gustav is near the tower” example presented in the previous subsection. It is easily shown that Zadeh conjunction and disjunction are consistent with set-theoretic intersection and union, respectively ([179], p. 132, Eqs. (4.61-4.62)): Σα (g) ∩ Σα (g ′ ) Σα (g) ∪ Σα (g ′ )

= =

Σα (g ∧ g ′ ) Σα (g ∨ g ′ ).

(22.41) (22.42)

However, fuzzy complementation is not consistent with the set-theoretic complement: Σα (g)c ̸= Σα (g c ). (22.43) 22.2.3

Uncertain Measurements

Suppose that we are given a Dempster-Shafer basic mass assignment (b.m.a.) o(T ) on Z. That is: • o(T ) ≥ 0 is a function defined on all closed subsets T ⊆ Z. • o(T ) ≥ 0 for all T .

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• o(T ) ̸= 0 for only a finite number of T (which are called the focal subsets of o). • The following is true:

∑

o(T ) = 1

(22.44)

T ⊆Z

where the summation is well defined because of the third property. Given this, the function o(T ) is an uncertain (also known as DempsterShafer or D-S) measurement. The semantic meaning of o(T ) is as follows: Each T is a hypothesis about the observed measurement z. Let T1 , ..., Tm be the focal subsets of o. Then one hypothesis is that z is constrained to being in T1 , z ∈ T1 , with weight o(T1 ). A second hypothesis is that z is constrained to being in T2 , z ∈ T2 , with weight o(T2 ). And so on. The number o(Z) is the weight of the hypothesis that we know nothing whatsoever about the measurement z (the “null hypothesis”). A fuzzy/vague measurement is thus an uncertain measurement whose focal subsets are nested (that is, linearly ordered under set-theoretic inclusion). The concept of an uncertain measurement can be generalized as follows. A fuzzy Dempster-Shafer (FDS) basic mass assignment (f.b.m.a.) o(g) is defined by the following properties: • o(g) ≥ 0 is a function defined on all fuzzy membership functions g(z) on Z. • o(g) ≥ 0 for all g. • o(g) ̸= 0 for only a finite number of g (called the focal fuzzy subsets of o). • The following is true:

∑

o(g) = 1.

(22.45)

g

The semantic meaning of o(g) can be illustrated as follows. Each g is a fuzzy hypothesis about the identity of z. Let g1 , ..., gm be the focal fuzzy subsets of o, and assume that they are finite-level. It is unclear that z is or is not constrained by a particular subset T1,1 , so one must regard T1,1 as an initial guess about the meaning of the first fuzzy hypothesis g1 . So as in Section 22.2.2.1, one chooses a nested sequence T1,1 ⊆ ... ⊆ T1,m1 of subsets that further elaborate the nature of the uncertainty involved in the hypothesis g1 .

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As in (22.32) and (22.33), the T1,1 ⊆ ... ⊆ T1,m1 define a finite-level fuzzy membership function g1 . But in general, g1 need not be finite-level. One can interpret the remaining fuzzy focal sets g2 , ..., gm similarly. Suppose that we are given FDS measurements o, o′ . Then the FDS combination of o and o′ is defined by ([179], p. 144, Eq. (4.129)) (o ∗ o′ )(g ′′ ) = 0 if g ′′ = 0 and, if g ′′ ̸= 0, ∑ (o ∗ o′ )(g ′′ ) = αFDS (o, o′ )−1 o(g) · o′ (g ′ ) (22.46) g·g ′ =g ′′

provided that αFDS (o, o′ ) ̸= 0, where the FDS agreement of o, o′ is ∑ αFDS (o, o′ ) = o(g) · o′ (g ′ ).

(22.47)

g·g ′ ̸=0

Here, (g · g ′ )(z) def. = g(z) · g ′ (z) and the event “g ̸= 0” means g(z) ̸= 0 for at least one z. When the focal subsets are actual subsets (“crisp”) then FDS combination reduces to Dempster’s rule of combination: ∑ (o ∗ o′ )(T ′′ ) = αDS (o, o′ )−1 o(T ) · o′ (T ′ ) (22.48) T ∩T ′ =T ′′

where the Dempster-Shafer agreement is ∑ αDS (o, o′ ) = o(T ) · o′ (T ′ ).

(22.49)

T ∩T ′ ̸=∅

The quantity 1 − αDS (o, o′ ) is the conflict between o and o′ . 22.2.3.1

Example of an Uncertain Measurement

Consider the following more complex natural-language report supplied by a human observer ([179], pp. 103-106): ζ

=

“Gustav is probably near the tower, but it could

(22.50)

be the smokestack, it’s so foggy I can’t say for sure.” In this case we are confronted not only with vagueness, but also with uncertainty in the form of three hypotheses. The alternative hypotheses are:

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• “Gustav is near the tower”. • “Gustav is near the smokestack”. • “I’m not sure what I’m seeing”. As in Section 22.2.2.1, we begin by modeling the first hypothesis as a closed circular disk T1 centered at the tower and the second as a closed circular disk T2 centered at the smokestack. The third hypothesis we model as the “null hypothesis” T0 = R2 —that is, there is no constraint at all on Gustav’s position. Suppose that we also know that there are several other landmarks in the vicinity that could be mistaken for the tower or the smokestack. We place disks T3 , ..., Td about them as well. We model the observer’s uncertainty with respect to these hypotheses by assigning weights τd ≥ ... ≥ τ2 > τ1 > τ0 to these three ∑d hypotheses, with j=0 τj = 1. Also as in Section 22.2.2.1, this model does not sufficiently capture the situation because of the vagueness of the concept “near.” That is, Ti is not a sufficiently nuanced representation of proximity to the ith landmark. Consequently, using the same reasoning as in Section 22.2.2.1, replace each Tj with a fuzzy membership function gj (z) that models the jth hypothesis, with probability τj that it represents the correct hypothesis. The collection g1 , ..., gd , together with the weights τ1 , ..., τd , collectively constitute the “measurement” that has been collected (given the uncertainties involved in its construction). Now define the random subset Θζ = {z| A ≤ gJ (z)}

(22.51)

where 1 ≤ J ≤ d is the random integer defined by Pr(J = j) = τj , with A, J being independent. This is the random set representation of the natural-language statement ζ. 22.2.3.2

Random Set Representation of Uncertain Measurements

Given a b.m.a. o(T ), let T1 , ..., Tm be the distinct focal subsets of o. Let Θo be a discrete random closed subset of Z such that Pr(Θo = T ) = o(T ).

(22.52)

Then Θo is called a random set representation of the uncertain measurement o. More generally, let us be given a f.b.m.a. o(g) and let g1 , ..., gm be the distinct focal fuzzy subsets of o. Let J ∈ {1, ..., m} be a random integer such

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that Pr(J = j) = o(gj ).

(22.53)

Define the random closed subset of Z by ΣA,J (o) = ΣA (gJ ) = {z| A ≤ gJ (z)}

(22.54)

where J, A are statistically independent and where ΣA (g) was defined in (22.40).1 Then ΣA,J (o) is called a random set representation of the uncertain measurement o. 22.2.4

Contingent Measurements (Inference Rules)

Let g(z) and g ′ (z) be two fuzzy measurements on Z. Then a (first-order) fuzzy rule has the form ([179], pp. 147-150) g ⇒ g′. It has the following semantic interpretation: if fuzzy measurement g is observed, then the fuzzy measurement g ′ also applies. As an example, g might be a representation of the statement g : “Gustav is near the large oak tree.”

(22.55)

and g ′ might be a representation of the statement g ′ : “Gustav is near the Tower.”

(22.56)

Given this, the rule g ⇒ g ′ indicates the following: we must conclude that Gustav is near the tower, if he was seen near the large oak tree. Therefore, he is near both the oak tree and the tower. 22.2.4.1

Random Set Representation of Contingent Measurements

The random set representation ΣΦ (g ⇒ g ′ ) of a fuzzy rule g ⇒ g ′ is defined only for a finite measurement space Z ([179], Eq. (4.162)): ΣΦ,A,A′ (g ⇒ g ′ ) def. = (ΣA (g ′ ) ∩ ΣA′ (g)) ∪ (ΣA′ (g)c ∩ Φ) 1

(22.57)

(22.54) is equivalent to, but much simpler than, the complicated definition that was presented in [179], pp. 145-147.

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where Φ is a uniformly distributed random subset of Z2 and where the notation ΣA (g) was defined in (22.40). 22.2.5

Generalized Fuzzy Measurements

The generalized fuzzy set concept of Y. Li (see [144] and [179], pp. 133-134) is the most general type of uncertainty representation. In particular, it subsumes all of those previously considered: imprecise, fuzzy, fuzzy Dempster-Shafer, and fuzzy rules. It also subsumes the “vague set” concept of Gau and Buehrer [90]. Let Z be a measurement space and [0, 1] the unit interval. Define Z∗ = Z × [0, 1].

(22.58)

Any subset W ⊆ Z∗ is called a generalized fuzzy measurement. The class of generalized fuzzy measurements is a Boolean algebra under the usual set-theoretic operations. In particular, and unlike the case for fuzzy measurements, • The law of the excluded middle is true for generalized fuzzy measurements. If g(z) is a fuzzy membership function on Z, then Wg = {(z, a)| a ≤ g(z)}

(22.59)

is a generalized fuzzy set. If ‘∧’ and ‘∨’ are the usual Zadeh conjunction and disjunction, then ([179], Eqs. (4.71,4.72)): Wg ∩ Wg′ = Wg∧g′ ,

Wg ∪ Wg′ = Wg∨g′

(22.60)

but, in general, Wgc ̸= Wgc . Every generalized fuzzy measurement gives rise to a fuzzy measurement ([179], Eq. (4.73)): ∫ 1 µW (z) = 1W (z, a)da (22.61) 0

provided that the integral exists. If W = Wg , then µWg (z) = g(z). 2

(22.62)

That Φ is uniformly distributed means that Pr(Φ = T ) = 2−M for every T ⊆ Z, where M is the number of elements in Z.

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22.2.5.1

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Random Set Representation of Generalized Fuzzy Measurements

The random set representation of a generalized fuzzy measurement is similar to that of a fuzzy measurement. Let αz be a uniform random scalar field on Z, as defined in Section 22.2.2.2. Then a random set representation of the generalized fuzzy measurement W ⊆ Z∗ is Σα (W ) = {z| (z, αz ) ∈ W }.

(22.63)

When αz = A is a fixed uniformly distributed random number in [0, 1], this becomes ([179], p. 134, Eq. (4.75)): ΣA (W ) = {z| (z, A) ∈ W }.

(22.64)

It can be shown that Σα (V ∩ W ) Σα (V ∪ W ) Σα (W c )

= = =

Σα (V ) ∩ Σα (W ) Σα (V ) ∪ Σα (W ) Σα (W )c

Σα (∅)

=

∅,

Σα (Z∗ ) = Z.

(22.65) (22.66) (22.67) (22.68)

That is: • The random set representation is completely compatible with the Boolean algebra on generalized fuzzy sets. Note that the random set representation of a fuzzy measurement g(z), (22.38), is a special case of (22.63): Σα (Wg ) = Σα (g).

22.3

(22.69)

GENERALIZED LIKELIHOOD FUNCTIONS (GLFS)

The random set representations of nontraditional measurements, as described in Section 22.2, are—despite being “random sets”—actually deterministic. To understand why, consider the following example. A natural-language statement ζ is deterministic. It is just one possible draw from a hypothetical

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random variable that ranges over all possible natural-language statements. Consequently, it follows that the random set representation Θζ of ζ must also be deterministic. That is, the randomness of Θζ is only a modeling artifact. But in general and in some fashion, ζ will be generated randomly. This randomness is due to the fact that a nontraditional measurement imposes some sort of constraint on some underlying random measurement process Z. Thus let Θ be the random set representation of a nontraditional measurement— a “generalized measurement.” Let Z = η(x) + V

(22.70)

be an underlying nonlinear-additive measurement model. Then the generalized likelihood of Θ is defined to be ρ(Θ|x) = Pr(Z ∈ Θ) = Pr(η(x) + V ∈ Θ).

(22.71)

If the measurement model is noiseless—that is, if V = 0—then the generalized likelihood of the deterministic generalized measurement Θ reduces to the form ρ(Θ|x) = Pr(η(x) ∈ Θ) = µΘ (η(x))

(22.72)

where µΘ (z) is the one-point covering function of Θ, as defined in (22.37). The purpose of this section is to describe generalized likelihoods in more detail. It is organized as follows: 1. Section 22.3.1: GLFs for the “nonnoisy” versions of the nontraditional measurements that were defined in Section 22.2. 2. Section 22.3.2: GLFs for the “noisy” versions of the nontraditional measurements that were defined in Section 22.2. 3. Section 22.3.3: Bayesian processing of generalized measurements using GLFs. 4. Section 22.3.4: Bayes optimality of the GLF approach. 22.3.1

GLFs for Nonnoisy Nontraditional Measurements

Suppose that the underlying sensor model is noiseless—meaning that the generalized measurement Θ is a deterministic representation of some nontraditional measurement. How can we derive a formula for the corresponding generalized

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likelihood ρ(Θ|x)? In this case (22.72) applies: ρ(Θ|x) = µΘ (η(z)).

(22.73)

Thus one need determine only the form of the one-point covering function µΘ (z). The following summarizes Table 5.1 of [179], p. 181, which lists the generalized likelihood functions for various special choices of the non-noisy nontraditional measurements Θ that were described in Section 22.2. ? Generalized likelihood function for fuzzy measurements g: ρ(g|x) = g(η(x)).

(22.74)

? Generalized likelihood function for generalized fuzzy measurements W :

ρ(W |x) = µW (η(x)) =

∫

1

1W (η(x), a)da.

(22.75)

0

? Generalized likelihood function for Dempster-Shafer measurements o: ρ(o|x) =

∑

o(T )

(22.76)

T ∋η(x)

where the summation is taken over all subsets T ⊆ Z that contain η(x). ? Generalized likelihood function for fuzzy Dempster-Shafer (FDS) measurements o: ∑ ρ(o|x) = o(g) · g(η(z)). (22.77) g

? Generalized likelihood function for fuzzy rules g ⇒ g ′ on fuzzy measurements g, g ′ that are represented by random subsets ΣA (g) and ΣA′ (g ′ ):3 1 ρ(g ⇒ g ′ |x) = (g ∧A,A′ g ′ )(η(x)) + (1 − g(η(x))) 2 ′ where the copula fuzzy conjunction ‘∧A,A ’ was defined in (22.28). 3

(22.78)

The derivation of (22.78) depends on the assumption that Z is finite. Since the formula for ρ(g ⇒ g ′ |x) does not depend on this fact, (22.78) can be accepted as a definition, valid for all measurement spaces.

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22.3.2

GLFs for Noisy Nontraditional Measurements

Now suppose that the underlying sensor model is noisy—meaning that the generalized measurement Θ, representing a deterministic representation of some nontraditional measurement, is perturbed by an underlying noisy measurement model Z = η(x) + V. Assume further that Θ, V are independent. Then the corresponding GLF ρ(Θ|x) can be constructed from the likelihood function f (z|x) for Z and the one-point covering function (22.37) for the deterministic Θ, as follows (see (22.93)): ∫ ρ(Θ|x) = µΘ (z) · f (z|x)dz. (22.79) As an example of a GLF of a noisy traditional measurement, consider a fuzzy Dempster-Shafer measurement o as defined in Section 22.2.3. Its one-point covering function was given in (22.77). Thus its GLF, with noise, is ∫ ∑ ρ(o|x) def. = ρ(Θo |x) = o(g) g(z) · f (z|x)dz. (22.80) g

In particular, the GLF for a noisy fuzzy measurement g is ∫ def. ρ(g|x) = ρ(Θg |x) = g(z) · f (z|x)dz. 22.3.3

(22.81)

Bayesian Processing of Generalized Measurements

Given (22.72), it becomes possible to generalize (2.26)—that is, Bayes’ rule for conventional measurements z—to a Bayes’ rule for generalized measurements Θ: fk+1|k+1 (x|Z k+1 ) =

ρk+1 (Θk+1 |x) · fk+1|k (x|Z k ) ρk+1 (Θk+1 |Z k )

(22.82)

where k

ρk+1 (Θk+1 |Z ) =

∫

ρk+1 (Θk+1 |x) · fk+1|k (x|Z k )dx

(22.83)

is the Bayes normalization factor. This means, in turn, that generalized measurements can be processed using the recursive Bayes filter. Equation (22.72) appears to be a heuristic definition, based on the intuitive notion of employing the intuitively conceived GLF ρk+1 (Θ|x) as though it were a

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conventional likelihood function. Consequently, one is led to ask the following question: Will (22.82) actually yield a mathematically rigorous, Bayes-optimal result? This question is answered in the affirmative in Section 22.3.4: • The single-sensor, single-target Bayes filter of Section 2.2.7 can be directly and rigorously extended to nontraditional measurements. As an additional consequence: • Multisensor-multitarget RFS filters, such as the PHD and CPHD filter, can also be directly and rigorously extended to nontraditional measurements (see Section 22.10). Example 10 Suppose that Θk+1 is the random set representation of a DempsterShafer b.m.a. o(T ) with focal subsets T1 , ..., Tm . Suppose that the measurement space is also the state space. Suppose that measurements are noiseless: fk+1 (z|x) = δz (x) where δz (x) is the Dirac delta function concentrated at z. Finally, assume that X = Z is finite and that fk+1|k (x|Z k ) is uniform. Then (22.82) becomes fk+1|k+1 (x|Z k+1 )

=

=

=

µΘk+1 (x) · fk+1|k (x|Z k ) µΘk+1 (y) · fk+1|k (y|Z k )dy ∑m o(Tj ) · 1Tj (x) ∫ ∑m j=1 o(T 1Tj (y)dy j) · j=1 ∑m j=1 o(Tj ) · 1Tj (x) ∑ . m j=1 o(Tj ) · |Tj | ∫

(22.84) (22.85)

(22.86)

This is the Voorbraak probability distribution of the b.m.a. o(T ) (see [179], p. 142, Eq. (4.115)). 22.3.4

Bayes Optimality of the GLF Approach

The purpose of this section is to summarize a proof, initiated in [161] and completed in [191], that the GLF approach is theoretically rigorous from a strict Bayesian point of view. The material in this section is somewhat more general and streamlined than that presented in [191]. Let Θ ⊆ Z be a generalized measurement that models a nontraditional measurement, as described in Section 22.2. By assumption, Θ is considered to be

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a constraint on an underlying random measurement process Z. Let Z = η(x) + V be the measurement model for this underlying process. We may assume that Θ, Z are statistically independent, or equivalently that Θ, V are independent. This is because the construction of Θ (as a model of a nontraditional measurement) should have no dependence on the randomness of the underlying measurement process Z. The statistical independence of a random (closed) subset Θ ⊆ Z and a random vector Z ∈ Z means the following. Define the measure pZ,Θ (T ) = Pr(Z ∈ T ∩ Θ). From the Radon-Nikod´ym theorem we know that there exists an a.e. unique density function fZ,Θ (z) such that pZ,Θ (T ) =

∫

fZ,Θ (z)dz. T

The independence of Θ and Z means that this density factors in the obvious fashion ([191], Eq. (74)): fZ,Θ (z) = fZ (z) · Pr(z ∈ Θ) = fZ (z) · µΘ (z).

(22.87)

Given these preliminaries, let ρ(Θ|x) = Pr(Z ∈ Θ|x) = Pr(η(x) + V ∈ Θ)

(22.88)

be the GLF. Imitating Bayes’ rule as in (22.82), construct the GLF-based posterior distribution conditioned on Θ (subject to the influence of V): ρ(Θ|x) · f0 (x) ρ(Θ)

(22.89)

ρ(Θ|x) · f0 (x)dx.

(22.90)

f (x|Θ) = where ρ(Θ) =

∫

As was demonstrated in [191], (22.89) is a theoretically rigorous Bayes posterior distribution, even though it is computed using the unconventional likelihood ρ(Θ|x). Specifically, the following can be demonstrated:

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1. Bayes Optimality of the GLF Approach: GLF approach

? ?? ? f (x|Θ) =

measure-theoretic approach

? ?? ? f (x|Z ∈ Θ)

(22.91)

where the right side is the conventional posterior distribution conditioned on the probabilistic event Z ∈ Θ, constructed using a measure-theoretic argument. 2. GLFs are Conditional Probabilities: The GLF ρ(Θ|x) is the conditional probability of the event Z ∈ Θ, given the event X = x: ρ(Θ|x) = Pr(Z ∈ Θ|X = x).

(22.92)

3. Integral Formula for GLFs: The GLF ρ(Θ|x) can be constructed from the likelihood function for Z and the one-point covering function (see (22.37)) for Θ: ∫ ρ(Θ|x) = µΘ (z) · f (z|x)dz. (22.93) The following is a sketch of the demonstration of these facts (a full proof is in Section K.33). First, let h(x) be a test function on states, and let ∫ E[h|Θ] = h(x) · f (x|Θ)dx (22.94) be its posterior expected value, as constructed using the GLF of Θ. Second, in [54], Curry, vander Velde, and Potter employ the measure-theoretic identity E[ h | E] = E[ E[h|·] | E] (22.95) where E is some probabilistic event. Here, E[h|·] abbreviates the function z ?→ E[h|z] defined by ∫ ∫ h(x) · f (z|x) · f0 (x)dx E[h|z] = h(x) · f (x|z)dx = . (22.96) f (z) That is, it is the posterior expectation of h with respect to the conventional measurement z, with Bayes normalization constant ∫ fZ (z) = f (z|x) · f0 (x)dx. (22.97)

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Also, for any test function g(z) on measurements (and, in particular, for the test function g(z) = E[h|z]), the quantity ∫ E[g | E] = g(z) · f (z|E)dz (22.98) is the posterior expected value of g with respect to the distribution f (z|E) conditioned on the event E. In our case, E is the event Z ∈ Θ, which results in E[ h | Z ∈ Θ] = E[ E[h|·] | Z ∈ Θ]

(22.99)

where, as is shown in Section K.33, E[g | Z ∈ Θ] =

∫

g(z) · µΘ (z) · fZ (z)dz ∫ . µΘ (w) · fZ (w)dw

Substituting (22.96) and (22.100) into (22.99) yields ∫ ∫ µΘ (z) · f (z|x)dz E[ h | Z ∈ Θ] = h(x) · ∫ · f0 (x)dx µΘ (w) · f (w)dw ∫ = h(x) · f (x|Θ)dx.

(22.100)

(22.101) (22.102)

Since this is true for all test functions h(x), it follows that the measure-theoretic posterior distribution has the form f (x|Z ∈ Θ) = ∫ where ρ˜(Θ|x) =

∫

ρ˜(Θ|x) · f0 (x) ρ˜(Θ|y) · f0 (y)dy

µΘ (z) · f (z|x)dz.

(22.103)

(22.104)

It can be shown that ρ(Θ|x) = ρ˜(Θ|x), from which follows f (x|Z ∈ Θ) = f (x|Θ).

(22.105)

Finally, since f (x|Z ∈ Θ) · Pr(Z ∈ Θ) = Pr(Z ∈ Θ|x) · f0 (x)

(22.106)

it follows that the GLF is the rigorously defined conditional probability ρ(Θ|x) = Pr(Z ∈ Θ|X = x).

(22.107)

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22.4

783

UNIFICATION OF EXPERT-SYSTEM THEORIES

Given the definition of a GLF as ρ(Θ|x) = Pr(Z ∈ Θ|x) = Pr(η(x) + V ∈ Θ),

(22.108)

it can be shown that the random set approach unifies the following aspects of expertsystems theory under a single Bayesian paradigm: • Bayes’ rule. • Fuzzy logic. • The Dempster-Shafer theory. • Rule-based inference. The purpose of this section is to summarize this unification. It is organized as follows. 1. Section 22.4.1: The random set approach unifies measurement fusion. 2. Section 22.4.2: Dempster’s rule of combination arises as a special instance of Bayes’ rule. 3. Section 22.4.3: The random set approach provides a Bayes-optimal approach for converting different kinds of measurements (uncertainty representations) to others: fuzzy to probabilistic, probabilistic to fuzzy, and so on 22.4.1

Bayesian Unification of Measurement Fusion

Measurement fusion refers to the process of combining multiple measurements, possibly from different information sources, into a single composite measurement whose information content is, in some sense, equivalent to that of the original measurements. The random set theory of measurements described in the previous sections in this chapter results in: • A unified Bayesian theory of measurements. • A unified Bayesian theory of measurement fusion. • A unification of expert-systems theory that encompasses measurement-fusion operators such as fuzzy conjunction, Dempster’s rule of combination, and rule-firing.

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The purpose of this section is to summarize this theory. The starting point is as follows. In the Bayesian paradigm: • All measurements are mediated by posterior probability distributions; and, consequently, • Measurement fusion of any kind must be expressed in terms of posterior distributions. Specifically, given conventional measurements z1 , ..., zm , all Bayes-relevant information is contained in the posterior distribution f (x|z1 , ..., zm ) = ∫

f (z1 , ..., zm |x) · f0 (x) f (z1 , ..., zm |y) · f0 (y)dy

(22.109)

where f (z1 , ..., zm |x) is the joint likelihood of z1 , ..., zm and where f0 (x) is the prior distribution. Our interest in the z1 , ..., zm is limited to how they constrain the possible values of the state x. Thus, suppose that we have a combination operator (a “fusion rule”) ‘⊙’ that allows us to fuse z1 , ..., zm into the single measurement z1 ⊙ ... ⊙ zm . To be a fusion rule, ‘⊙’ must be commutative—z1 ⊙ z2 = z2 ⊙ z1 —since otherwise the fused measurement would depend on the order in which information was fused. For the same reason, it must also be associative—(z1 ⊙ z2 ) ⊙ z3 = z1 ⊙ (z2 ⊙ z3 ). A fusion rule is a Bayes combination operator, or is a Bayes-optimal fusion rule, if it loses no Bayes-relevant information. In other words, the following relationship between posterior distributions must be satisfied ([179], pp. 111-112, 182): f (x|z1 ⊙ ... ⊙ zm ) = f (x|z1 , ..., zm ) (22.110) for all z1 , ..., zm and any m, where: f (x|z1 ⊙ ... ⊙ zm ) = ∫

f (z1 ⊙ ... ⊙ zm |x) · f0 (x) . f (z1 ⊙ ... ⊙ zm |y) · f0 (y)dy

(22.111)

Equivalently, ‘⊙’ is a Bayes combination operator if z1 ⊙ ... ⊙ zm is a sufficient statistic, that is, if f (z1 ⊙ ... ⊙ zm |x) = Kz1 ,...,zm · f (z1 , ..., zm |x)

(22.112)

for all z1 , ..., zm and any m, for some Kz1 ,...,zm that is independent of x. Stated differently:

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• ‘⊙’ is a Bayes-optimal fusion rule if measurement fusion using ‘⊙’ is equivalent to measurement fusion using Bayes’ rule alone. The following are examples of Bayes-optimal fusion rules. It is assumed that the underlying noise model is noiseless (V = 0), so that the generalized measurements are deterministic. 1. Bayes-optimal measurement fusion of generalized measurements. The settheoretic intersection ‘∩’ of generalized measurements is a Bayes-optimal fusionrule. For the definition of a joint generalized likelihood function, (22.12) yields the following relationship: (22.113)

ρ(Θ1 , ..., Θm |x) = ρ(Θ1 ∩ ... ∩ Θm |x).

This relationship takes different special forms depending on the particular type of generalized measurement. 2. Bayes-optimal measurement fusion of fuzzy measurements (Section 22.2.2): Let (g ∧ g ′ )(z) = Pr(A ≤ g(z), A′ ≤ g ′ (z)) (22.114) define a copula conjunction, where A, A′ are uniformly distributed random variables on [0, 1]. Then ‘∧’ is a Bayes-optimal measurement fusion rule since ρ(g1 ∧ ... ∧ gm |x) = ρ(g1 , ..., gm |x). (22.115) •

In particular, let ‘∧’ and ‘∧’ be, respectively, the Zadeh and prodsum fuzzy conjunctions of (22.23) and (22.26); and let g1 , ..., gm be fuzzy • measurements. Then both ‘∧’ and ‘∧’ are Bayes-optimal measurement 4 fusion rules since ρ(g1 ∧ ... ∧ gm |x) •

=

ρ(g1 , ..., gm |x)

(22.116)

=

ρ(g1 , ..., gm |x).

(22.117)

•

ρ(g1 ∧ ... ∧ gm |x)

3. Bayes-optimal measurement fusion of generalized fuzzy measurements (Section 22.2.5): Let W1 , ..., Wm be generalized fuzzy measurements with respective random set representations ΣA (W1 ), ..., ΣA (Wm ). Then (22.118)

ρ(W1 ∩ ... ∩ Wm |x) = ρ(W1 , ..., Wm |x) 4

•

•

Caution: This does not mean, however, that ρ(g1 ∧ ... ∧ gm |x) = ρ(g1 ∧ ... ∧ gm |x), since the underlying models are different.

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and so ‘∩’ is a Bayes-optimal measurement fusion rule. 4. Bayes-optimal measurement fusion of FDS measurements (Section 22.2.3): Let ‘∗’ denote FDS combination as defined in (22.46), and let o1 , ..., om be FDS measurements. Then ([179], p. 185, Eq. (5.119)): ρ(o1 ∗ ... ∗ om |x) = Ko1 ,...,om · ρ(o1 , ..., om |x)

(22.119)

for some Ko1 ,...,om . Thus FDS combination ‘∗’ of fuzzy Dempster-Shafer measurements is a Bayes-optimal measurement fusion rule. 5. Bayes-optimal firing of fuzzy rules on measurements (Section 22.2.4): Let g, g ′ be fuzzy measurements with random set representations ΣA (g) and ΣA′ (g ′ ) and let g ⇒ g ′ be the fuzzy rule with antecedent g and consequent g ′ . Then ([179], p. 187, Eq. (5.135)): ρ(g, g ⇒ g ′ |x) = ρ(g ∧A,A′ g ′ |x).

(22.120)

That is, the firing of the rule g ⇒ g ′ by its antecedent g is equivalent to knowing g, g ′ simultaneously. Stated differently: the logical rule modus ponens is Bayes-optimal. 6. Bayes-optimal partial firing of fuzzy rules on measurements: The following generalization, for the “partial firing” of the rule g ⇒ g ′ by a partial antecedent g ′′ , is also true ([179], p. 187, Eq. (5.134)): ρ(g ′′ , g ⇒ g ′ |x) = ρ(g ′′ ∧ (g ⇒ g ′ )|x)

(22.121)

where in this case ‘∧’ denotes the conjunction operator in the GMN conditional event algebra of rules. 22.4.2

Dempster’s Rule Arises as a Particular Instance of Bayes’ Rule

It is well known that the Kalman filter arises as a special case of the single-sensor, single-target recursive Bayes filter ([179], pp. 33-41). Specifically, suppose that the sensor likelihood function is assumed to be linear-Gaussian: fk+1 (z|x) = NRk+1 (z − Hk+1 x).

(22.122)

Then ask the following question: What additional assumptions must be made in order for the Bayes filter to be expressible using closed-form formulas? The answer

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is this: Further assume that the Markov transition density and the initial distribution are linear-Gaussian: fk+1|k (x|x′ ) = NQk (x − Fk x′ ),

f0|0 (x|Z 0 ) = NP0|0 (x − x0|0 ). (22.123)

Then all of the predicted and posterior distributions are linear-Gaussian: fk+1|k+1 (x|Z k+1 ) k

fk+1|k (x|Z )

=

NPk+1|k+1 (x − xk+1|k+1 )

(22.124)

=

NPk+1|k (x − xk+1|k )

(22.125)

where xk|k , Pk|k and xk+1|k , Pk+1|k are as specified by the Kalman filter predictor and corrector equations. In particular, it follows that −1 Pk+1|k+1 −1 xk+1|k+1 Pk+1|k+1

=

−1 −1 T Pk+1|k + Hk+1 Rk+1 Hk+1

(22.126)

=

−1 Pk+1|k xk+1|k

(22.127)

+

−1 T Hk+1 Rk+1 zk+1 .

That is, the information form of the Kalman corrector arises as a special instance of Bayes’ rule. This is true even though the Kalman corrector looks nothing whatsoever like Bayes’ rule. As was shown in [157], • The same assertion is true for Dempster’s rule of combination and its fuzzy generalization—that is, both naturally arise as special cases of Bayes’ rule. Specifically, suppose that the likelihood function is not as in (22.122) but as in (22.77): ∑ ρ(o|x) = o(g) · g(η(z)). (22.128) g k

Let Z : o1 , ..., ok be a time sequence of FDS measurements. Then ask the same question as before: What additional assumptions must be made in order for the Bayes filter to be expressible using closed-form formulas? The answer is this: • the initial distribution must have the form f0|0 (x|Z 0 ) = f (x|ξ0|0 )

(22.129)

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where ξ0|0 is an “FDS state”; and where f (x|ξ) denotes the probability density induced by an FDS state ξ.5 Given this, suppose that fk+1|k+1 (x|Z k+1 ) = ∫

ρ(ok+1 |x) · fk+1|k (x|Z k ) ρ(ok+1 |y) · fk+1|k (y|Z k )dy

(22.130)

is the posterior distribution conditioned on a new FDS measurement ok+1 . Then it can be shown that fk+1|k+1 (x|Z k+1 ) = f (x|ξk+1|k+1 ).

(22.131)

Here, the measurement-updated FDS state is −1 ξk+1|k+1 = ηk+1 ok+1 ∗ ξk+1|k

where ‘∗’ is FDS combination as defined in (22.46); where ∑ −1 (ηk+1 o)(h) = o(g);

(22.132)

(22.133)

g◦ηk+1 =h

and where the summation is taken over all g such that g(ηk+1 (x)) = h(x) for all x; and where (g ◦ ηk+1 )(x) = g(ηk+1 (x)). In particular, if X = Z and h(x) = x for all x, then (22.128) reduces to FDS combination: ξk+1|k+1 = ok+1 ∗ ξk+1|k . (22.134) Thus: since fk+1|k+1 (x|Z k+1 ) is the result of applying Bayes’ rule, it follows that: 5

An FDS state is a fuzzy b.m.a. ξ(h) on fuzzy membership functions h(x) on the state space X whose focal fuzzy subsets have finite integrals. In this case f (x|ξ) is defined by ∑ ξ(h) · h(x) f (x|ξ) = ∑ h ′ ′ h′ ξ(h ) · |h | ∫ where |h′ | = h′ (x)dx. Note that ξ is not uniquely determined by f (x|ξ). Given ξ, define the FDS state ξ ′ by ξ ′ (hξ ) = 1 and ξ ′ (h) = 0 for h ̸= hξ , where ∑ hξ (x) = ξ(h) · h(x). h

Then

ξ′

is an FDS state and f (x|ξ) = f (x|ξ ′ ).

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• Equation (22.132) is just the form that Bayes’ rule becomes when the likelihood function has the form in (22.128); or stated differently, • FDS combination ‘∗’ arises naturally as a special instance of Bayes’ rule. See [157] for more details. This reasoning is the basis of the Kalman evidential filter (KEF), a generalization of the Kalman filter that processes both conventional and FDS measurements ([179], pp. 199-209). 22.4.3

Bayes-Optimal Measurement Conversion

The following question has engendered an ongoing controversy in the expert-system literature: • How does one correctly convert one uncertainty representation to another— fuzzy to probabilistic, Dempster-Shafer to probabilistic, Dempster-Shafer to fuzzy, and so on? In principle, such conversions would appear to be impossible to accomplish in general. For example, any conversion of a Dempster-Shafer b.m.a. o(T ) to a probability distribution fo (z) would, or so it would appear, result in a considerable and unacceptable loss of information ([179], pp. 189-190). For, suppose that the measurement space Z is finite with M elements. Then the specification of a b.m.a. o(T ) on all nonempty T ⊆ Z requires the specification of 2M − 1 numbers. By way of contrast, the specification of a probability distribution fo (z) requires only M − 1 numbers. Thus it would seem that any conversion of o to fo must result in a huge loss of information. A second issue involves the compatibility of fusion rules. Fusion of FDS measurements is accomplished using FDS combination, whereas fusion of fuzzy measurements is accomplished using fuzzy conjunction. Thus if fusion in the FDS realm is to be consistent with fusion in the fuzzy logic realm, one must have a relationship of the form µo∗o′ = µo ∧ µo′ (22.135) where o ?→ µo denotes the conversion of an FDS measurement o to a fuzzy measurement µo . That is, the conversion of the fusion of two measurements must produce the same answer as the fusion of the conversions of those two measurements. These conundrums are easily resolved if one adopts the Bayesian viewpoint advocated in this chapter. That is, they are easily resolved if one insists that

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generalized measurements Θ must always be mediated by posterior distributions f (x|Θ) ([179], pp. 189-194). Specifically: a conversion ζ ?→ cζ from one measurement type ζ to another measurement type cζ does not lose Bayes-relevant information if (22.136)

f (x|cζ1 , ..., cζm ) = f (x|ζ1 , ..., ζm )

for all ζ1 , ..., ζm and all m ≥ 1. ˆ is the Similarly, if ‘⊙’ is the fusion rule for measurements ζ and ‘⊙’ fusion rule for measurements cζ , then the conversion ζ ?→ cζ is Bayes-invariant if ˆ ⊙c ˆ ζm ), f (x|cζ1 ⊙...⊙ζm ) = f (x|cζ1 ⊙... (22.137) a fact which will be automatically true if ˆ ⊙c ˆ ζm cζ1 ⊙...⊙ζm = cζ1 ⊙...

(22.138)

for all ζ1 , ..., ζm and all m. The following discussion summarizes various Bayes-optimal measurementconversion rules. It is assumed that the underlying noise model is noiseless (V = 0), so that the generalized measurements are deterministic. The following format will be observed ([179], pp. 191–193): fusion rule (pre) ‘⊙’

conversion rule ζ→ ? cζ = formula

fusion rule (post) ˆ ‘⊙’

where (1) the left side specifies the fusion rule being used preconversion, (2) the center specifies the fusion rule being used after conversion, and (3) the right side specifies the formula for the conversion rule. 1. Converting fuzzy measurements to FDS measurements. Let g be a fuzzy measurement that is to be converted into an FDS measurement og . Then the conversion rule is fusion rule (pre) prodsum conjunction • ‘∧’

conversion rule ′ { g ?→ og (g′ ) 1 if g =g = 0 if otherwise

fusion rule (post) FDS combination ‘∗’

where the prodsum conjunction operator was defined in (22.26); and where FDS combination was defined in (22.46).

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2. Converting FDS measurements to fuzzy measurements. Let o be an FDS measurement that is to be converted into an fuzzy measurement µo . Then the conversion rule is fusion rule (pre) FDS combination ‘∗’

conversion rule o ?→ µo (z) ∑ = g o(g) · g(z)

fusion rule (post) prodsum conjunction • ‘∧’

3. Converting fuzzy measurements to generalized fuzzy measurements. Let g be a fuzzy measurement that is to be converted into a generalized fuzzy measurement Wg . Then the conversion rule is fusion rule (pre) Zadeh conjunction ‘∧’

conversion rule g ?→ Wg = {(z, a)| a ≤ g(z)}

fusion rule (post) intersection ‘∩’

where the Zadeh conjunction was defined in (22.23); and where the settheoretic intersection of generalized fuzzy measurements was introduced in (22.65). 4. Converting FDS measurements to probabilistic measurements. Let o be an FDS measurement that is to be converted into a probability distribution φo . Then the conversion rule is fusion rule (pre) FDS combination ‘∗’

conversion rule o∑ ?→ φo (z)

=

∑

o(g)·g(z) ∫ g(w)dw g o(g) g

fusion rule (post) parallel combination • ‘∗’

where the Bayesian parallel combination ([179], pp. 137, 186, 272) of two probability distributions f1 (x) and f2 (x) is defined as6 • (f1 ∗ f2 )(x) = ∫

f1 (x) · f2 (x) . f1 (y) · f2 (y)dy

(22.139)

5. Converting probabilistic measurements to fuzzy measurements. Let φ be a probability distribution that is to be converted into a fuzzy measurement µφ . 6

This is actually a parallel combination assuming an improper uniform prior distribution.

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Then the conversion rule is fusion rule (pre) parallel combination • ‘∗’

22.5

conversion rule φ ?→ µφ (z) = supφ(z) φ(w) w

fusion rule (post) prodsum conjunction • ‘∧’

GLFS FOR IMPERFECTLY CHARACTERIZED TARGETS

The following implicit assumption has been made in the preceding sections: • The measurement function η(x) is known with complete accuracy for all choices of x. That is: given the presence of a target with state x, it is known with certainty that the (nonnoisy) measurement it will generate is η(x). However, in practical application this assumption is not valid in general, as will be demonstrated shortly using simple examples. There is an additional difficulty: some targets may be completely uncharacterized because they do not belong to our knowledge base of known targets. That is, they are of the “none of the above (NOTA)” type or class. The generalized likelihood function (GLF) approach described in previous sections can be extended to address such situations. This extension comes with a price, however: • It is not known if this more general GLF approach is strictly Bayes-optimal, in the sense of Section 22.3.4. • Only a more modest form of the expert-system unification results of Section 22.4.1 are possible. The purpose of this section is to summarize the random set GLF approach to nontraditional measurements when targets are imperfectly characterized ([179], pp. 213-222). It is organized as follows: 1. Section 22.5.1: A motivational example involving imperfectly known target types. 2. Section 22.5.2: A motivational example involving applications involving received signal strength (RSS). 3. Section 22.5.3: Random set modeling of imperfectly characterized targets.

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4. Section 22.5.4: Generalized likelihood functions for imperfectly characterized targets. 5. Section 22.5.5: Bayes filtering for imperfectly characterized targets. 22.5.1

Example: Imperfectly Characterized Target Types

Let the target state x = c be the class identifier c of a ground vehicle (a truck, tank, and so on); and let η(c) be some attribute that can be used to distinguish different targets from each other—for example, the number of tires/hubs on the vehicle. Ideally, it is known a priori that a truck of a certain type c0 has exactly η(c0 ) = 6 tires, and that—likewise—the targets of all other class types have been similarly characterized. In actuality, some targets may have been inconclusively characterized. For example, target-type c0 may be believed to have six tires. But it could, with a lesser degrees of certitude, also have only four tires—or, with even lesser certitude, as many as eight tires. 22.5.2

Example: Received Signal Strength (RSS)

Ristic has described an example of an imprecisely described measurement function, which occurs in practical applications involving received signal strength (RSS) [248], [249] for the localization of energy-emitting sources. Suppose that a reference signal source located at (x, y) has RSS as measured at a short reference distance d0 from (x, y). Then its unknown state is x = (x, y, A)T . Let RSS-measuring sensors be located at imperfectly-known locations xi = (xj , yj )T for j = 1, ..., m. (By “imperfectly known” is meant that xj ∈ [ˆ x j − εx , x ˆj + εx ] and yj ∈ [ˆ yj − εy , yˆj + εy ] where (ˆ xj , yˆj ) are the nominal sensor locations and εx , εy are confidence bounds.) Given this, a measurement zj of the RSS (in decibels) collected by the jth sensor can be modeled as zj = ηj (x, Vj ) = ηj (x) + Vj

(22.140)

where Vj is zero-mean white Gaussian noise with known standard deviation; where the measurement function has the form ( ) dj (x, y) ηj (x, y, A) = A − 10 · θj · log (22.141) d0

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and where dj (x, y) =

√

(xj − x)2 + (yj − y)2

(22.142)

for j = 1, ..., m is the distance between the source and the jth sensor; and where θj is the propagation loss from the source to that sensor. Because of multipath and shadowing effects, it has been experimentally determined that θj can have any value in the interval [2, 4]. Consequently, the measurement function for the jth sensor is actually random set-valued. That is, (22.140) assumes the form { ( ) } dj (x, y) abbr. j ηj (x, y, A) = Σ(x,y,A) = A − 10 · θ · log + Vj | θ ∈ [2, 4] d0 (22.143) where for each x, y, A, ηj (x, y, A) is a random set rather than a random function. In this case, the generalized measurement model has the form zj ∈ Σj(x,y,A) . 22.5.3

(22.144)

Modeling Imperfectly Characterized Targets

The purpose of this section is to show how the random set modeling approach can be extended to targets whose measurement functions η(x) are not precisely characterized. Suppose first that the measurement function η(x) is known to within containment in some subset H0,x ⊆ Z: η(x) ∈ H0,x

(22.145)

where H0,x is an initial guess about the possible values that η(x) can have. In this case, one could equally well regard η(x) as being a set-valued function: η(x) = H0,x .

(22.146)

Since H0,x is just a guess, we can hedge by specifying a nested sequence of subsets η(x) ∈ H0,x ⊆ H1,x ⊆ ... ⊆ Hn,x

(22.147)

with ∑n associated probabilities ηi,x that Hi,x is the correct hypothesis, with i=1 ηi,x = 1. In this case, η(x) could be regarded as being nested-set-valued: η(x) = {Hi,x }1≤i≤n .

(22.148)

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More generally still, the Hi,x need not be nested. Either way, {Hi,x }1≤i≤n can be represented as a discrete random subset Σx of Z with Pr(Σx = Hi,x ) = ηi,x ,

(22.149)

in which case η(x) can be regarded as random-set valued: η(x) = Σx .

(22.150)

Most generally, • the target model η(x) = Σx can be taken to be any nonempty random (closed) subset Σx of Z, for every x—that is, any Σx such that Pr(Σx ̸= ∅) = 1.

(22.151)

The random set Σx is a model of a target with state x, taking into account all of the uncertainties associated with the specification of η(x). If the underlying measurement model has noise (that is, V ̸= 0), then the generalized measurement model will have the form Σx + V = {z + V| z ∈ Σx }. 22.5.4

(22.152)

GLFs for Imperfectly Characterized Targets

When the measurement function η(x) is precisely known and Θ is a noiseless nontraditional measurement, the corresponding measurement model is η(x) ∈ Θ. What should it be when η(x) is replaced by Σx ? The answer is the following heuristic formula: Θ ∩ Σx ̸= ∅. (22.153) That is: • the generalized measurement Θ “matches” the target model Σx if it does not flatly contradict it. Note that if Θ = {z} is a conventional measurement then the expression Θ ∩ Σx ̸= ∅ becomes z ∈ Σx . (22.154)

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If we further assume that Σx = {η(x) + V} is a conventional measurement function, it becomes z = η(x) + V. (22.155) If the underlying noise process is not deterministic—that is, if V ̸= 0—then the generalized measurement model would have the form Θ ∩ (Σx + V) ̸= ∅.

(22.156)

However, in what follows it will be assumed for conceptual clarity that V = 0. Given this, the generalized likelihood function (GLF) for imperfectly characterized targets is defined to be: ρ(Θ|x) = Pr(Θ ∩ Σx ̸= ∅).

(22.157)

That is, it is the probability that the measurement matches the target model. Joint GLFs are defined in the obvious manner: ρ(Θ1 , ..., Θm |x) = Pr(Θ1 ∩ Σ1x ̸= ∅, ..., Θm ∩ Σm x ̸= ∅)

(22.158)

where Σ1x , ..., Σm x are independent, identically distributed (i.i.d.) versions of Σx . When Θ1 , Σ1x are independent of Θ2 ∩ Σ2x then ρ(Θ1 , Θ2 |x)

= =

Pr(Θ1 ∩ Σ1x ̸= ∅) · Pr(Θ2 ∩ Σ2x ̸= ∅) ρ(Θ1 |x) · ρ(Θ2 |x).

(22.159) (22.160)

Explicit formulas for (22.157) can be derived in the following special cases: 1. Generalized likelihood function for fuzzy measurements and fuzzy models: Let Θ = ΣA (g) and Σx = ΣA (ηx ) where g(z) and ηx (z) are fuzzy membership functions on Z, and where the notation was defined in (22.40). Then the GLF of g can be shown to be ([179], p. 217, Eq. (6.11)): ρ(g|x) def. = ρ(ΣA (g)|x) = sup min{g(z), ηx (z)}.

(22.161)

z

2. Generalized likelihood function for FDS measurements and FDS models: Let Θo be the random set corresponding to the FDS measurement o(g), and let Σx be the random set corresponding to the FDS measurement σx (g). Then the GLF of o can be shown to be ([179], p. 220, Eq. (6.24)): ρ(o|x) def. = ρ(Θo |x) = αFDS (o, σx )

(22.162)

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797

where the FDS agreement αFDS (o, o′ ) was defined in (22.47). When o is Dempster-Shafer—that is, when all of the focal fuzzy subsets of o are crisp—then ρ(o|x) = αDS (o, σx ) (22.163) where αDS (o, o′ ) is the Dempster-Shafer agreement—that is, αDS (o, o′ ) = 1 − Ko,o′ where Ko,o′ is the Dempster-Shafer conflict between o and o′ . 3. Generalized likelihood function for generalized fuzzy measurements and generalized-fuzzy models: Let Θ = ΣA (W ) and Σx = ΣA (Wx ) where W, Wx are generalized fuzzy subsets of Z and the notation ΣA (W ) was defined in (22.64). Then the GLF of W can be shown to be ([179], p. 219, Eq. (6.17)): ρ(W |x)

= =

ρ(ΣA (W )|x) ∫ 1 sup 1W (z, a) · 1Wx (z, a)da z

=

(22.165)

0

sup z

(22.164)

∫

1

(22.166)

1W ∩Wx (z, a)da. 0

Example 11 Consider Ristic’s received signal strength (RSS) generalized measurement model (22.143): z ∈ Σ(x,y,A) (22.167) where Σ(x,y,A) =

{

A − 10 · θ · log

(

d(x, y) d0

)

+ V | θ ∈ [2, 4]

}

(22.168)

and where the sensor index j has been suppressed for the sake of conceptual clarity. The corresponding GLF is [248], [249]: ρ(z|x, y, A)

=

Pr(z ∈ Σ(x,y,A) ) ( ) ∫ z−A+40·log d(x,y) d

(22.169)

0

=

z−A+20·log

(

d(x,y) d0

where fV (z) is the probability distribution of V .

)

fV (w)dw

(22.170)

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22.5.5

Bayes Filtering with Imperfectly Characterized Targets

The Bayes filter can be used to process generalized measurements, generated by imperfectly characterized targets, exactly as in (22.82). Simulations have shown that Bayes filters perform appropriately when GLFs as in (22.157) are employed— see, for example, pp. 221-232 of [179].

22.6

GLFS FOR UNKNOWN TARGET TYPES

As noted earlier, some targets may not be included in the knowledge base of known target types—they are “NOTA (none of the above).” The GLF approach provides a natural, albeit heuristic, methodology for addressing such problems. The are two cases: modeled targets are completely characterized (the measurement function η(x) is precisely known), or they are not. 22.6.1

Unmodeled Target Type

Let c1 , ..., cM be the modeled target types. Introduce a new NOTA target type c0 and define its GLF to be ([179], pp. 196-199): ρ(Θ|c0 ) = Pr(η(c1 ) ∈ / Θ, ...., η(cM ) ∈ / Θ).

(22.171)

That is, • A generalized measurement Θ is consistent with the NOTA target type if it is inconsistent with all of the modeled target types. As an example, suppose that Θo is the random set model of an FDS measurement o(g). Then the GLF of the NOTA target type is ([179], p. 198, Eq. (5.193)): ρ(Θ|c0 ) =

∑ g

o(g) · min{1 − g(c1 ), ..., 1 − g(cM )}.

(22.172)

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22.6.2

799

Unmodeled Target Types—Imperfectly Characterized Measurement Function

In this case the situation is more complicated ([179], pp. 232-247). The analog of (22.171) is ([179], p. 232, Eq. (6.41)): ρ(Θ|c0 )

=

Pr(Θ ∩ Σc1 = ∅, ..., Θ ∩ ΣcM = ∅)

=

Pr(Θ ⊆ Σc0 )

(22.173)

where Σc0 def. = Σcc1 ∩ ... ∩ ΣccM

(22.174)

is the Boolean-algebra definition of the NOTA type (“NOTA = not-c1 and not-c2 and....and not-cM ”). Alternatively, by analogy with (22.157) one could define ρ(Θ|c0 ) = Pr(Θ ∩ Σc0 ̸= ∅).

(22.175)

Equation (22.173) is the “strong” definition of the NOTA target type, whereas (22.175) is the “weak” definition. An example of Bayes filtering using the weak definition is given in [179], pp. 244-245.

22.7

GLFS FOR INFORMATION WITH UNKNOWN CORRELATIONS

When two or more information sources are independent, their joint GLF can be written as a product: 1 1..s

s

1 1

s s

ρ (Θ, ..., Θ|x) = ρ(Θ|x) · · · ρ(Θ|x)

(22.176)

j

where Θ is the generalized measurement collected by the jth source and where j j

ρ(Θ|x) is its generalized likelihood. In general, information sources will not be independent. For example, if the 1

s

Θ, ..., Θ are features of different kinds drawn from the same camera image, then they will be correlated in unknown ways. The random set GLF approach provides a heuristic means of addressing such correlations ([179], pp. 195-196)).

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1

2 1

2

1

2

The simplest instance occurs if Θ = ΣA1 (g) and Θ = ΣA2 (g) where g, g are fuzzy membership functions defined on possibly different measurement spaces; where A1 , A2 are uniformly distributed random numbers in [0, 1]; and where the notation ΣA (g) was defined in (22.40). Given this, the joint GLF can be explicitly computed to be: 1

2

1

12

1

ρ (Θ, Θ|x)

2 2

(22.177)

Pr(η(x) ∈ Θ, η(x) ∈ Θ)

=

1

1

2

2

Pr(A1 ≤ g(η(x)), A2 ≤ g(η(x)))

=

1

1

2

(22.179)

g(η(x)) ∧A1 ,A2 g(η(x))

=

1

2

1

2

(22.180)

ρ(Θ|x) ∧A1 ,A2 ρ(Θ|x)

=

(22.178)

2

where the copula notation ‘∧A1 ,A2 ’ was defined in (22.28). Thus the joint likelihood can be expressed in terms of a copula that defines the nature of the statistical correlation of the information sources. By analogy, for arbitrary generalized measurements, let 1

s

1..s

1 1

s s

ρ (Θ, ..., Θ|x) = ρ(Θ|x) ∧ · · · ∧ ρ(Θ|x)

(22.181)

where ‘∧’ is some fuzzy conjunction operator. This operator heuristically expresses the intersource correlation that is believed to exist between the information sources. A simple example is the Hamacher fuzzy conjunction ([179], p. 131): a ∧ a′ ≜

aa′ . a + a′ − aa′

(22.182)

It models the correlation between two random variables that are nearly statistically independent. An example of Bayes filtering using this approach is given in [179], pp. 238244.

22.8

GLFS FOR UNRELIABLE INFORMATION SOURCES

Suppose that it is known, via some procedure, that an information source is unreliable. Further suppose that the degree of reliability of this source has been characterized as a number 0 ≤ α ≤ 1, with α = 0 indicating complete unreliability and α = 1 indicating complete reliability. Suppose that the GLF for this information

RFS Filters for Ambiguous Measurements

801

source is ρ(Θ|x). How can we modify ρ(Θ|x) to take this assessment of reliability into account? Furthermore, how do we do so as rigorously as possible? The answer is the reliability-discounted GLF:7 ρ(Θ|x, α) = 1 − α + α · ρ(Θ|x).

(22.183)

This equation has the effect of “flattening” the GLF ρ(Θ|x) and thus increasing the uncertainty in the variable x. If α = 1 then ρ(Θ|x, 1) = ρ(Θ|x). If α is very small, then ρ(Θ|x, α) ∼ = 1 for all x. Discounting the GLF in this fashion is analogous to the strategy of increasing the noise covariance of a conventional sensor in order to address greater uncertainty in the measurements. This approach applies to an information source that is regarded as generally unreliable. But what if the source is reliable in some respects but unreliable in others—that is, it is biased. Such situations can be addressed if the measurement function η(x) is precisely specified. Suppose that it is believed that any measurement in the set B ⊆ Z is unreliable with reliability factor α, as delivered by the information source. Let Θ be a generalized measurement. Then any measurement in Θ ∩ B is unreliable, whereas any measurement in Θ ∩ B c can be regarded as reliable. In this case, the GLF ρ(Θ|x) has the form ρ(Θ|x)

= =

Pr(Z ∈ Θ|x) Pr(Z ∈ Θ ∩ B|x) + Pr(Z ∈ Θ ∩ B c |x)

(22.184) (22.185)

=

ρ(Θ ∩ B|x) + ρ(Θ ∩ B c |x).

(22.186)

The first term should be discounted, leading to the discounted GLF ρ(Θ|x, α, B) = 1 − α + α · ρ(Θ ∩ B|x) + ρ(Θ ∩ B c |x).

(22.187)

The reasoning remains valid if we replace the precisely-specified constraint B with a random-set constraint Ω. Equation (22.183) is the consequence of the following reasoning. First suppose that Θo is the random set representation of a Dempster-Shafer b.m.a. o(T ), as described in Section 22.2.3, with focal subsets T1 , ..., Tm and with Pr(Θo = Tj ) = o(Tj ).

(22.188)

If o is unreliable with reliability factor α, then it is possible to discount it to reflect this additional uncertainty (see Eq. (2) of [200]). Since the weight o(Tj ) of 7

This model was proposed by Bishop and Ristic in [22], but as a definition rather than as a theorem.

802

Advances in Statistical Multisource-Multitarget Information Fusion

a non-null hypothesis Tj ̸= Z is a measure of the confidence in Tj , reduce it by the factor α, oα (Tj ) = α · o(Tj ), (22.189) and shift the sum of the missing weights to the null hypothesis (the hypothesis indicating complete uncertainty): oα (Z) = 1 − α · (1 − o(Z)). Let βo (T ) =

∑

o(W ) = Pr(Θo ⊆ T )

(22.190)

(22.191)

W ⊆T

be the belief-mass function corresponding to o. Then it is easily shown that the belief-mass function corresponding to oα is { ∑ α · βo (T ) if T = ̸ Z α βoα (T ) = o (W ) = . (22.192) 1 if otherwise W ⊆T

Now suppose that Θ is any generalized measurement and let βΘ (T ) = Pr(Θ ⊆ T ) be its belief-mass function. Then we must determine what generalized measurement Θα has the property that { α · βΘ (T ) if T ̸= Z βΘα (T ) = Pr(Θα ⊆ T ) = . (22.193) 1 if otherwise The answer is the following: Θ α = Θ ∪ Zα where Zα is the discrete random subset of Z   1−α α Pr(Zα = T ) =  0

(22.194)

defined by if if if

T =Z T =∅ otherwise

(22.195)

and where Θ, Σx , Zα are assumed to be independent. That is, Θα = Θ with probability α and Θα = Z with probability 1 − α. Suppose that (22.194) is known to be true. Define the reliability-discounted GLF to be the GLF of the discounted generalized measurement Θα : ρ(Θ|x, α) def. = ρ(Θα |x).

(22.196)

RFS Filters for Ambiguous Measurements

803

Abbreviate Σx,α = Σx ∩ Θα . Then (22.183) follows from ρ(Θα |x)

= =

Pr(Σx,α ̸= ∅) (22.197) α α Pr(Σx,α ̸= ∅, Z = Z) + Pr(Σx,α ̸= ∅, Z ̸= Z) (22.198)

where Pr(Σx,α ̸= ∅, Zα = Z) = Pr(Σx ̸= ∅) · (1 − α) = 1 − α

(22.199)

and Pr(Σx,α ̸= ∅, Zα ̸= Z) = Pr(Σx ∩ Θ ̸= ∅) · α

(22.200)

and where Pr(Σx ̸= ∅) = 1 because of (22.151). To prove (22.194), note that

=

Pr(Θα ⊆ T ) Pr(Θ ∪ Zα ⊆ T )

= =

Pr(Θ ∪ Zα ⊆ T, Zα = Z) + Pr(Θ ∪ Zα ⊆ T, Zα = ∅) (22.203) Pr(Z ⊆ T ) · (1 − α) + Pr(Θ ⊆ T ) · α (22.204)

(22.201) (22.202)

which equals 1 if T = Z and α · βΘ (T ) otherwise.

22.9

USING GLFS IN MULTITARGET FILTERS

The discussion in this chapter thus far has been oriented towards tracking a single target using a single nontraditional information source. However, if the measurement function η(x) is precisely known then the single-target approach can be readily extended to multisource-multitarget detection and tracking. Furthermore, this extension is theoretically rigorous. This is because (according to (22.92)) if the GLF ρ(Θ|x) = Pr(Z ∈ Θ) = Pr(η(x) + V ∈ Θ) (22.205) is employed in Bayes’ rule like a conventional likelihood function, f (x|Z ∈ Θ) =

Pr(Z ∈ Θ|x) · f0 (x) , Pr(Z ∈ Θ)

(22.206)

then the GLF is a rigorously defined conditional probability that behaves like a likelihood function in Bayes’ rule: Pr(η(x) + V ∈ Θ) = Pr(Z ∈ Θ|X = x).

(22.207)

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Advances in Statistical Multisource-Multitarget Information Fusion

Nearly all of the multitarget filters discussed in this book require simplifying independence assumptions in the multitarget measurement model. A typical consequence of these assumptions is that the multitarget filtering formulas are constructed in terms of single-sensor likelihoods Lz (x) = fk+1 (z|x). The same is true for the clutter intensity function κk+1 (z). Thus in what follows, it will be assumed that: • The measurement function ηk+1 (x) is precisely known. • The single-target likelihood function has the nonlinear-additive form fk+1 (z|x) = fVk+1 (z − ηk+1 (x)).

(22.208)

• Any generalized measurement Θ is independent of the underlying random measurement Z = ηk+1 (x) + Vk+1 . Given these assumptions and (22.79), the corresponding GLF has the form ∫ LΘ (x) = ρk+1 (Θ|x) = µΘ (z) · fk+1 (z|x)dz (22.209) where µΘ (z) is the one-point covering function of Θ as defined in (22.37). In similar fashion, κk+1 (z) can be thought of as a likelihood function of the form κk+1 (z|c) where c is the state of a clutter generator, but in which κk+1 (z|c) = κk+1 (z) does not functionally depend on c. Given this, one can extend κk+1 (z) to generalized measurements in the obvious manner: ∫ κk+1 (Θ) = µΘ (z) · κk+1 (z)dz. (22.210) More generally, suppose that fk+1 (Z|X) is a multitarget likelihood function and κk+1 (Z) is the distribution of a clutter RFS. Then these can be extended to generalized measurements as follows: ∫ fk+1 ({Θ1 , ..., Θm }|X) = µΘ1 (z1 ) · · · µΘm (zm ) (22.211)

κk+1 ({Θ1 , ..., Θm })

=

·fk+1 ({z1 , ..., zm }|X)dz1 · · · dzm ∫ µΘ1 (z1 ) · · · µΘm (zm ) (22.212) ·κk+1 ({z1 , ..., zm })dz1 · · · dzm .

RFS Filters for Ambiguous Measurements

805

Consequently, to apply the techniques in the preceding sections of this chapter to multitarget scenarios, all one has to do is: • Substitute LΘ (x) in place of Lz (x) wherever Lz (x) occurs. • Substitute κk+1 (Θ) in place of κk+1 (z) wherever κk+1 (z) occurs. In what follows, two situations will be considered in turn: • Processing generalized measurements using RFS-based multitarget filters (Section 22.10). • Processing generalized measurements using conventional multitarget filters (Section 22.11).

22.10

GLFS IN RFS MULTITARGET FILTERS

The section is organized as follows: 1. Section 22.10.1: Using GLFs in PHD filters. 2. Section 22.10.2: Using GLFs in CPHD filters. 3. Section 22.10.3: Using GLFs in CBMeMBer filters. 4. Section 22.10.4: Using GLFs in Bernoulli filters. 5. Section 22.10.5: Implementations of RFS multitarget filters for use with generalized measurements. 22.10.1

Using GLFs in PHD Filters

The measurement-update equations for the classical PHD filter were described in Section 8.4.3. Let Zk+1 = {z1 , ..., zm } be a set of conventional measurements. Then the measurement-update formula for the classical PHD filter is ∑ Dk+1|k+1 (x) pD (x) · Lz (x) = 1 − pD (x) + Dk+1|k (x) κk+1 (z) + τk+1 (z)

(22.213)

z∈Zk+1

where κk+1 (z) is the clutter intensity function and where ∫ τk+1 (z) = pD (x) · Lz (x) · Dk+1|k (x)dx.

(22.214)

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Advances in Statistical Multisource-Multitarget Information Fusion

Suppose instead that Zk+1 = {Θ1 , ..., Θm } is a set of generalized measurements. Then (22.213) assumes the form Dk+1|k+1 (x) = 1 − pD (x) + Dk+1|k (x)

∑

pD (x) · LΘ (x) κk+1 (Θ) + τk+1 (Θ)

(22.215)

Θ∈Zk+1

where LΘ (x) was defined in (22.209), κk+1 (Θ) was defined in (22.210) and where ∫ τk+1 (Θ) = pD (x) · LΘ (x) · Dk+1|k (x)dx. (22.216) 22.10.1.1

PHD Filters with Vague Measurements

Suppose in particular that Θj = ΣA (gj ) where gj (z) are fuzzy measurements. Then if |Zk+1 | = m then (22.215) becomes m ∑ Dk+1|k+1 (x) pD (x) · Lgj (x) = 1 − pD (x) + Dk+1|k (x) κ (g ) + τk+1 (gj ) j=1 k+1 j

(22.217)

where by (22.209) and (22.210) Lg (x) κk+1 (g)

=

∫

g(z) · Lz (x)dz

(22.218)

=

∫

g(z) · κk+1 (x)dz.

(22.219)

Example 12 (Classical PHD filter is limiting case) Suppose that gj (z) = 1Ej (z) where Ej is a very small neighborhood of zj , with hypervolume |Ej | = V for all j = 1, ..., m. Then Lgj (x) ∼ (22.220) = V · Lzj (x) and κk+1 (gj ) ∼ = V · κk+1 (zj ).

(22.221)

Thus in the limit as |Ej | ↘ 0, (22.217) reduces to the conventional PHD filter measurement-update, (22.213):   m ∑ pD (x) · Lzj (x)  · Dk+1|k (x). Dk+1|k+1 (x) −→ 1 − pD (x) + κ (z ) + τ (z ) k+1 j k+1 j j=1 (22.222)

RFS Filters for Ambiguous Measurements

807

Thus the PHD filter for generalized measurements is consistent with the PHD filter for conventional measurements. 22.10.2

Using GLFs in CPHD Filters

Recall from (8.105) through (8.115) that the measurement-update equations for the classical CPHD filter are pk+1|k+1 (n)

=

ℓZ (n) · pk+1|k (n) ∑ k+1 l≥0 ℓZk+1 (l) · pk+1|k (l)

(22.223)

Dk+1|k+1 (x)

=

LZk+1 (x) · Dk+1|k (x)

(22.224)

where ( ∑ min{m,n}

) (m − j)! · pκk+1 (m − j) ·j! · Cn,j · ϕn−j · σj (Zk+1 ) k ( ∑m ) (22.225) κ l=0 (m − l)! · pk+1 (m − l) (l) ·σl (Zk+1 ) · Gk+1|k (ϕk )   ND pD (x)) · L Zk+1 1  ∑ (1 − (22.226) m pD (x)·Lzj (x) D Nk+1|k + j=1 ck+1 (zj ) · LZk+1 (zj ) j=0

ℓZk+1 (n)

=

LZk+1 (x)

=

( ∑m

) − j)! · pκk+1 (m − j) (j+1) ·σj (Zk+1 ) · Gk+1|k (ϕk ) ( ∑m ) (22.227) κ l=0 (m − l)! · pk+1 (m − l) (l) ·σl (Zk+1 ) · Gk+1|k (ϕk ) ( ∑m−1 ) κ i=0 (m − i − 1)! · pk+1 (m − i − 1) (i+1) ·σi (Zk+1 − {zj }) · Gk+1|k (ϕk ) ( ∑m ) (22.228) κ (m − l)! · p (m − l) k+1 l=0 (l) ·σl (Zk+1 ) · Gk+1|k (ϕk ) j=0 (m

ND

L Zk+1

=

LZk+1 (zj )

=

D

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Advances in Statistical Multisource-Multitarget Information Fusion

and where σi ({z1 , ..., zm })

=

ϕk

=

τˆk+1 (z)

=

(

τˆk+1 (z1 ) τˆk+1 (zm ) , ..., ck+1 (z1 ) ck+1 (zm )

)

σm,i ∫ (1 − pD (x)) · sk+1|k (x)dx ∫ pD (x) · Lz (x) · sk+1|k (x)dx.

(22.229) (22.230) (22.231)

Suppose instead that Zk+1 = {Θ1 , ..., Θm } with |Zk+1 | = m is a set of generalized measurements. Then the CPHD filter equations can be applied by simply substituting LΘ (x)

=

∫

µΘ (z) · fk+1 (z|x)dz

(22.232)

ck+1 (Θ)

=

∫

µΘ (z) · ck+1 (z)dz

(22.233)

in place of Lz (x) and ck+1 (z). Remark 88 (Classical CPHD filter is limiting case) Let Θj = ΣA (gj ) be fuzzy measurements such that gj (z) = 1Ej (z), where Ej is a very small neighborhood of zj with hypervolume |Ej | = V for all j = 1, ..., m. Note that in the CPHD filter measurement-update equations, Lgj (x) and ck+1 (gj ) are always paired together in a ratio. Thus as |Ej | ↘ 0, Lgj (x) ck+1 (gj )

∼ = ∼ =

V · Lzj (x)

(22.234)

V · ck+1 (zj )

(22.235)

and so sk+1|k [pD Lgj ] sk+1|k [pD Lzj ] −→ . ck+1 (gj ) ck+1 (zj )

(22.236)

Thus in the limit, the CPHD filter measurement-update equations for fuzzy measurements reduce to the CPHD filter equations for conventional measurements.

RFS Filters for Ambiguous Measurements

22.10.3

809

Using GLFs in CBMeMBer Filters

Recall from (13.51) through (13.58) in Section 13.4.3, that the measurement-update equations for the CBMeMBer filter are 1 − sik+1|k [pD ]

qiL

=

i qk+1|k ·

sL i (x)

=

sik+1|k (x) ·

i 1 − qk+1|k · sik+1|k [pD ]

1 − pD (x) 1 − sik+1|k [pD ]

(22.237) (22.238)

and ∑νk+1|k

i i qk+1|k (1−qk+1|k )·sik+1|k [pD Lzj ] i (1−qk+1|k ·sik+1|k [pD ])2

i=1

qjU

= κk+1 (zj ) +

sU j (x)

∑

∑νk+1|k i=1

i νk+1|k qk+1|k i i=1 1−qk+1|k

=

∑νk+1|k i=1

i qk+1|k ·sik+1|k [pD Lzj ]

(22.239)

i 1−qk+1|k ·sik+1|k [pD ]

· sik+1|k (x) · pD (x) · Lzj (x)

i qk+1|k i 1−qk+1|k

(22.240) · sik+1|k [pD Lzj ]

where sik+1|k [pD ] sik+1|k [pD Lzj ]

=

∫

pD (x) · sik+1|k (x)dx

(22.241)

=

∫

pD (x) · Lzj (x) · sik+1|k (x)dx.

(22.242)

Suppose instead that Zk+1 = {Θ1 , ..., Θm } with |Zk+1 | = m is a set of generalized measurements. Then the CBMeMBer filter equations can be applied by substituting ∫ LΘ (x) = µΘ (z) · fk+1 (z|x)dz (22.243) ∫ κk+1 (Θ) = µΘ (z) · κk+1 (z)dz. (22.244) Remark 89 (CBMeMBer filter is limiting case) Let Θj = ΣA (gj ) be fuzzy measurements such that gj (z) = 1Ej (z), where Ej is a very small neighborhood

810

Advances in Statistical Multisource-Multitarget Information Fusion

of zj with hypervolume |Ej | = V for all j = 1, ..., m. As with the CPHD filter, in the CBMeMBer filter measurement-update equations Lgj (x) and ck+1 (gj ) always occur as a ratio. Thus as |Ej | ↘ 0, Lgj (x) ∼ = V · Lzj (x),

κk+1 (gj ) ∼ = V · κk+1 (zj )

(22.245)

and so in the limit, the CBMeMBer filter equations for fuzzy measurements reduce to the CBMeMBer filter equations for conventional measurements. 22.10.4

Using GLFs in Bernoulli Filters

The filtering equations for the Bernoulli filter were given in (13.7) and (13.8). If the clutter process is extended to generalized measurements as in (22.212), the Bernoulli filter equations can likewise be extended as follows: (22.246)

pk+1|k+1 =

∑

(Zk+1 −{Θ}) 1 − sk+1|k [pD ] + Θ∈Zk+1 sk+1|k [pD LΘ ] · κk+1 κk+1 (Zk+1 ) ∑ κk+1 (Zk+1 −{Θ}) p−1 z∈Zk+1 sk+1|k [pD LΘ ] · k+1|k − sk+1|k [pD ] + κk+1 (Zk+1 )

and sk+1|k+1 (x) sk+1|k (x) =

∑ (Zk+1 −{Θ}) 1 − pD (x) + pD (x) z∈Zk+1 LΘ (x) · κk+1 κk+1 (Zk+1 ) ∑ (Zk+1 −{Θ}) 1 − sk+1|k [pD ] + z∈Zk+1 sk+1|k [pD LΘ ] · κk+1 κk+1 (Zk+1 )

where κk+1 (Zk+1 ) was defined in (22.212) and where ∫ sk+1|k [pD ] = pD (x) · sk+1|k (x)dx ∫ sk+1|k [pD LΘ ] = pD (x) · LΘ (x) · sk+1|k (x)dx. 22.10.5

(22.247)

(22.248) (22.249)

Implementations of RFS Filters for Nontraditional Measurements

The purpose of this section is to describe performance results for various implementations of the GLF approach described in this chapter. The following implementations are considered:

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1. Section 22.10.5.1: RF transmitter identification with imprecise attribute data. 2. Section 22.10.5.2: Target classification using formatted-message data. 3. Section 22.10.5.3: Bernoulli filtering with imprecise natural-language statements. 4. Section 22.10.5.4: Estimation of the parameters of a static received signal strength (RSS) source. 22.10.5.1

RF Transmitter Identification with Imprecise Attribute Data

In 1999, Mahler, Leavitt, Warner, and Myre applied GLF techniques to the problem of identifying imperfectly characterized, frequency-hopping, radio frequency (RF) transmitters, using imprecise measurements of the observed frequencies ([193]; [179], pp. 244-245). Since the transmitters were imperfectly characterized, fuzzy target models were devised using the methodology described in Section 22.5. A Bayes classifier using these techniques was constructed. The measurements were assumed to be frequency-intervals of fixed width, with randomly-varying centers. Five possible transmitter types were possible. One would intuitively expect that the classifier would perform better if the degrees of imprecision and randomness were smaller, than if they were large. But the opposite behavior was observed. When both the interval width and the variance of the interval center were decreased, the classifier exhibited greater difficulty in identifying the transmitters. Indeed, the fourth and fifth transmitters could not be distinguished at all. The reason for this was a peculiar interaction between uncertainty due to imprecision and uncertaintly due to randomness. The center frequencies of the fourth and fifth transmitters were closer together than those of other possible transmitter pairs. Because of the imprecision of the measurements, the fourth and fifth transmitters could be distinguished from each other only if a statistically outlying measurement was collected. Since this was more likely to happen when the randomness of the center frequency was larger, the classifier was able to perform better under such conditions. 22.10.5.2

Target Classification Using Formatted-Message Data

In 2001, Sorensen, Brundage, and Mahler applied GLF techniques to the problem of identifying poorly characterized ground and air targets of 75 different types ([273];

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Advances in Statistical Multisource-Multitarget Information Fusion

[179], pp. 238-247). The data was assumed to be derived from human operators observing sensor data of various types, who created data in the form of formatted alphanumeric strings. A simulated scenario called “LONEWOLF-98” contained ground targets of 16 different types, observed using various sensors: moving target indicator radar (MTI), electronic intelligence (ELINT), communications intelligence (SIGINT), and imaging sensors. A Bayes classifier was used to process 100 formatted messages. The authors reported that the classifier identified the correct target in all instances, with very high confidence. A “weak NOTA” version of the classifier was also tested, by including a target type not contained in its data base. The authors reported that the classifier successfully identified the presence of the target of unknown type. 22.10.5.3

Bernoulli Filtering with Imprecise Natural-Language Statements

Bishop and Ristic [22] and Ristic, Vo, Vo, and Farina ([262], Section IX-E) report the application of GLFs to the problem of detecting and tracking at most one target ([262], Section VII) using a particle filter implementation of a Bernoulli filter. The GLF-based Bernoulli filter was also tested in simple two-dimensional simulations ([262], Section IX-E). In [22], Bishop and Ristic report a detailed analysis in which the person of interest moves in a field containing the following landmarks: a wall, an L-shaped building, a rectangular swimming pool, a tower, and a square-shaped garage. Five different observers report the following natural-language statements: ζ1

=

“The target is in the field.”

(22.250)

ζ2

=

(22.251)

ζ3

=

“If the sun is shining then the target is near the pool or the garage.” “I do not see the target.”

ζ4 ζ5

= =

“The target is in front of the tower.” “The target is at one o’clock.”

(22.253) (22.254)

(22.252)

All these statements require prior knowledge of the current positions of the observers. Also, note that ζ2 has the form of an inference rule. The authors report that the filter achieved remarkable accuracy in localizing the target, “given nothing but rather vague individual statements about its possible location” ([22], p. 939). In the simulations reported by Ristic, Vo, Vo, and Farina, a pedestrian-filled corridor is first prepared by identifying two dozen “anchors” (landmarks). Multiple

RFS Filters for Ambiguous Measurements

813

human observers attempt to spot and track an appearing and disappearing individual of interest, who is moving through the corridor. When they spot the target, they provide reports of the form “The target is near the anchor A.” It is assumed that a speech recognition system and language parser are applied to the observer’s reports to extract the semantic content and produce an imprecise measurement. The preprocessed reports are then fed to a Bernoulli filter, which has been implemented using particle methods. It is assumed that probability of detection is pD = 0.9, and the false-alarm rate is λ = 0.15 with Poisson clutter. The authors reported that their filter detected and tracked the subject with reasonable accuracy, given the imprecise nature of the measurements. 22.10.5.4

Received Signal Strength (RSS) Estimation

Ristic’s GLF approach for estimating source locations using received signal strength (RSS) measurements was described in (22.143) and (22.169). Recall that this involved GLFs of the form zj ∈ Σj(x,y,A) (22.255) for m RSS-measuring sensors at imperfectly-known locations (xj , yj ) for j = 1, ..., m. Ristic assumed that the RSS source is static and that there were m = 12 sensors, arranged approximately in a circle around the RSS source. He implemented a Bayes filter (actually, multiple iterations of Bayes’ rule, since the source is static) using particle methods. The prior distribution was assumed to be uniform in some region. The GLF filter was compared to a similar implementation of a Bayes filter using a conventional likelihood function. Both filters were run using a sequence of M = 5000 RSS measurements [248], [249]. Ristic observed that, at the conclusion of the scenario, the GLF-filter’s positional particle cloud and its histogram for A were both reasonably wellconcentrated. The estimates of position and A derived from these were both biased—though contained in the support of the particle cloud histogram. The filter with the conventional likelihood function, had a more tightly concentrated positional particle cloud and histogram. However, neither the particle cloud nor the histogram contained the correct parameter values of the source—thus indicating high certainty about inaccurate estimates. Finally, Ristic ran 1000 Monte Carlo trials to determine the percentage of times that the support of the final GLF-posterior contained the true parameter values. This percentage rose steadily as the number M of measurements increased, reaching 100% for about M = 1000 and above.

814

22.11

Advances in Statistical Multisource-Multitarget Information Fusion

USING GLFS WITH CONVENTIONAL MULTITARGET FILTERS

Measurement-to-track association (MTA), the basis of the conventional approach to multitarget detection and tracking, was reviewed in Section 7.2.4. It is possible to generalize the mathematics described in that section so that both nontraditional and traditional measurements can be processed in the same way using MTA methods. The purpose of this section is to describe these methods, which were first described in 2011 in [164]. The section is organized as follows: 1. Section 22.11.1: Measurement-to-track association (MTA) with generalized measurements. 2. Section 22.11.2: An exact closed-form example of MTA with fuzzy measurements. 3. Section 22.11.3: MTA with joint kinematic and nonkinematic measurements. 22.11.1

Measurement-to-Track Association (MTA) with Nontraditional Measurements

Suppose that, at time tk+1 , we are in possession of n predicted tracks with respective predicted track distributions fk+1|k (x|1), ..., fk+1|k (x|n). Also suppose that, at time tk+1 , the sensor collects a set Zk+1 = {Θ1 , ..., Θm } of generalized measurements with |Zk+1 | = m. Then as in Section 7.2, a measurement-to-track association (also known as association hypothesis) is a function θ : {1, ..., n} → {0, 1, ..., m} such that θ(i) = θ(i′ ) > 0 implies i = i′ . Let Zθ = {Θθ(i) | θ(i) > 0} (22.256) denote the set of target-generated generalized measurements; and let mθ = |Zθ | denote the number of those measurements. Given this, the MTA formulas of Section 7.2.4 can be extended to generalized measurements. As in that section, assume that the probability of detection is constant: pD (x) = pD . Recall from (22.209) and (22.210) that the GLFs for target-generated measurements and for Poisson clutter are, respectively, ∫ ρk+1 (Θ|x) = µΘ (z) · fk+1 (z|x)dz (22.257) ∫ κk+1 (Θ) = µΘ (z) · κk+1 (z)dz (22.258)

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where µΘ (z) is the one-point covering function of Θ as defined in (22.37). Equations (7.37) and (7.35) can then be generalized to ∏ κk+1 (θ) = e−λk+1 κk+1 (Θ) (22.259) ℓ˜k+1 (Θ|i)

=

∫

Θ∈Zk+1 −Zθ

ρk+1 (Θ|x) · fk+1|k (x|i)dx.

(22.260)

We then get: • Global association likelihood for generalized measurements: Equation (7.36) generalizes to ∏ n−mθ θ ℓZk+1 |Xk+1|k (θ) = κk+1 (θ) · pm (1 − p ) ℓ˜k+1 (Θθ(i) |i). D D i:θ(i)>0

(22.261) • Global association probability for generalized measurements: (7.38) generalizes to pZk+1 |Xk+1|k (θ) = ∑ 22.11.2

ℓZk+1 |Xk+1|k (θ) θ ′ ℓZk+1 |Xk+1|k (θ

. ′)

Equation

(22.262)

A Closed-Form Example: Fuzzy Measurements

The purpose of this section is to show that MTA using fuzzy measurements can, under suitable simplifying assumptions, be accomplished using exact closed-form formulas. The approach is therefore potentially suitable for practical implementation. The material follows that of Section 6.1 of [164]. The approach can be extended to fuzzy Dempster-Shafer (FDS) measurements, but at the price of increased computational complexity (see Section 6.2 of [164]). The concept of a fuzzy (“vague”) measurement was introduced in Section 22.2.2.2. The GLF for a noisy fuzzy measurement g(z) was given in (22.81): ∫ ρk+1 (g|x) = g(z) · fk+1 (z|x)dz. (22.263) Assume that at time tk+1 : • The distributions of the predicted tracks are linear-Gaussian: fk+1|k (x|i) = NPi (x − xi ).

(22.264)

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Advances in Statistical Multisource-Multitarget Information Fusion

• The information source collects m fuzzy measurements Z = {g1 , ..., gm } that are linear-Gaussian in form: √ gj (z) = det 2πGj · NGj (z − gj ). (22.265) • The conventional likelihood function is linear-Gaussian: fk+1 (z|x) = NRk+1 (z − Hk+1 x).

(22.266)

• The conventional Poisson clutter intensity function is linear-Gaussian in form: κk+1 (z) = λk+1 · NCk+1 (z − ck+1 ). (22.267) (In particular, if ∥Ck+1 ∥ is very large then κk+1 (z) is effectively uniform.) Then it is easily shown that: • The generalized likelihoods of the fuzzy measurements g1 , ..., gm are: √ ρk+1 (gj |x) = det 2πGj · NRk+1 +Gj (gj − Hk+1 x). (22.268) • The clutter generalized likelihoods of the g1 , ..., gm are: κk+1 (gj ) = λk+1 · NCk+1 +Gj (gj − ck+1 ).

(22.269)

• The local association likelihoods (22.260) of the g1 , ..., gm become √ T ℓ˜k+1 (gj |i) = det 2πGj · NRk+1 +Gj +Hk+1 Pi Hk+1 (gj − Hk+1 xi ). (22.270) Let Jθ ⊆ {1, ..., m} be the set of indices j of the fuzzy measurements that are not target-generated—that is, of those j such that gj ∈ Zk+1 − Zθ . Then the global association likelihood, see (22.261), is ℓZk+1 |Xk+1|k (θ)

=

mθ θ e−λk+1 λm−m · pD (1 − pD )n−mθ (22.271) k+1 κ ·Qk+1|k (θ) · Qk+1|k (θ) ( ) 1 κ 1 2 2 · exp − dZk+1 |Xk+1|k (θ) − dZk+1 |Xk+1|k (θ) 2 2

where dκZk+1 |Xk+1|k (θ)2 =

∑ j∈Jθ

(gj − ck+1 )T (Ck+1 + Gj )−1 (gj − ck+1 )

(22.272)

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817

and dZk+1 |Xk+1|k (θ)2   (gθ(i) − Hk+1 xi )T ∑ T  ·(Rk+1 + Gθ(i) + Hk+1 Pi Hk+1 )−1  ·(gθ(i) − Hk+1 xi ) i:θ(i)>0

=

(22.273)

and where Qκk+1|k (θ) = ∏

j∈Jθ

√

1 det 2π(Ck+1 + Gj )

(22.274)

and (22.275)

Qk+1|k (θ) 1 =

∏

i:θ(i)>0

√

. T ) det 2π(Rk+1 + Gθ(i) + Hk+1 Pi Hk+1

For, first note that (22.259) becomes Z

κk+1 (θ)

= =

k+1 e−λk+1 κk+1

−Zθ

(22.276) 1 θ √ e−λk+1 λm−m ·∏ (22.277) k+1 det 2π(Ck+1 + Gj ) j∈Jθ    (gj − ck+1 )T ∑ 1  ·(Ck+1 + Gj )−1  · exp − 2 ·(gj − ck+1 ) j∈Jθ

and thus (22.261) becomes, as claimed, ℓZk+1 |Xk+1|k (θ)

=

n−mθ θ κk+1 (θ) · pm D (1 − pD ) ∏ · ℓ˜k+1 (gθ(i) |i)

(22.278)

i:θ(i)>0

=

n−mθ θ κk+1 (θ) · pm D (1 − pD ) 1 √ ·∏ T ) det 2π(Rk+1 + Gθ(i) + Hk+1 Pi Hk+1 i:θ(i)>0 ∑   − 12 i:θ(i)>0 (gθ(i) − Hk+1 xi )T T · exp  ·(Rk+1 + Gθ(i) + Hk+1 Pi Hk+1 )−1  ·(gθ(i) − Hk+1 xi )

(22.279)

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Advances in Statistical Multisource-Multitarget Information Fusion

=

22.11.3

n−mθ θ θ e−λk+1 λm−m · pm · Qκk+1|k (θ) · Qk+1|k (θ) D (1 − pD ) k+1 ( ) 1 κ 1 2 2 · exp − dZk+1 |Xk+1|k (θ) − dZk+1 |Xk+1|k (θ) . (22.280) 2 2

MTA with Joint Kinematic and Nonkinematic Measurements

Practical applications that involve nontraditional measurements typically involve target states of the form (c, x) where: • x is a kinematic state vector. • c∈C on).

is a discrete identity state variable (target type, target class, and so

In such applications, the measurements typically have the form (ϕ, z) where • z is a kinematic measurement of x (such as position). • ϕ is a nonkinematic measurement of c (a target feature). The purpose of this section is to extend the results of the previous sections to this situation. The material follows that of Section 7 of [164]. Assume that probability of detection is constant: pD (x) = pD . Then under current assumptions, the local association likelihood of (22.260) has the form ∑∫ ˜ ℓk+1 (Θϕ , z|i) = ρk+1 (Θϕ , z|c, x) · fk+1|k (c, x|i)dx (22.281) c∈C

where Θϕ is the random set representation of the feature ϕ. Assume that: • Kinematic measurements z and nonkinematic features ϕ approximately, statistically independent, in the sense that

are, at least

ρk+1 (Θϕ , z|c, x) κk+1 (Θϕ , z)

= =

ρk+1 (Θϕ |c) · fk+1 (z|x) kin κfea k+1 (Θϕ ) · κk+1 (z)

(22.282) (22.283)

fk+1|k (c, x|i)

=

fk+1|k (c|i) · fk+1|k (x|i)

(22.284)

µΘϕ (z) · ck+1 (z)dz

(22.285)

where κfea k+1 (Θϕ )

=

∫

κkin k+1 (z)

=

λk+1 · ck+1 (z).

(22.286)

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The section is organized as follows: 1. Section 22.11.3.1: Local association likelihoods for joint kinematic and nonkinematic measurements. 2. Section 22.11.3.2: Global association probabilities for joint kinematic and nonkinematic measurements. 3. Section 22.11.3.3: Processing joint kinematic and nonkinematic measurements. 22.11.3.1

Local Association Likelihoods for Joint kinematic and nonkinematic Measurements

Let Zk+1 kin Zk+1

= =

{(Θϕ1 , z1 ), ..., (Θϕm , zm )} {z1 , ..., zm }

(22.287) (22.288)

fea Zk+1

=

{Θϕ1 , ..., Θϕm }.

(22.289)

In this case (22.259) becomes κk+1 (θ)

=

∏

e−λk+1

(22.290)

κk+1 (Θ, z)

(Θ,z)∈Zk+1 −Zθ



=

θ e−λk+1 λm−m · k+1



·  =



(Θ,z)∈Zk+1 −Zθ

fea −Z fea Θ∈Zk+1 θ



(Θ,z)∈Zk+1 −Zθ

 cfea k+1 (Θ)

(22.291)



∏

∏



∏

 ckin k+1 (z) 

 cfea k+1 (Θ)

θ · e−λk+1 λm−m k+1

∏ kin −Z kin z∈Zk+1 θ

(22.292) 

 ckin k+1 (z)

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Advances in Statistical Multisource-Multitarget Information Fusion

and so kin κk+1 (θ) = cfea k+1 (θ) · κk+1 (θ)

where cfea k+1 (θ)

∏

=

cfea k+1 (Θ)

(22.293)

∏

(22.294)

fea −Z fea Θ∈Zk+1 θ

κkin k+1 (θ)

e−λk+1

=

κkin k+1 (z)

kin −Z kin z∈Zk+1 θ

Also, (22.260) can be simplified to ℓ˜k+1 (Θϕ , z|i)

=

∑∫

ρk+1 (Θϕ |c) · ρk+1 (z|x)

(22.295)

c∈C

·fk+1|k (c|i) · fk+1|k (x|i)dx

=

(

∑

ρk+1 (Θϕ |c) · fk+1|k (c|i)

)

(22.296)

c∈C

· =

(∫

fk+1 (z|x) · fk+1|k (x|i)dx

)

˜kin ℓ˘fea k+1 (Θϕ |i) · ℓk+1 (z|i)

(22.297)

where ℓ˘fea k+1 (Θϕ |i)

=

∑

ρk+1 (Θϕ |c) · fk+1|k (c|i)

(22.298)

fk+1 (z|x) · fk+1|k (x|i)dx.

(22.299)

c∈C

ℓ˜kin k+1 (z|i)

=

∫

Thus the kinematic and nonkinematic parts of the problem can be dealt with separately.

RFS Filters for Ambiguous Measurements

22.11.3.2

821

Global Association Probabilities for Joint Kinematic/Nonkinematic Measurements

Given the previous considerations, the global association probability of (22.262) becomes

pZk+1 |Xk+1|k (θ) = ∑

kin pfea Zk+1 |Xk+1|k (θ) · pZk+1 |Xk+1|k (θ) θ′

kin ′ ′ pfea Zk+1 |Xk+1|k (θ ) · pZk+1 |Xk+1|k (θ )

(22.300)

where pfea Zk+1 |Xk+1|k (θ) pkin Zk+1 |Xk+1|k (θ)

=

∑

=

∑

ℓfea Zk+1 |Xk+1|k (θ)

(22.301)

fea ′′ θ ′′ ℓZk+1 |Xk+1|k (θ )

ℓkin Zk+1 |Xk+1|k (θ) kin ′′ θ ′′ ℓZk+1 |Xk+1|k (θ )

.

(22.302)

For, the global association likelihood of (22.261) separates into kinematic and nonkinematic parts: ℓZk+1 |Xk+1|k (θ)

=

n−mθ θ κk+1 (θ) · pm D (1 − pD ) ∏ · ℓ˜k+1 (Θθ(i) , zθ(i) |i)

(22.303)

i:θ(i)>0

=

mθ kin n−mθ κfea k+1 (θ) · κk+1 (θ) · pD (1 − pD )   ∏ · ℓ˘k+1 (Θθ(i) |i) i:θ(i)>0



·

∏ i:θ(i)>0



ℓ˜k+1 (zθ(i) |i)

(22.304)

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=

κfea k+1 (θ) ·

∏

ℓ˘k+1 (Θθ(i) |i)

(22.305)

i:θ(i)>0 n−mθ θ ·pm · κkin k+1 (θ) D (1 − pD ) ∏ · ℓ˜k+1 (zθ(i) |i) i:θ(i)>0

=

kin ℓfea Zk+1 |Xk+1|k (θ) · ℓZk+1 |Xk+1|k (θ)

(22.306)

where ℓfea Zk+1 |Xk+1|k (θ)

κfea k+1 (θ) ·

=

∏

ℓ˘k+1 (Θθ(i) |i)

(22.307)

i:θ(i)>0

ℓkin Zk+1 |Xk+1|k (θ)

n−mθ θ pm · κkin k+1 (θ) D (1 − pD ) ∏ · ℓ˜k+1 (zθ(i) |i).

=

(22.308)

i:θ(i)>0

Thus, as claimed,

pZk+1 |Xk+1|k (θ)

=

=

=

22.11.3.3

∑

ℓZk+1 |Xk+1|k (θ)

′ θ ′ ℓZk+1 |Xk+1|k (θ ) ℓfea (θ) · ℓkin Zk+1 |Xk+1|k (θ) Zk+1 |X ∑ fea k+1|k kin ′ ′ θ ′ ℓZk+1 |Xk+1|k (θ ) · ℓZk+1 |Xk+1|k (θ )

∑

kin pfea Zk+1 |Xk+1|k (θ) · pZk+1 |Xk+1|k (θ) θ′

kin ′ ′ pfea Zk+1 |Xk+1|k (θ ) · pZk+1 |Xk+1|k (θ )

(22.309)

(22.310)

. (22.311)

Processing Joint Kinematic/Nonkinematic Measurements

The purpose of this section is to show how the formulas derived in earlier sections allow us to propagate the joint kinematic-nonkinematic tracks in a MTA-based algorithm, using single-object filters. • Time update of the kinematic part of a track: This is accomplished in the usual fashion, via the time-update equations for the extended Kalman filter (EKF).

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823

• Time update of the nonkinematic part of a track: This is accomplished using the prediction step of a discrete Bayes filter:8 pk+1|k (c) =

∑

pk+1|k (c|c′ ) · pk|k (c′ ).

(22.312)

c′ ∈C

• Measurement update of the kinematic part of a track: This is accomplished in the usual fashion, with the measurement-update equations for the extended Kalman filter (EKF). • Measurement update of the nonkinematic part of a track: This is accomplished using the Bayes’ rule step of a discrete Bayes filter: pk+1|k+1 (c) = ∑

ρk+1 (Θϕk+1 |c) · pk+1|k (c) ′ ′ c′ ∈C ρk+1 (Θϕk+1 |c ) · pk+1|k (c )

(22.313)

where pk+1|k (c|c′ ) is the Markov transition matrix.

8

As was noted in Remark 33 in Section 9.5.8.1, it is not necessarily the case that target identity does not change with time.

Part V

Sensor, Platform, and Weapons Management

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Chapter 23 Introduction to Part V Automated sensor management is the process of “directing the right sensors to the right targets at the right times.” Since sensors are often carried on moving platforms, sensor management inherently subsumes automated platform management—“directing the right sensors, on the right platforms, to the right targets, at the right times.” Moreover, modern “smart” weapons are typically themselves sensorcarrying platforms, since in their terminal phase they are guided to targets by onboard “seeker” sensors. Thus, sensor management also inherently subsumes much of automated weapons management. When treated with this degree of generality, sensor management is usually referred to as resource management or “Level 4 data fusion.” The purpose of this chapter is to describe a statistically unified approach to resource management, viewed as a generalized form of sensor management. The core concepts (closed-loop multitarget Bayes filter, multitarget control theory, multitarget information-theoretic objective functions, single-sample hedging-optimization) were first introduced in 1996 [162]. They were further elaborated in 2005 [169], and most recently summarized in 2011 [184]. Otherwise, the approach has been extended and refined in a dispersed, somewhat scattershot fashion in various publications over a period of fifteen years. The material in Part V is its first extended, systematic description. Let us begin by reviewing the challenges that are peculiar to the problem. A sensor management system must, if it is to accomplish its goals, simultaneously take into account a large number of disparate factors that affect performance: • The strengths and weaknesses of the sensors, and of the platforms that carry them—for example:

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– Platform availability, location, and body-frame orientation. – Platform aerodynamic constraints and fuel/power levels. – Platform environmental constraints (terrain, weather,and so on). – Sensor gimbal orientation, slew rate, and range of motion. – Sensor resolution, and false alarm and/or clutter characteristics. – Physical extent and shape of sensor Field of View (FoV). • The current and anticipated future characteristics of targets—for example, – Constraints on target motion due to target dynamical limitations. – Environmental constraints on target motion (terrain, obscuration). – Estimated target fuel/power supply. – The likelihood that targets will appear or disappear, in what numbers and in what locations. • Constraints associated with communication networks—for example: – Bandwidth. – Signal dropouts. – Time latencies. • Dynamic and unpredictably changing scenarios, including the unpredictable evolution of operators’ priorities and commanders’ tactical goals. • The highly complex, nondeterministic, and nonlinear interactions between all of these factors. Ultimately, this means that resource management requires a seamless integration of several interrelated functionalities: • Detailed and accurate multisensor-multitarget statistical modeling. • Multisensor-multitarget information fusion. • Multitarget area or volume search. • Multitarget detection, localization, tracking, and identification. • Multitarget-multisensor sensor management.

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• Multisensor-multitarget platform management. • Tactical prioritization of targets according to situational significance. • Operator contextual guidance, including evolving commanders’ priorities. The approach described in this chapter is based on three insights: • Resource management is inherently an optimal nonlinear control problem. • However, it differs from standard control problems in that it is also inherently a stochastic combinatoric multiobject problem, involving: – Randomly varying finite sets of targets. – Randomly varying finite sets of measurements. – Randomly varying finite sets of sensors/platforms. • Consequently, the practical implementation of resource management requires drastic but principled approximations. Indeed, principled computability is the overriding emphasis of Part V. The approach summarized in this chapter inherently accounts for these issues, using the following top-down, system-level, Bayesian paradigm: 1. Model all platforms, sensors, and targets as a single, joint dynamically evolving multiobject stochastic system. 2. Encapsulate the characteristics of all sensors in the form of multisensormultitarget likelihood functions. 3. Encapsulate the dynamic characteristics of all targets in the form of multitarget Markov transition densities. 4. Encapsulate subjective, mission-specific imperatives in the form of tactical importance functions (TIFs). 5. Using the information in the previous four items, propagate the state of the joint system using a multisensor-multiplatform-multitarget Bayes filter. 6. Apply information-theoretic objective functions that codify global goals for sensor management, but which are also physically intuitive and therefore mission-intuitive.

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7. Apply sensor/platform optimization strategies that hedge against the inherent unknowability of future observation collections. 8. Devise principled approximations of this general (but usually intractable) formulation, including: (a) Approximate multisensor-multiplatform-multitarget filters. (b) Approximate sensor management objective functions. (c) Approximate sensor management hedging and optimization strategies. The remainder of this introduction is devoted to an overview of the approach. It is organized as follows: 1. Section 23.1: A summary of some of the basic issues in sensor management. 2. Section 23.2: An introduction to one possible approach for unifying information theory with operational reality: the posterior expected number of targets (PENT); its generalization, the posterior expected number of targets of interest (PENTI); and its variant, the cardinality variance. 3. Section 23.3: A summary of the general approach and its approximations. 4. Section 23.4: The organization of Part V.

23.1

BASIC ISSUES IN SENSOR MANAGEMENT

Research in sensor management has been preoccupied with three major procedural dichotomies. Should sensor management be: 1. Top-down (control-theoretic) or bottom-up (rule-based)? 2. Single-step look-ahead (“temporally myopic”) or multistep look-ahead (“model myopic”)? 3. Information-theoretic (theoretically principled) or mission-oriented (subjective/heuristic/intuitive)? Each of these dichotomies is considered in turn.

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Top-Down or Bottom-Up?

Bottom-up (BU) sensor management methodologies are based on attempts to (1) partition a resource management problem into a hierarchy of subtasks, and then (2) to employ some amalgam of mathematics and heuristics to direct transitions from one subtask in the hierarchy to another. These transitions can be entirely rule-based (“if this occurs, do that”), or can employ the optimization of subtask-by-subtask utility functions. Top-down (TD) sensor management methodologies are, broadly speaking, based on control-theoretic paradigms. The basic idea behind them is to (1) determine optimal or near-optimal allocation of sensor/platform assets via (2) maximization or minimization of some overall global objective function. BU versus TD approaches have complementary strengths and weaknesses: • Heuristic versus mathematically defensible. • Deterministic versus stochastic. • Fast versus computationally intensive. • Stable versus potentially unstable (for example, contention between sensors). On the one hand, the accumulation of locally optimized decisions in a BU system does not necessarily roll up into desired global sensor/platform behavior. TD systems, on the other hand, inherently achieve globally near-optimal decisions. Similarly, BU systems can suffer from rigidity due to finite rule bases that are incapable of addressing the infinite number of possible contingencies that arise in practice. TD systems, by way of contrast, exhibit greater robustness because they are capable of addressing this infinitude of contingencies. It is for these reasons that, in this chapter and throughout Part V, the emphasis will be on top-down—specifically, control-theoretic—resource management. 23.1.2

Single-Step or Multistep?

Single-step look-ahead ( “myopic”) TD systems attempt to determine the optimal allocation of sensors and platforms only at the next time-step. Multistep look-ahead TD systems, on the other hand, attempt to optimally allocate sensors throughout an entire future time-window. Multistep TD sensor management is often portrayed as inherently superior— indeed, the very terminology “nonmyopic” implies as much. However, this portrayal is both inaccurate and misleading. The multistep look-ahead approach

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is feasible only when all target motions can be predicted with sufficient accuracy throughout the entire future time-window. If this is the case, then multistep systems will indeed tend to produce smooth, accurate sensor/platform allocations. Singlestep systems, on the other hand, will often produce inefficient, “zig-zaggy” solutions. Suppose, however, that the motions of some targets are unpredictable, because of factors such as evasive target maneuvers or transits by targets of road intersections. In such cases, multistep systems will be model-myopic rather than temporally myopic. Because of the faulty target predictions caused by blind adherence to inaccurate motion models, they will recommend correspondingly inaccurate sensor/platform placements. Single-step systems, on the other hand, will adapt more nimbly to unanticipated target behaviors. A final issue: multistep systems are also far more computationally demanding than single-step systems, especially in multisensor-multitarget contexts. It is for these reasons that, throughout Part V, the emphasis will be on singlestep TD systems. Nevertheless, the approach can be extended to multistep look-ahead, as is briefly shown in Section 23.3.6. 23.1.3

Information-Theoretic or Mission-Oriented?

TD sensor management systems are based on the optimization of some sort of “global” objective function. There are two extremal approaches: • Objective functions based on some formal mathematical definition of “information,” for example: – Shannon entropy. – Kullback-Leibler cross-entropy (which subsumes entropy). – The R´enyi α-divergence family (which subsumes Kullback-Leibler cross entropy). – The vast Csisz´ar divergence family, described in Section 6.3 (which subsumes α-divergence). • “Mission-oriented” objective functions, which attempt to devise mathematical formulas that, at least approximately, capture the intent underlying subjective tactical objectives. These objective functions are often assumed to have

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an additive structure that is amenable to solution using “dynamic programming” methods.1 Mission-oriented objective functions tend to be heuristic. Yet even so, they often do not end up adequately modeling actual mission objectives. Informationtheoretic objective functions are, on the other hand, theoretically rigorous. Yet they have no obvious relationship with operational requirements. Indeed, they typically do not even have a clear intuitive, physical interpretation. What reason is there to believe, for example, that maximizing some abstract Shannon-type information functional will result in sufficiently optimal collection of missionrelevant information? Perhaps the biggest challenge is the fact that the very definition of missionoriented information typically depends on subjective factors such as operators’ priorities and commanders’ intentions. There is a further difficulty: the fact that there is a literal infinitude of information-theoretic objective functions. For example, each choice of α results in a different α-divergence objective function. Similarly, each choice of a convex weighting function c(x) results in a different Csisz´ar-divergence objective function. This creates a “Tower of Babel” dilemma: which information functional should one choose, and when, and why? One argument raised in support of the α-divergence family is that one can resolve this dilemma by empirically determining the optimal value of α. But, ultimately, this only begs the question.2 Why not go one better and try to determine the optimal choice of c(x)—a seeming impossibility? More to the point: What is the “optimal” criterion for optimality? How does one know that this optimal choice, if it even exists or could be determined, has anything to do with mission requirements? The position taken in this chapter, as in earlier publications [160], [184], is as follows: • We must somehow establish strong connections between two seemingly incommensurable realms: – Abstract information theory on the one hand. 1 2

Good descriptions of the dynamic programming methodology an be found in [240], pp. 369-378; and [107], pp. 18-23. A number of criticisms of the α-entropy approach have been raised in [3] and [4]. Simulation comparisons can be found in Aughenbaugh and La Cour [12] and in Chun, Kadar, Blasch, and Bakich [27].

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– Subjective but intuitively accessible operational reality. • Foundational multisensor-multiplatform-multitarget statistics provides a good starting point for achieving this goal.

23.2

INFORMATION THEORY AND INTUITION: AN EXAMPLE

The purpose of this section is to introduce at least one concept that establishes common ground between abstract information theory and physical intuition. In basic surveillance applications, a “natural” or “inherent” goal of resource management should be to, at any moment: • Maximize the number of well-resolved targets; and, more generally. • Maximize the number of well-resolved targets of interest. In fact, these could be regarded as the minimal objectives of any mission. They obviously have an immediate, intuitive, physical meaning. But as we shall see, both: • Can be formulated in a mathematically rigorous manner (Section 25.9.4). • Are approximate information-theoretic objective functions—specifically, they are rather drastic approximations of Kullback-Leibler cross-entropy, αdivergence, and other information-theoretic functionals (Section 25.9.4). These approximate objective functions are: • The posterior expected number of targets (PENT) Nk|k at a given time tk . • A generalization of Nk|k , the posterior expected number of targets of interest ι (PENTI) Nk|k at a given time tk , given a tactical importance function (TIF) ιk|k (x). • A variant of Nk|k , the cardinality variance (the variance of Nk|k ). • The Cauchy-Schwartz divergence, which reduces to a simple formula when used with PHD filters. The remainder of the section is organized as follows: 1. Section 23.2.1: PENT for “cookie cutter” sensor fields of view (FoVs).

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2. Section 23.2.2: PENT for general sensor FoVs. 3. Section 23.2.3: Characteristic behaviors of PENT. 4. Section 23.2.4: The cardinality-variance objective function. 5. Section 23.2.5: The Cauchy-Schwartz divergence objective function. 23.2.1

PENT for “Cookie Cutter” Sensor Fields of View (FoVs)

Consider the following thought experiment, which is represented schematically in Figure 23.1. We are given the following: • Four poorly localized ground targets. • A single high-resolution sensor. • The sensor has a disk-shaped “cookie cutter” Field of View (FoV)—that is, pD = 1 in the disk and pD = 0 outside it. • The sensor has no missed detections and no clutter, so that any measurement has to be target-generated. Given this, we desire an answer to the following question: Over time, how should the sensor FoV be positioned in the most informative manner? Suppose that the FoV can be positioned so that it covers three targets (Nk|k = 3) rather than merely two (Nk|k = 2). Then this would clearly be the better choice. We could then collect measurements from three targets and use these measurements to better localize them. If no larger number of targets can be covered, then this will be the best possible placement of the FoV. But at the next time-step, information will not be further increased by placing the FoV over the same three targets. This is because these targets have already been sufficiently localized, while the remaining two targets have not been. This time, it would be better to reposition the FoV to cover the remaining poorly-detected target. This will result in an increase in PENT from Nk|k = 3 to Nk+1|k+1 = 4. Finally, since we did not collect measurements from the first three targets, their error ellipses will increase in size. We therefore run some risk of losing them altogether. So at the next time-step, the better choice would be to reposition the FoV over the first three targets and collect additional measurements from them. Thus as time progresses, successive maximization of Nk|k will tend to result in a “reciprocating” placement of the FoV, in which it tends to “bounce back and forth” between one group of targets and another.

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Figure 23.1 An illustration of the PENT (posterior expected number of targets) objective function for sensor management.

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This “cookie cutter” example will be revisited in greater detail in Sections 26.3.1.4 and 26.3.2.5. 23.2.2

PENT for General Sensor Fields of View

In general, the FoV of a sensor will not be a “cookie cutter” but, rather, a statedependent probability of detection ∗

∗

pD (x, x) abbr. = pD,K+1 (x, x)

(23.1)

∗

where x is the state of the target and x is the state of the sensor (see Figure 23.2). ∗ ∗ Ideally pD (x, x) = 1 if the sensor FoV corresponding to x is precisely located ∗ at the target location corresponding to the target state x. If pD (x, x) = 0 then the sensor FoV is placed as poorly as possible. Thus Nk|k will be larger or smaller depending on how the FoV is placed over targets. If the probability of detection does not depend on the target state—that is, ∗ ∗ if pD (x, x) = pD (x)—then sensor management will be difficult. This is because measurements will be collected from all targets with equal probability, regardless ∗ of the sensor state x. There is no FoV to relocate. ∗ ∗ Sensor management is still possible even if pD (x, x) = pD (x), at least in ∗ principle. This is because one could, for example, choose x to cause sensor ∗ resolution—that is, the covariance of the measurement density fk+1 (z|x, x)—to be as small as possible. ∗ If pD (x, x) is constant in one or more dimensions—as, for example, is the case with bearing-only or range-only sensors—then sensor management will also be more difficult. In this case, at least two sensors will be required so that targets can be triangulated. In such cases, it is unlikely that the PENT objective function will exhibit good performance. For these reasons, in Part V it will be generally assumed that: • If the PENT or PENTI objective functions are to be effective, the sensor FoVs ∗ pD (x, x) must be sufficiently “concentrated” on some bounded region of the ∗ target state space. (Note that, in such a case, it is still allowable for pD (x, x) to have multiple peaks.) If this assumption is not true, PENT can be expected to behave poorly.

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Figure 23.2 A schematic depiction of a sensor field of view (FoV). The FoV is a probability of detection that varies with the state x of the target. It also depends ∗ on the current state x of the sensor.

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Characteristics of PENT

The following five points about the “natural,” “mission-oriented” objective function Nk|k require special emphasis: 1. Nk|k can be defined in a mathematically rigorous manner from the multitarget statistics of the multitarget system. 2. Nk|k is a computational approximation of abstract information-theoretic objective functions, including Kullback-Leibler discrimination and R´enyi αdivergence. 3. Nk|k is more computationally tractable than these objective functions. 4. Nk|k can be modified so as to cause sensors to preferentially dwell on targets of tactical interest (ToIs). In this case it is called the posterior expected number of targets of interest (PENTI). 5. PENT and PENTI are, therefore, concrete instances of resource management objective functions that are: (a) Abstract information-theoretic. (b) Subjective-intuitive mission-oriented. 23.2.4

The Cardinality-Covariance Objective Function

The PENT and PENTI objective functions were devised with two primary goals in mind: computational tractability and physical intuitiveness. They are also oriented towards use with PHD filters, since they use only first-order (in a point process sense) information. They are also easily modified to account for sensor and platform dynamics. Suppose, however, that a nontrivial cardinality distribution is also available, as is the case with CPHD and CBMeMBer filters. Then similar but potentially more effective approaches can be contemplated. As this book was nearing completion, Hung Gia Hoang and B.-T. Vo [108] proposed one such objective function: cardinality covariance—that is, the variance of PENT rather than PENT itself. In this approach, the goal is to minimize the error in the target-number estimate rather than to maximize the target-number estimate. Under certain circumstances, this objective function performs quite effectively (see [108] and Section 26.6.4.2).

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Like PENT, cardinality covariance has certain limitations. Most obviously, if the number of targets is known a priori to be n0 , then the cardinality distribution will be pk|k (n) = δn,n0 . In this case the variance of pk|k (n) will be identically zero and thus there is nothing to minimize. In addition, at the current time it is not clear how to tractably incorporate sensor and platform dynamics into the cardinality variance. See Section 25.9.3 for more details. 23.2.5

The Cauchy-Schwartz Objective Function

As this book was nearing completion, Hung Gia Hoang, Vo, Vo, and Mahler [108] proposed a new intuitive and tractable objective function: the Cauchy-Schwartz information functional, which was introduced in (6.73). It has an inherent intuitive interpretation, as the negative log-cosine of the angle between two multitarget distributions. When these distributions are Poisson, it has an even more direct physical interpretation, since it then reduces to the L2 norm of the difference of the corresponding PHDs. In this case it can then be computed in exact closed form, if the PHDs are approximated as Gaussian mixtures—see (25.99). Since this work was late-breaking at the time of this writing, limited emphasis will be placed on it—see Section 25.9.1 and Section 26.6.4.3.

23.3

SUMMARY OF RFS SENSOR CONTROL

The purpose of this section is to provide a “road map” for the chapters in Part V. It is organized as follows: 1. Section 23.3.1: A general approach to RFS single-step look-ahead control, based on the closed-loop multisensor-multitarget Bayes filter. 2. Section 23.3.2: A simpler special case: when sensor dynamics are “ideal”— that is, sensors can be reallocated to have any desired sensor-state between time-steps. 3. Section 23.3.3: A simplified form of RFS control for sensors whose dynamics are not ideal. 4. Section 23.3.4: Multisensor-multitarget control using PHD filters and CPHD filters.

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5. Section 23.3.5: The “pseudosensor” approximation for multisensor control with PHD and CPHD filters. 6. Section 23.3.6: A general approach to RFS multistep look-ahead control. (This will be the only consideration of multistep control in Part V.) 23.3.1

RFS Control Summary: General Approach (Single-Step)

This section is a summary of Chapter 25. Assume that: ∗1

∗s

• There are s sensors with respective states x, ..., x, and these can be chosen and changed to address some future goal. • The dynamics of the jth sensor between times tk and tk+1 are governed ∗ ∗j ∗j by a Markov density of the form f k+1|k ( x| x ′ , uk ), which determines which ∗j

∗j

states x at time tk+1 are “reachable” by the sensor if it had state x ′ at time tk ; and where – uk is the “control action” (or “control” in brief) at time tk , which is to be selected in order to determine which Markov density results in the most efficacious reallocation of the sensor at time tk+1 . ∗j

• The state x of the jth sensor is observed by its own “actuator sensor,” ∗j which collects an actuator-measurement z at each measurement-collection time. Let: • Z (k) : Z1 , ..., Zk be the time sequence of measurement sets at time tk , with Zj consisting of all measurements collected by all sensors at time tj . ∗

∗

∗

• Z (k) : Z 1 , ..., Z k be the time sequence of actuator-measurement sets at time ∗ tk , with Z j consisting of all measurements collected by all actuator sensors at time tj . • U (k) : U0 , ..., Uk be the time sequence of control sets (the “multisensor control policy”) at time tk , with Ui being chosen at time ti and where Ui consists of all controls for all sensors at time ti . Given this, the general approach to multisensor-multitarget sensor management is a closed-loop version of the multisensor, multitarget Bayes filter. It has the

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following general structure:

...

multisensor-multitarget measurement-update ∗ ∗ → fk|k (X, X|Z (k) , Z (k) , U (k−1) ) optimal selection of next multisensor controls time-projection ∗ ∗

→

fk+1|k+1 (X, X|Z (k) , Z (k) , U (k−1) , ∗ Z, Z, U ) objectivization and hedging

↓ select next control set

Uk = arg supU Ok+1 (U ) multisensor-multitarget time & measurement updates ∗ ∗ fk+1|k+1 (X, X|Z (k+1) , Z (k+1) , U (k) )

→ ∗

Uk ,Zk+1 ,Z k+1 ∗

→ ...

↗ ∗

Here, fk|k (X, X|Z (k) , Z (k) , U (k−1) ) is the probability (density) that, at time tk , ∗

the targets have state set X and the sensors have state set X, given the time∗ histories Z (k) of target-measurements, Z (k) of actuator-sensor measurements, and U (k−1) of controls. The sequence of sensor management steps to be taken (schematically portrayed in Figure 23.3) are as follows: 1. Time-projection: Use the multitarget filter predictor equation to time∗ ∗ extrapolate fk|k (X, X|Z (k) , Z (k) , U (k−1) ) to ∗

∗

fk+1|k (X, X|Z (k) , Z (k) , U (k−1) , U ) where U is the to-be-determined set of controls for the sensors at time tk . Then use the multitarget filter corrector equations to measurement-update it to ∗

∗

fk+1|k+1 (X, X|Z (k) , Z (k) , U (k−1) , ∗

Z, Z, U ) ∗

where Z and Z are the unknowable sensor and actuator-sensor measurement sets at the future time tk+1 .

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2. Objectivization: From the marginal distributions ∗

∗

fk+1|k+1 (X|Z (k) , Z (k) , U (k−1) , Z, Z, U ), ∗

fk+1|k (X|Z (k) , Z (k) , U (k−1) , U ) ∗

construct an objective function Ok+1 (U, Z, Z) that measures the increase in information about targets, as contained in the posterior distribution ∗

∗

fk+1|k+1 (X|Z (k) , Z (k) , U (k−1) , Z, Z, U ), over and above that contained in the prior distribution ∗

fk+1|k (X|Z (k) , Z (k) , U (k−1) , U ). ∗

3. Hedging: The future measurement sets Z, Z at time tk+1 are unknowable. ∗ Devise a means of eliminating them from Ok+1 (U, Z, Z) to get a “hedged” objective function Ok+1 (U ). 4. Optimization: If larger values of Ok+1 (U ) indicate better measurementcollection, solve the optimization problem (23.2)

Uk = arg sup Ok+1 (U ) U

to determine the best set Uk of controls at time tk . Otherwise, if smaller values of Ok+1 (U ) indicate better measurement-collection, solve the optimization problem Uk = arg inf Ok+1 (U ). (23.3) U

5. Prediction: Given Uk , use the multitarget filter predictor to time-update ∗ ∗ ∗ ∗ fk|k (X, X|Z (k) , Z (k) , U (k−1) ) to fk+1|k (X, X|Z (k) , Z (k) , U (k) ). ∗

6. Correction: Collect the measurement sets Zk+1 , Z k+1 at time tk+1 and use the multitarget filter corrector equation to measurement-update ∗ ∗ ∗ ∗ fk+1|k (X, X|Z (k) , Z (k) , U (k) ) to fk+1|k+1 (X, X|Z (k+1) , Z (k+1) , U (k) ). Remark 90 (“Separable control” approximation) Note that, at the outset, an approximation has already been made. Even in the single-sensor, single-target case,

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Figure 23.3 A schematic representation of single-step look-ahead sensor management for multiple sensors. The Fields of View (FoVs) of two sensors are to be placed, in the most informative manner possible, at the next measurement time. One must hedge against the fact that the future measurements are unknowable.

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provably optimal control systems are in general nonlinear. In this case they do not obey the “separation principle” ([89], p. 363). That is, they cannot be recursively structured as a filtering step followed by a control-optimization step. Thus the separation principle is here being assumed to be true, as an approximation. Besides this approximation, we face the following practical difficulties: • All six of the above sensor-management steps will be computationally intractable in general. • Consequently, each and every step will require drastic approximations to render it practicable. Devising these approximations is one of the primary goals of Part V. The approximations that will be described in this section are as follows: 1. For time-projection: replace the multitarget Bayes filter predictor with the predictor for an approximate filter: a PHD, CPHD, or CBMeMBer filter. In addition, use a pseudosensor approximation to recast the multisensor management problem as a single-sensor management problem. This will be summarized in Section 23.3.5 and will be described in more detail in Section 26.3.2.3. 2. For objectivization: replace the conventional information-theoretic objective function, such as Kullback-Leibler cross-entropy, with an approximate and physically intuitive information-theoretic objective function: the posterior expected number of targets (PENT); the expected number of targets of interest (PENTI); or the cardinality variance; or the Cauchy-Schwartz divergence. 3. For hedging: Replace the usual hedging strategies, such as expected-value hedging (“hedging against the average measurement”) ∗

Ok+1 (U ) = E

∗

[Ok+1 (U, Z, Z)]

Z,Z

or minimum-value hedging (“hedging against the worst-case measurement”) ∗

Ok+1 (U ) = inf∗ Ok+1 (U, Z, Z), Z,Z

with an approximation. Two approximations will be considered:

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(a) Multisample approximation, in which multiple representative samples Z1,k+1 ,...,Zν,k+1 are drawn from the distribution fk+1 (Z|Z (k) ); and ∗

∗

∗ multiple representative samples Z 1,k+1 ,...,Z ν,k+1 are drawn from the ∗

∗

distribution fk+1 (Z|Z (k) ); and then these are used to approximate the expected value: ∗

ν ν ∗ 1 ∑∑ ∼ E ∗ [Ok+1 (U, Z, Z)] = ∗ Ok+1 (U, Zj,k+1 , Z j ′ ,k+1 ). Z,Z ν ν j=1 j ′ =1 (23.4) ∗

(b) Single-sample approximation, in which a single “most representative” ! sample Zk+1 is drawn from fk+1 (Z|Z (k) ) and, likewise, a single ∗

∗

∗

sample Z !k+1 is drawn from fk+1 (Z|Z (k) ): ∗

! EZ [Ok+1 (U, Z)] ∼ , Z !k+1 ). = Ok+1 (U, Zk+1

(23.5)

For target measurements, the single sample that will be proposed is the ! P IM S predicted ideal measurement set Zk+1 = Zk+1 : P IM S Ok+1 (U ) = Ok+1 (U, Zk+1 ).

(23.6)

P IM S The PIMS Zk+1 is a generalization of the concept of a predicted measurement to the multitarget case, taking the collectability of measurements into account (see Section 25.10.2).

4. For optimization: Replace the general optimization problem3 Uk = arg sup Ok+1 (U )

(23.7)

U

with an approximate one—namely, a restriction of the possible values of U to a small finite number of “admissible” (that is, sufficiently representative) controls (Section 25.11). 5. For prediction: Replace the multitarget Bayes filter predictor with the predictor for an approximate filter: a PHD, CPHD, or CBMeMBer filter. 3

If an objective function such as the cardinality variance is used, the optimization would use an ‘arg inf’ rather than an ‘arg sup’.

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6. For correction: Replace the multitarget Bayes filter corrector with the corrector for an approximate filter: a PHD, CPHD, or CBMeMBer filter. Unfortunately, these approximations will still not be sufficient to achieve computational tractability. Two further approximations (actually, special cases) will prove both computationally and conceptually useful: • Control with ideal sensor dynamics (Section 25.12), which will be summarized shortly in Section 23.3.2. It is assumed that: – The controls are sensor states. ∗

– Any sensor with state x′ at time tk can be arbitrarily redirected to ∗ any state x at time tk+1 . – Any sensor state at the next time-step is reachable during the timeinterval [tk , tk+1 ]. • Control with simplified nonideal sensor dynamics (Section 25.13), which will be summarized shortly in Section 23.3.3. It is assumed that: – Controls are sensor states. – The dynamics of a sensor between times tk and tk+1 are described ∗ ∗j ∗j by an a priori Markov density f k+1|k ( x| x ′ ), rather than governed by ∗

∗j ∗j

a controlled Markov density f k+1|k ( x| x ′ , uk ). 23.3.2

RFS Control Summary: Ideal Sensor Dynamics

This section is a summary of Section 25.12. The general approach summarized in Section 23.3.1 will be too computationally demanding for most practical situations. As an initial simplifying assumption, suppose that: • Between times tk and tk+1 , all sensors can be redirected to dwell anywhere desired. Phased-array radars are approximately ideal in this sense, since their beams are electronically steered and thus can be rapidly redirected.

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j

Given the assumption of ideal sensor dynamics, the control vector uk at time ∗j tk for the jth sensor is the same thing as the sensor state x k+1 at time tk+1 : ∗j

(23.8)

uk = x k+1 .

The single-step sensor management scheme of Section 23.3.1 then assumes the following form: multitarget measurement-update ∗ ... → fk|k (X|Z (k) , X (k) ) optimal selection of next multisensor state time-projection ∗

fk+1|k+1 (X|Z (k) , X (k) , ∗ Z, X)

→

objectivization and hedging

↓ select next sensor state set ∗

∗

X k+1 = arg sup ∗ Ok+1 (X) → ∗

X k+1 ,Zk+1

X multitarget time & measurement updates ∗ fk+1|k+1 (X|Z (k+1) , X (k+1) )

→ ...

↗

The control scheme consists of the following steps: 1. Target prediction: Use the multitarget filter predictor equation to time-update ∗ ∗ fk|k (X|Z (k) , X (k) ) to fk+1|k (X|Z (k) , X (k) ) using the multisensor-multitarget Bayes filter predictor. 2. Time-projection: Use the multitarget filter corrector equations to measure∗ ment update fk+1|k (X|Z (k) , X (k) ) to ∗

fk+1|k+1 (X|Z (k) , X (k) , ∗

Z, X) where Z is the unknowable target-generated measurement set at time tk+1 ; ∗ ∗1 ∗s and where X = { x, ..., x} is the to-be-determined set of sensor states at time tk+1 .

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∗

∗

3. Objectivization: From fk+1|k+1 (X|Z (k) , X (k) , Z, X), construct the PENT ∗

or PENTI objective function Nk+1|k+1 (X, Z); or the cardinality-covariance ∗

2 objective function σk+1|k+1 (X, Z); or perhaps also the Cauchy-Schwartz ∗

divergence CSk+1|k+1 (X, Z). 4. Hedging: Eliminate Z using either multisample approximation (23.4), or the single-sample predicted ideal measurement set (PIMS) approximation, which will be described in Section 25.10.2 : ideal

∗

N k+1|k+1 (X) ideal

∗

=

P IM S Nk+1|k+1 (X, Zk+1 )

=

2 P IM S σk+1|k+1 (X, Zk+1 ).

∗

σ 2 k+1|k+1 (X)

(23.9)

∗

(23.10)

5. Optimization: Using a finite set of admissible multisensor states, solve an approximate version of the optimization problem ideal

∗

X k+1

=

∗

arg sup N k+1|k+1 (X)

(23.11)

∗

X ideal

∗

X k+1

=

∗

arg inf σ 2 k+1|k+1 (X) ∗

(23.12)

X ideal

∗

X k+1

=

∗

arg sup CS k+1|k+1 (X).

(23.13)

∗

X ∗

6. Target correction: Given X k+1 , collect the next target-generated measure∗ ment set Zk+1 , and measurement-update fk+1|k (X|Z (k) , X (k) ) to ∗

fk+1|k+1 (X|Z (k+1) , X (k+1) ) using the multisensor-multitarget Bayes filter corrector. 23.3.3

RFS Control Summary: Simplified Nonideal Sensor Dynamics

This section is a summary of Section 25.13. The control scheme in this case is a modification of the ideal-sensor approach described in Section 23.3.2. It allows nonideal sensor dynamics to be taken into account, while preserving the conceptually and computationally simplified structure

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of the ideal-sensor case. As with the ideal-dynamics case, selecting the current control is the same as selecting a future sensor state: ∗j

(23.14)

uk = x k+1 .

Unlike the ideal-sensor case, it is assumed that the dynamics of the jth are gov∗j

∗j ∗j

erned by an a priori Markov density f k+1|k ( x| x ′ ). This density mathematically ∗j

∗j

encapsulates an assumption about the reachability of x from x ′ . The single-step sensor management scheme of Section 23.3.1 then assumes the following two-filter form: multisensor/target measurement-updates ∗ ∗ ∗ ... → f k|k (X|Z (k) ) ∗

fk|k (X|Z (k) , X (k) )

... →

optimal selection of next multisensor state ∗ ∗ ∗ → f k+1|k (X|Z (k) )

↓ time-projection ∗

fk+1|k+1 (X|Z (k) , X (k) , ∗ Z, X)

→

objectivization and hedging

↓ select next sensor state set ∗

∗

X k+1 = arg sup ∗ Ok+1 (X) ∗

Z k+1

X multisensor/target time & measurement updates ∗ ∗ ∗ ↗ f k+1|k+1 (X|Z (k+1) )

→ ...

∗

fk+1|k+1 (X|Z (k+1) , X (k+1) )

→ ∗

X k+1 ,Zk+1

→ ...

↗

The bottom filter in each of the three boxes is a conventional multitarget Bayes filter on the multitarget state X. The top filter describes the Time evolution of all of the ∗ ∗1 ∗s sensors. If there are a constant number s of sensors, then X = { x, ..., x} with ∗ ∗ ∗ ∗ |X| = s and so f k|k (X|Z (k) ) will have the form ∗

∗

∗1

∗

f k|k (X|Z (k) ) = s! · δ

∗

s,|X|

∗1 ∗1

∗s

∗s ∗s

· f k|k ( x|Z k ) · · · f k|k ( x|Z k )

(23.15)

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∗j

851

∗j ∗j

where the probability distribution f k|k ( x|Z k ) describes the state of the jth sensor at time tk . The control scheme consists of the following steps: ∗j

∗j

∗j ∗j

∗j ∗j

1. Sensor prediction: Time update each f k|k ( x|Z k ) to f k+1|k ( x|Z k ) using the single-target Bayes filter predictor. This results in a time-update of ∗ ∗ ∗ ∗ ∗ ∗ f k|k (X|Z (k) ) to f k+1|k (X|Z (k) ). ∗

∗

2. Target prediction: Time update fk|k (X|Z (k) , X (k) ) to fk+1|k (X|Z (k) , X (k) ) using the multisensor-multitarget Bayes filter predictor. ∗

3. Time-projection: Measurement update fk+1|k (X|Z (k) , X (k) ) to ∗

fk+1|k+1 (X|Z (k) , X (k) , ∗

Z, X), where Z is the unknowable target-generated measurement set at time tk+1 ; ∗ ∗1 ∗s and where X = { x, ..., x} is the to-be-determined set of sensor states at time tk+1 . ∗

∗

4. Objectivization: From fk+1|k+1 (X|Z (k) , X (k) , Z, X), construct the PENT ∗

or PENTI objective function Nk+1|k+1 (X, Z); or perhaps the Cauchy∗

Schwartz divergence CSk+1|k+1 (X, Z). 5. Hedging: Eliminate Z using either multisample approximation (23.4), or the single-sample predicted ideal measurement set (PIMS) approximation, which will be described in Section 25.10.2 : ideal

∗

∗

P IM S N k+1|k+1 (X) = Nk+1|k+1 (X, Zk+1 ).

(23.16)

6. Dynamicization: Account for sensor and platform dynamics: nonideal

N

∗

k+1|k+1 (X)

ideal

∗

∗

∗

∗

= N k+1|k+1 (X) · f k|k (X|Z (k) ).

(23.17)

That is, optimization of PENT must account for the reachability of the optimal multisensor state. (At the current time, no analogous equation is known for the cardinality-variance objective function.)

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7. Optimization: Using a finite representative set of multisensor states, solve an approximate version of the optimization problem: nonideal

∗

X k+1

=

arg sup N

∗

k+1|k+1 (X)

(23.18)

∗

X nonideal

∗

X k+1

=

∗

arg sup CS k+1|k+1 (X).

(23.19)

∗

X ∗j

∗j

8. Sensor correction: Collect the next set Z k+1 with |Z k+1 | ≤ 1 of actuatorsensor measurements for each sensor,4 and then measurement-update each ∗j

∗j ∗j

∗j

∗j ∗j

f k+1|k ( x|Z (k) ) to f k+1|k+1 ( x|Z (k+1) ). ∗

9. Target correction: Given X k+1 , collect the next target-generated measure∗ ment set Zk+1 , and measurement-update fk+1|k (X|Z (k) , X (k) ) to ∗

fk+1|k+1 (X|Z (k+1) , X (k+1) ) using the multisensor-multitarget Bayes filter corrector. 23.3.4

RFS Control Summary: Control with PHD and CPHD Filters

This section is a summary of Chapter 26. Control with PHD filters differs from that in Section 23.3.3 only in that the multisensor-multitarget Bayes filter is replaced by a multisensor PHD filter (or a 4

An actuator-sensor measurement can be the empty set because of the model for transmission dropouts described in Section 25.6.4.

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single-sensor PHD filter with the pseudosensor approximation): multisensor/target measurement-updates ∗ ∗ ∗ ... → f k|k (X|Z (k) ) ∗

... →

Dk|k (x|Z (k) , X (k) )

→

optimal selection of multisensor state ∗ ∗ ∗ f k+1|k (X|Z (k) )

↓ time-projection ∗

Dk+1|k+1 (x|Z (k) , X (k) , ∗ Z, X)

→

objectivization and hedging

↓ select next sensor state set nonideal

∗

X k+1 = arg sup ∗ N ∗

Z k+1

∗

k+1|k+1 (X)

X multisensor/target time & measurement updates ∗ ∗ ∗ ↗ f k+1|k+1 (X|Z (k+1) )

→ ...

∗

Dk+1|k+1 (x|Z (k+1) , X (k+1) )

→ ∗

X k+1 ,Zk+1

→ ...

↗

Similarly, control with CPHD filters differs from that in Section 23.3.3 only in that the multisensor-multitarget Bayes filter is replaced by a multisensor CPHD filter (or a single-sensor PHD filter using the pseudosensor approximation). Sensor control using Bernoulli filters and CBMeMBer filters will also be considered (see Sections 26.2 and 26.5). 23.3.5

RFS Control Summary: “Pseudosensor” Approximation for Multisensor Control

For the multisensor problem, the tracking filter may be an approximate multisensor PHD or CPHD filter, such as those considered in Chapter 10. In principle, therefore, the sensor management “time-projection” step should use the same filters. However, it is desirable to have a more computationally attractive approximation. The purpose of this section is to describe one such approximation, the “pseudosensor approximation.” For greater detail, see Sections 26.3.2.2 and 26.3.2.3.

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The concept is most easily illustrated by examining the simplest case: two sensors observing a single target. Suppose that these sensors have the following models: ∗1

1

∗2

2

• Respective sensor probabilities of detection pD (x, x) and pD (x, x). 1

∗1

2

∗2

• Respective sensor likelihood functions Lz1 (x, x) and Lz2 (x, x). • Both sensors are clutter-free. 12

At time tk+1 , the two sensors collect a joint measurement set Z from the target. There are only four collections that are possible: 12

• Z = ∅ (neither sensor detects the target). 12

1

• Z = {z} (the first sensor detects the target but the second does not). 12

2

• Z = {z} (the second sensor detects the target but the first does not). 12

1

2

• Z = {(z, z)} (both sensors detect the target). Because there is no clutter and only one target, the two sensors behave as though they were a single sensor—here called the “pseudosensor.” The probability of detection of this sensor is 12

∗1 ∗2

1

∗1

2

∗2

p D (x, x, x) = 1 − (1 − pD (x, x))(1 − pD (x, x)). 12

∗1 ∗2

(23.20)

That is, p D (x, x, x) is the probability that at least one of the two sensors collects a measurement. If a measurement is collected by the pseudosensor, this measurement will 12 1 12 2 12 1 2 have three possible forms: z = z or z = z or z = (z, z). The likelihood

Introduction to Part V

12

855

∗1 ∗2

function L12 (x, x, x) for the pseudosensor is given by the formula: z 12

=

∗1 ∗2

12

∗1 ∗2

p D (x, x, x) · L 12 (x, x, x) z  1 ∗1 ∗2 2  pD (x, x) · (1 − pD (x, x))   1  ∗1   ·Lz1 (x, x)    ∗2 2  (1 − p1 (x, ∗1 x)) · pD (x, x)) D 2 ∗2  ·Lz2 (x, x)    ∗1 ∗2 1 2   pD (x, x) · pD (x, x))   1 2  ∗1 ∗2  ·L (x, x) · L (x, x) 1

2

z

z

(23.21) 12

if

1

z =z 12

if 12

if

2

z =z 1

.

(23.22)

2

z = (z, z)

Given these preliminaries, it becomes possible to transform multisensor sensor management into single-sensor sensor management, by applying single-sensor sensor management to the pseudosensor. Now modify the modeling assumptions as follows: • The resolution of the sensors is good. • There are multiple targets that are well separated. • The sensors are corrupted by relatively sparse clutter. Then the target-generated measurements will still behave, at least approximately, as though generated by the pseudosensor. To see why, suppose that the measurements are positions. If both sensors collect measurements from a target, then these measurements will typically appear as pairs located near that target and will be clearly associated with it. As the number of targets increases and the clutter density increases, the pseudosensor approximation will become less and less valid. However, it will suffice for the purpose of reducing the computational complexity of the sensor management “time-projection” step. The pseudosensor approximation can be generalized to an arbitrary number of targets—see Section 26.3.2.3. 23.3.6

RFS Control Summary: General Approach (Multistep)

Multistep look-ahead control is expected to be computationally formidable in most multitarget situations, and thus will not be examined in detail in Part V. The

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purpose of this section is, for the sake of conceptual completeness, to summarize the extension to the approach in Section 23.3.1 to multistep control. In multistep look-ahead control, we are to determine the control policy for all sensors throughout a future time window that begins at time tk+1 and ends at time tk+k′ for some choice of k ′ ≥ 1. As with multistep control in general, the approach crucially depends on the following assumption: • The Time evolution of the targets in the future time-window tk , ..., tk+k′ −1 can be accurately predicted by the multitarget Markov densities fk+i+1|k+i (X|X ′ ) for i = 0, 1, ..., k + k ′ − 1. The control policy is single-step look-ahead if k ′ = 1, two-step look-ahead if k = 2, and so on. Let: ′

• Z (k) : Z1 , ..., Zk be the time sequence of multisensor target measurement sets at time tk . ∗

∗

∗

• Z (k) : Z 1 , ..., Z k be the time sequence of multi-actuator measurement sets for all sensors at time tk . ∗

• U k : U0 , ..., Uk be the time sequence of multisensor states at time tk . At time tk , the control policy U0 , ..., Uk−1 has previously been chosen. Before we can collect the measurement set Zk+1 at time tk+1 , we must choose the control set Uk at time tk . For multistep control, Uk must be chosen as the initial control set of an optimal predicted control policy Uk , ..., Uk+k′ −1 . This will result in more accurate and smoothly evolving control. At any given time tk the predicted control policy Uk , ..., Uk+k′ −1 is, in addition, the current optimal resource management plan. Thus, to a great extent, multistep control subsumes the basic elements of mission planning.

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The sensor management scheme for multistep look-ahead control has the following general structure:

...

multisensor-multitarget measurement-update ∗ ∗ → fk|k (X, X|Z (k) , Z (k) , U (k−1) ) optimal selection of multisensor controls in time-window time-projection ∗

∗

fk+k′ |k+k′ (X, X|Z (k) , Z (k) , U (k−1) , Zk+1 , ..., Zk+k′ , ∗ ∗ Z k+1 ..., Z k+k′ , Vk , ..., Vk+k′ −1 ) → objectivization and hedging ↓ select multisensor controls in future time-window

(Uk , ..., Uk+k′ −1 ) = arg supVk ,...,Vk+k′ −1 Ok+k′ (Vk , ..., Vk+k′ −1 ) → ∗

Uk ,Zk+1 ,Z k+1

multisensor-multitarget time & measurement updates ∗ ∗ fk+1|k+1 (X, X|Z (k+1) , Z (k+1) , U (k) )

→ ...

↗

The sequence of sensor management steps to be taken are as follows: 1. Time-projection: Use successive iterations of the multitarget filter predictor ∗ ∗ and corrector equations to update fk|k (X, X|Z (k) , Z (k) , U (k−1) ) to ∗

∗

fk+k′ |k+k′ (X, X|Z (k) , Z (k) , U (k−1) , ∗

∗

Zk+1 , ..., Zk+k′ , Z k+1 ..., Z k+k′ , Vk , ..., Vk+k′ −1 ) ∗

∗

where Zk+1 , ..., Zk+k′ and Z k+1 ..., Z k+k′ are the time sequences of unknowable sensor and actuator-sensor measurement sets during the timewindow; and where Vk , ..., Vk+k′ −1 is the to-be-determined time sequence of controls for the sensors throughout the time-window. 2. Objectivization: From the marginal distributions ∗

fk+k′ |k+k′ (X|Z (k) , Z (k) , U (k−1) , ∗

∗

Zk+1 , ..., Zk+k′ , Z k+1 ..., Z k+k′ , Vk , ..., Vk+k′ −1 )

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and ∗

fk+1|k (X|Z (k) , Z (k) , U (k−1) , Vk ) construct an objective function Ok+k′ (Vk , ..., Vk+k′ −1 , ∗

∗

Zk+1 , ..., Zk+k′ , Z k+1 ..., Z k+k′ ) that measures the increase in information about targets. 3. Hedging: Determine a method for eliminating the unknowable future mea∗ ∗ surement sets Zk+1 , ..., Zk+k′ and Z k+1 ..., Z k+k′ from Ok+k′ (Vk , ..., Vk+k′ −1 , ∗

∗

Zk+1 , ..., Zk+k′ , Z k+1 ..., Z k+k′ ) to get a “hedged” objective function Ok+k′ (Vk , ..., Vk+k′ −1 ). Two approximate forms of expected-value hedging include multisample hedging and single-sample PIMS hedging. 4. Optimization: Solve the optimization problem

=

(Uk , ..., Uk+k′ −1 ) arg sup Ok+k′ (Vk , ..., Vk+k′ −1 )

(23.23)

Vk ,...,Vk+k′ −1

to determine the future optimal control policy Uk , ..., Uk+k′ −1 . 5. Prediction: Given Uk , use the multitarget predictor equation to time-update ∗ ∗ ∗ ∗ fk|k (X, X|Z (k) , Z (k) , U (k−1) ) to fk+1|k (X, X|Z (k) , Z (k) , U (k) ). ∗

6. Correction: Collect the measurement sets Zk+1 , Z k+1 at time tk+1 and use the multitarget filter corrector equation to measurement-update ∗ ∗ ∗ ∗ fk+1|k (X, X|Z (k) , Z (k) , U (k) ) to fk+1|k+1 (X, X|Z (k+1) , Z (k+1) , U (k) ).

23.4

ORGANIZATION OF PART V

Part V is organized as follows:

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1. Chapter 24: Illustration of the concepts in the simplest case: single-sensor, single-target sensor management, including a simple algebraically closedform example (Section 24.9). 2. Chapter 25: General multisensor-multitarget sensor management, including tactical target prioritization for sensor management. 3. Chapter 26: Sensor management using approximate filters—Bernoulli filters, PHD filters, CPHD filters, and CBMeMBer filters—including their implementations and applications.

Chapter 24 Single-Target Sensor Management 24.1

INTRODUCTION

The purpose of this section is to introduce the basic concepts of RFS sensor management by illustrating how they are applied in the simplest possible nontrivial case. Assume that: • A single target has already been detected and is to be tracked. • Measurements of the target are collected by a single sensor with no clutter/false alarms but with a known field of view (FoV), as specified by a state∗ dependent probability of detection pD (x, x). • The sensor is successively repositioned through time via a sequence U k : u1 , ..., uk of control commands (“actions”). • The sensor is to be repositioned with the aim of maximizing information about the target, compared to what was known previously. 24.1.1

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter: • In the approach described here, the primary goal of sensor management is the optimal placement of the sensor’s field of view (Sections 24.2 and 24.4). Of lesser concern is sensor placement in order to maximally improve sensor resolution.

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• The target and the sensor should be analyzed as a jointly evolving stochastic ∗ system (x, x), whose Time evolution is governed by a joint recursive Bayes filter. This filter propagates a joint target and sensor probability distribution ∗

∗

fk|k (x, x|Z (k) , Z (k) , U k−1 ), where U k−1 : u0 , ..., uk−1 is a sequence of control actions—a “control policy” (Sections 24.2 and 24.4). • In a Bayesian approach, the sensor’s future state should be chosen so as to maximize the information contained in the target’s posterior track distribution, as compared to the information contained in the track’s prior track distribution (Section 24.5). • Single-sensor, single-target control can be greatly simplified if we assume that the sensor’s dynamical behavior is ideal—that is, the sensor’s state can be changed without any constraint (Section 24.8). • The ideal-sensor assumption can be modified so as to allow the dynamical behavior of a nonideal sensor to be taken into account (Section 24.10). • The basic concepts of the Bayesian single-sensor, single-target sensor management approach can be illustrated using a simple, closed-form analytical example based on linear-Gaussian models (Section 24.9). 24.1.2

Organization of the Chapter

The section is organized as follows: 1. Section 24.2: A typical example of single-sensor, single-target sensor management: missile tracking cameras. 2. Section 24.3: Target and sensor modeling for nonlinear single-sensor, singletarget sensor management. 3. Section 24.4: The single-step look-ahead version of single-sensor, singletarget sensor management. 4. Section 24.5: Objective functions for single-sensor, single-target sensor management.

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5. Section 24.6: Accounting for the unknown future measurement using multisample hedging or single-sample hedging (specifically, predicted measurement (PM) hedging). 6. Section 24.7: Optimization of the sensor management objective function. 7. Section 24.8: A special case: ideal sensor dynamics. That is, the sensor can ∗ be redirected without constraint from its state xk at time tk to any possible ∗ future state xk+1 at time tk+1 . 8. Section 24.9: A simple closed-form example: ideal sensor dynamics with linear-Gaussian models. 9. Section 24.10: A generalization of the ideal-sensor case that allows nonideal sensor dynamics to be taken into account.

24.2

EXAMPLE: MISSILE-TRACKING CAMERAS

A missile-tracking camera provides an easily understood illustration of the basic concepts of control-theoretic sensor management. Two situations will be considered: single-camera tracking (Section 24.2.1) and two-camera tracking (Section 24.2.2). 24.2.1

Single-Camera Missile Tracking

A missile is launched. A gimbaled camera must optimally point itself so as to continually keep the missile centered within its optical Field of View (FoV). The FoV centroid consists of the range centroid (focal length) and angular centroid (bearing centroid). The control system must achieve this goal by periodically estimating the current target position (azimuth, elevation, and range) from the target’s camera image, and then predicting where the target will be at the time of the next camera-image collection. Thus the control system must integrate two items: a target tracker that predicts the future locations of the missile; and a sensor management objective function that minimizes the distance between the predicted target position and the FoV centroid. This process is complicated by the fact that the camera—and not just the missile—is itself a dynamically evolving physical object. Its motion is limited by physical constraints such as slew rate and, possibly, a restricted range of motion. At any given time tk , changes in its bearing and range are effected by actuator motors.

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These actuators are observed by internal actuator sensors that measure quantities such as voltages. The camera’s bearing, range, and angular velocity—its state—is not known a priori and so must be estimated from these actuator measurements. This means that the control system requires a second, parallel, tracking algorithm— one that tracks and predicts the state of the camera itself. Taken together, the missile and the camera comprise a jointly evolving dynamical control system. Most commonly this is modeled as a linear control system ([140], pp. 257-259, 402-404). This is what will be assumed in the following discussion, and will be revisited in the form of a concrete example in Section 24.9. ∗ The state x of the target and the state x of the sensor evolve as a joint ∗ state (x, x). The laws of motion of the target and the sensor are both assumed to be linear-Gaussian: Xk+1|k

=

∗

∗

Xk+1|k

=

(24.1)

Fk x + W k ∗

∗

∗

∗

(

∗

F k x + Dk u + Wk = F k x +

∗

∗

F −1 k Dk u

)

∗

+ Wk . (24.2)

Here, • Fk is the state transition matrix for the target. ∗

• F k is the state transition matrix for the sensor. • u (the “control input”) models the effect of applying input signals to the sensor actuators at time tk in order to effect a sensor reallocation at time tk+1 . ∗

∗

• F −1 k Lk uk selects the starting point of the sensor state transition. At each time tk , the camera collects a measurement zk of the current target ∗ state xk ; and the actuator sensor collects a measurement zk of the current sensor ∗ ∗ state xk . This results in a joint measurement (zk , zk ). The models for both the camera and the actuator sensor(s) are assumed to be linear-Gaussian: ∗

Zk+1

=

∗

Zk+1

∗

=

∗

Hk+1 x + J k+1 x + Vk+1 ∗

(24.3)

∗

H k+1 x + Vk+1

where: • Hk+1 is the measurement matrix for the sensor.

(24.4)

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∗

• H k+1 is the measurement matrix for the actuator sensor. ∗

∗

• J k+1 x models the influence of the sensor state on the target-generated measurements. Let the “reference vector” (24.5)

rk = Ak xk|k

be the target position corresponding to the estimated target state xk|k , where Ak is a matrix that transforms target state-vectors into target-position vectors. Let the “controlled vector” ∗ ∗ ∗ rk = Ak xk|k (24.6) ∗

∗

be the FoV-centroid corresponding to the estimated camera state xk , where Ak is a matrix that transforms sensor state-vectors into position (FoV centroid) vectors. ∗ Given this, an optimal camera control system should keep rk as close as possible to rk for all k. Moreover, it must do so while also not exceeding the physical dynamical limitations of the camera. Mathematically, this means that the magnitudes of the controls, as measured by a Mahalanobis norm, ∥u∥2k = uT Ck−1 u

(24.7)

must not be too large. For conceptual clarity, consider the single-step look-ahead case. Then the linear control system consists of the following recursive steps: 1. Target prediction: A Kalman filter predictor is used to time-update the current target state xk|k to the future state xk+1|k . 2. Sensor prediction with unknown control: A Kalman filter predictor is used ∗ ∗ to time-update the current sensor state xk|k to the future state xk+1|k (u), which because of (24.2), depends on the (yet-to-be-determined) control u. 3. Target measurement-update with a random measurement: If Zk+1 is the future random target-generated measurement, use a Kalman filter corrector to measurement-update xk+1|k to xk+1|k+1 (Zk+1 ), which is a random vector. The reference vector Rk+1 (Zk+1 ) = Ak+1 xk+1|k+1 (Zk+1 ) is therefore also random.

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∗

4. Sensor measurement-update: If Zk+1 is the future actuator measurement, ∗ use the Kalman filter corrector equations to measurement-update xk+1|k (u) ∗

∗

to xk+1|k+1 (u, Zk+1 ). Thus the controlled vector ∗

∗

∗

∗

∗

Rk+1 (u, Zk+1 ) = Ak+1 xk+1|k+1 (u, Zk+1 ) is a random vector. 5. Objectivization: Define an objective function, which is usually a Mahalanobis square-distance: ∗

(24.8)

Ok+1 (u, Zk+1 , Zk+1 ) ∗

=

∗

∗

( Rk+1 (u, Zk+1 ) − Rk+1 (Zk+1 ))T E −1 i ∗

∗

·( Rk+1 (u, Zk+1 ) − Rk+1 (Zk+1 )) −1 +uT Ck+1 u. −1 (Recall that the term uT Ck+1 u ensures that the magnitude of the control will be kept as small as possible.) ∗

6. Hedging: Eliminate the unknowable future measurements Zk+1 , Zk+1 . This is most typically accomplished by taking the expected value: ∗

Ok+1 (u) = E[Ok+1 (u, Zk+1 , Zk+1 )].

(24.9)

7. Optimization: Minimize the objective function: uk+1 = arg inf Ok+1 (u).

(24.10)

u

Under our current linear-Gaussian assumptions, this can be shown to have an exact, closed-form solution. 8. Measurement update: At the next time-step, collect the joint measurement ∗ (zk+1 , zk+1 ), and repeat.

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For multistep look-ahead, (24.9) is replaced by a sum of single-step objective functions, taken over a window of time: Ok+1 (uk+1 , ..., uk+k′ )  ) ′ ( ∗ ∗ k+k ∑ T −1 ( Ri − Ri ) Ei ( Ri . E −1 − Ri ) + uTi−1 Ci−1 ui−1 i=k+1

=

24.2.2

(24.11)

Two-Camera Missile Tracking

In this case, the missile is tracked by two jointly cooperating cameras. Range-totarget, which in the single-camera case must be inferred from focal length, can now be more accurately estimated via triangulation. However, the cameras must not only track the missile individually, but must also share and coordinate their information. In this case the target-sensor system has a joint state variable of the form ∗1 ∗2 ∗1 ∗2 (x, x, x), where x, x are the respective states of the first and second cameras. 1 2 ∗1 ∗2 1 2 Similarly, the joint measurement will have the form (z, z, z , z ), where z, z are the respective target-generated measurements collected by the two cameras, and ∗1 ∗2 z , z are the respective actuator-sensor measurements. Given this, the single-step look-ahead procedure for a single camera is generalized as follows: 1. Target prediction: A Kalman filter predictor is used to time-update the current target state xk|k to the future state xk+1|k . 2. First-sensor prediction: A Kalman filter predictor is used to time-update the ∗1 ∗1 1 1 current sensor state x k|k to the future state x k+1|k (u), where u is the yet-to-be-determined control for the first sensor. 3. Second-sensor prediction: A Kalman filter predictor is used to time-update ∗2 ∗2 2 2 the current sensor state x k|k to the future state x k+1|k (u), where u is the yet-to-be-determined control for the second sensor. 1

2

4. Target update: If Zk+1 , Zk+1 are the future random target-generated measurements for the two sensors, use a two-sensor Kalman filter corrector 1 2 to measurement-update xk+1|k to xk+1|k+1 (Zk+1 , Zk+1 ). Let 1

2

1

2

Rk+1 (Zk+1 , Zk+1 ) = Ak+1 xk+1|k+1 (Zk+1 , Zk+1 ) be the random reference vector.

(24.12)

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∗1

5. First-sensor update: If Z k+1 is the future actuator measurement, use a ∗1 1 Kalman filter corrector to measurement-update x k+1|k (u) to ∗1

1

∗1

x k+1|k+1 (u, Z k+1 ).

Let ∗1

1

∗1

∗1

∗1

1

∗1

(24.13)

Rk+1 (u, Z k+1 ) = Ak+1 x k+1|k+1 (u, Z k+1 ) be the random controlled vector for the first sensor. ∗2

6. Second-sensor update: If Z k+1 is the future actuator measurement, use a ∗2 2 Kalman filter corrector to measurement-update x k+1|k (u) to ∗2

2

∗2

x k+1|k+1 (u, Z k+1 ).

Let ∗2

2

∗2

∗2

∗2

2

∗2

(24.14)

Rk+1 (u, Z k+1 ) = Ak+1 x k+1|k+1 (u, Z k+1 ) be the random controlled vector for the second sensor.

7. Objectivization: Define a suitable distance-based objective function from ∗1

∗2

Rk+1 , Rk+1 , and Rk+1 , such as 1

∗1

1

2

∗2

2

(24.15)

Ok+1 (u, u, Zk+1 , Z k+1 , Zk+1 , Z k+1 ) 1

=

∗1

1

1

2

O k+1 (u, Zk+1 , Z k+1 , Zk+1 ) 2

1

2

∗2

2

+O k+1 (u, Zk+1 , Zk+1 , Z k+1 ) 1

2

where O k+1 and O k+1 are defined as in (24.8). 1

∗1

8. Hedging: Eliminate the unknowable future measurements Zk+1 , Z k+1 , 2

∗2

Zk+1 , Z k+1 , typically by taking the expected value: 1

2

1

2

1

∗1

∗2

2

Ok+1 (u, u) = E[Ok+1 (u, u, Zk+1 , Z k+1 , Zk+1 , Z k+1 )].

(24.16)

9. Optimization: Minimize the objective function to determine the optimal controls for the two sensors: 1

2

1

2

(uk+1 , uk+1 ) = arg inf Ok+1 (u, u). 1 2

u,u

10. Repeat.

(24.17)

Single-Target Sensor Management

24.3

869

SINGLE-SENSOR, SINGLE-TARGET CONTROL: MODELING

Now turn to the general single-step, single-sensor, single-target control problem, but without clutter. In this case: • The target is known a priori to exist. • The clutter process is trivial: κk+1 (Z) = δ0,|Z| , where κk+1 (Z) is the multiobject clutter probability distribution. • The sensor probability of detection (the sensor FoV) is general: ∗

∗

pD (x, x) abbr. = pD,k+1 (x, x).

(24.18)

• There are no target appearances or disappearances. • The evolution of the joint target-sensor system is governed by a Markov transition density that factors as follows: ∗

∗

∗

∗

∗

fk+1|k (x, x|u, x′ , x′ ) = fk+1|k (x|x′ ) · f k+1|k (x|x′ , u).

(24.19)

That is, the target’s dynamical behavior is independent of the dynamical behavior of the sensor. Also (as an approximation) the sensor’s dynamical behavior is assumed to be independent of the dynamical behavior of the target; but does depend on a control variable u that parametrizes a family ∗ ∗ ∗ f k+1|k (x|x′ , u) of Markov densities that governs the sensor’s motion. The control u, which is chosen at time tk , determines the possible constraints on the sensor’s state at the future time tk+1 . • If a measurement is collected, the likelihood function of the joint system factors as ∗ ∗ ∗ ∗ ∗ ∗ fk+1 (z, z|x, x) = fk+1 (z|x, x) · f k+1 (z|x). (24.20) That is, the measurement z collected by the sensor from the target, depends ∗ on the state x of the target and the state x of the sensor. The actuator ∗ ∗ measurement z, however, depends only on x. • Thus the total likelihood function for the target-generated measurements is:  ∗ 1 − pD (x, x) if Z=∅  ∗ ∗ ∗ fk+1 (Z|x, x) = pD (x, x) · fk+1 (z|x, x) if Z = {z} . (24.21)  0 if otherwise

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Let: • Z (k) : Z1 , ..., Zk be the time sequence of target measurement sets at time tk , where each Zi is either a singleton or the empty set. ∗

∗

∗

• Z k : z1 , ..., zk be the time sequence of actuator measurements at time tk . • U k : u0 , ..., uk be the time sequence of controls (control policy) at time tk , with ui being chosen at time ti . The first control u0 is chosen at the initial time t0 , because it is required to ∗ ∗ ∗ choose the sensor’s first Markov density f 1|0 (x|x′ , u0 ). The Bernoulli filter of Section 13.2 is a general solution for the problem of detecting and tracking a single target using a single sensor with missed detections and clutter. Given the above assumptions, it reduces to a single-target Bayes filter of the form ∗

∗

∗

∗

fk|k (x, x|Z (k) , Z k , U k−1 )

... → →

fk+1|k (x, x|Z (k) , Z k , U k )

→

fk+1|k+1 (x, x|Z (k+1) , Z k+1 , U k )

∗

∗

→ ...

where: • Time update: ∗

∗

=

fk+1|k (x, x|Z (k) , Z k , U k−1 , uk ) ∫ ∗ ∗ ∗ fk+1|k (x|x′ ) · f k+1|k (x|x′ , uk ) ∗

∗

(24.22)

∗

·fk|k (x′ , x′ |Z (k) , Z k , U k−1 )dx′ dx′ . • Measurement update with Zk+1 = {zk+1 }: ∗

∗

=

fk+1|k+1 (x, x|Z (k+1) , Z k+1 , U k−1 , uk ) ( ) ∗ ∗ ∗ ∗ ∗ pD (x, x) · fk+1 (zk+1 |x, x) · f k+1 (zk+1 |x) ∗ ∗ ·fk+1|k (x, x|Z (k) , Z k , U k−1 , uk ) ∗

∗

fk+1 ({zk+1 }, zk+1 |Z (k) , Z k , U k−1 , uk )

(24.23)

.

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• Measurement update with Zk+1 = ∅: ∗

∗

=

fk+1|k+1 (x, x|Z (k+1) , Z k+1 , U k−1 , uk ) ( ) ∗ ∗ ∗ ∗ (1 − pD (x, x)) · f k+1 (zk+1 |x) ∗ ∗ ·fk+1|k (x, x|Z (k) , Z k , U k−1 , uk ) ∗

∗

∗

∗

(24.24)

fk+1 (∅, zk+1 |Z (k) , Z k , U k−1 , uk ) and where

=

f ({z }, zk+1 |Z (k) , Z k , U k−1 , uk ) ∫k+1 k+1 ∗ ∗ ∗ ∗ pD (x, x) · fk+1 (zk+1 |x, x) · fk+1 (zk+1 |x) ∗

∗

(24.25)

∗

·fk+1|k (x, x|Z (k) , Z k , U k−1 , uk )dxdx and ∗

∗

=

f (∅, zk+1 |Z (k) , Z k , U k−1 , uk ) ∫k+1 ∗ ∗ ∗ ∗ (1 − pD (x, x)) · f k+1 (zk+1 |x) ∗

∗

(24.26)

∗

·fk+1|k (x, x|Z (k) , Z k , U k−1 , uk )dxdx. At time tk for k ≥ 1 the control-sequence U k−1 : u0 , ..., uk−1 has been chosen. At time tk , the next control uk must be chosen. However, in a Bayesian approach: • Determination of uk distribution

requires that the information in the future posterior ∗

∗

∗

fk+1|k+1 (x, x|Z (k) , Z k , U k−1 , Zk+1 , zk+1 , uk ) must be maximized, compared to the information in the prior distribution ∗

∗

fk+1|k (x, x|Z (k) , Z k , U k−1 , uk )

(24.27)

despite the fact that the future measurements are inherently unknowable.

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24.4

SINGLE-SENSOR, SINGLE-TARGET CONTROL: SINGLE-STEP

The previous control policy U k−1 has been determined, and we are to determine the next control uk . It should be chosen so that the placement of the FoV ∗ pD (x, x) is as efficacious as possible, given the influence of the sensor noise ∗ (sensor resolution) as specified by the likelihood function fk+1 (z|x, x). The control scheme has the form of a closed-loop Bayes filter:

...

single-sensor/target measurement-update ∗ ∗ → fk|k (x, x|Z (k) , Z k , U k−1 ) optimal selection of next single-sensor control time-projection ∗

∗

→

fk+1|k+1 (x, x|Z (k) , Z k , U k−1 , ∗ Z, z, u) hedging

↓ select next control

uk = arg supu Ok+1 (u) → ∗

uk ,zk+1 ,zk+1

single-sensor/target time & measurement updates ∗ ∗ fk+1|k+1 (x, x|Z (k+1) , Z k+1 , U k )

→ ...

↗ ∗

Here, Z is the unknown sensor measurement set at time tk+1 , z is the unknown actuator measurement at time tk+1 , and Ok+1 (u) is the sensor management objective function that will be used to determine the control u at time tk . This control scheme is explained in more detail in the following subsections.

24.5

SINGLE-SENSOR, SINGLE-TARGET CONTROL: OBJECTIVE FUNCTIONS

The purpose of an optimal sensor control u at time tk is to maximally increase the amount of information known about the target state x at time tk+1 . In a Bayesian formulation, there are two ways one could go about doing this. In the first formulation we: • Measure the amount of information in the marginal posterior distribution ∗ ∗ fk+1|k+1 (x|Z (k) , Z k , U k , Z, z, u) following the action.

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• Do so in comparison to the information contained in the marginal distribution ∗ fk|k (x|Z k , Z k , U k ) prior to the action. This method for measuring information gain includes two different effects: the effect of the time-update and the effect of the measurement-update. Our goal is to measure the information gain that is attributable solely to the information contained in the new measurements. Thus in the second formulation we: • Maximize the amount of information in the marginal posterior distribution ∗ ∗ fk+1|k+1 (x|Z (k) , Z k , U k , Z, z, u) following the action. • Do so in comparison to the prior information contained in the marginal ∗ predicted distribution fk+1|k (x|Z k , Z k , U k , u). This is the formulation that will be adopted in the remainder of Part V. ∗ Begin with the following observation: the future observations Z, z are unknowable, but it is clear that: • Good control has not occurred unless the sensor succeeds in actually collecting a measurement from the target. That is, we may assume that Z ̸= ∅ and thus Z = {z} for some z. In what follows, the following choices for objective function are considered: Kullback-Leibler information Gain, Csisz´ar information Gain, and CauchySchwartz information Gain. 24.5.1

Kullback-Leibler Information Gain

This is the cross-entropy (see Section 6.3) ∗

=

Ok+1 (u, z, z) ∫ ∗ ∗ fk+1|k+1 (x|Z (k) , Z k , U k−1 , {z}, z, u) ( ) ∗ ∗ fk+1|k+1 (x|Z (k) , Z k , U k−1 , {z}, z, u) dx · log ∗ fk+1|k (x|Z (k) , Z k , U k−1 , u)

(24.28)

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of the marginal distributions on the target state, ∗

=

∗

fk+1|k+1 (x|Z (k) , Z k , U k−1 , {z}, z, u) ∫ ∗ ∗ ∗ ∗ fk+1|k+1 (x, x|Z (k) , Z k , U k−1 , {z}, z, u)dx

(24.29)

and ∗

= 24.5.2

fk+1|k (x|Z (k) , Z k , U k−1 , u) ∫ ∗ ∗ ∗ fk+1|k (x, x|Z (k) , Z k , U k−1 , u)dx.

(24.30)

Csisz´ar Information Gain

More generally, we could use any Csisz´ar discrimination ∗

=

Ok+1 (u, {z}, z) ( ) ∗ ∫ ∗ fk+1|k+1 (x|Z (k) , Z k , U k−1 , {z}, z, u) ck+1 ∗ fk+1|k (x|Z (k) , Z k , U k−1 , u)

(24.31)

∗

·fk+1|k (x|Z (k) , Z k , U k−1 , u)dx where ck+1 (x) is a convex kernel—see (6.62). 24.5.3

Cauchy-Schwartz Information Gain

The multitarget version of this information functional was introduced in (6.73). The single-target version is ∗

=

Ok+1 (u, {z}, z) (24.32) ( ∫ ) ∗ ∗ fk+1|k+1 (x|Z (k) , Z k , U k−1 , {z}, z, u) ∗ ·fk+1|k (x|Z (k) , Z k , U k−1 , u)dx  − log  √∫ ∗ ∗ (k) k k−1 2 fk+1|k+1 (x|Z , Z , U , {z}, z, u) dx   √∫ ∗ (k) k k−1 · fk+1|k (x|Z , Z , U , u)2 dx

Further study is required to determine if this objective function produces good sensor management behavior.

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24.6

875

SINGLE-SENSOR, SINGLE-TARGET CONTROL: HEDGING ∗

Whatever objective function Ok+1 (u, z, z) is employed, it depends on the future ∗ measurements z, z. Thus we must “hedge” against the fact that they are inherently unknowable. The following methods are considered: • Expected-value hedging. • Minimum-value hedging. • Multisample approximate expected-value hedging. • Single-sample approximate expected-value hedging, in the form of “maxPM” hedging. • A combination of expected-value hedging and PM single-sample hedging. 24.6.1

Expected-Value Hedging

In this approach—the one that is most commonly employed in control theory—we hedge against (intuitively understood) the “average measurement”: [ ] ∗ Ok+1 (u) = E Ok+1 (u, Zk+1 , Zk+1 ) . (24.33) 24.6.2

Minimum-Value Hedging

In this approach we hedge against the worst-case measurement: ∗

Ok+1 (u) = inf∗ Ok+1 (u, z, z).

(24.34)

z,z

24.6.3

Multisample Approximate Hedging

Both expected-value and minimization hedging will be computationally intensive, so approximations are desirable. One such approach is as follows. Multiple representative samples z1,k+1 ,...,zν,k+1 are drawn from the distribution fk+1 (z|Z k ), ∗ ∗ ∗ and multiple representative samples z1,k+1 ,...,zν,k+1 are drawn from the distribu∗

∗

tion fk+1 (z|Z k ). Then they are used to approximate the expected value: ∗

ν ∑ ν ∑ ∗ ∼ 1∗ Ok+1 (u, zj,k+1 , zj ′ ,k+1 ). E[Ok+1 (u, Z, Z)] = ν ν j=1 j ′ =1 ∗

(24.35)

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24.6.4

Single-Sample Approximate Hedging

If the multisample approach proves too computationally difficult, one can instead draw a single sample z!k+1 that is a “most representative” measurement in some sense. An obvious approach—one that foreshadows the “predicted ideal measurement -set (PIMS)” technique of Section 25.10.2 —is as follows. Suppose that the Markov densities have the additive forms fk+1|k (x|x′ ) ∗

∗

∗′

f k+1|k (x|x )

=

fWk (x − φk (x′ )) ∗

=

∗

f

∗

Wk

∗

(24.36)

∗′

(24.37)

(x − φk (x )).

If the target prediction models are sufficiently accurate, then the best-possible sensor and actuator-sensor measurements that can be collected at time tk+1 are the predicted measurements (PMs): M zP k+1 ∗PM

zk+1

= =

(24.38)

φk (xk+1|k ) ∗

∗

(24.39)

φk (xk+1|k )

∗

where xk+1|k and xk+1|k are, respectively, estimates of the predicted target-state and predicted sensor-state at time tk+1 . Thus, as a computational approximation, we could use “PM hedging” as an approximation of expected-value hedging: ∗

M PM Ok+1 (u) = Ok+1 (u, zP k+1 , zk+1 ).

(24.40)

However, this line of reasoning overlooks a subtle difficulty. Suppose that the probability of detection is a “cookie cutter”: ∗

(24.41)

∗ (x) pD (x, x) = 1S(x)

∗

∗

where S(x) is a bounded subset of the target state space. If xk+1|k ∈ / S(x), then M it is impossible to collect the predicted measurement zP , because the predicted k+1 track xk+1|k is undetectable by the sensor. It would be a bad idea to try to induce the sensor to collect a measurement that cannot possibly be collected. More generally: M • The more improbable it is that zP k+1 should try to collect it.

can be collected, the less the sensor

Single-Target Sensor Management

M • The more probable it is that zP k+1 should try to collect it.

877

can be collected, the more the sensor

Some approach will have to be devised to take these considerations into account. This issue will be addressed in general detail in Section 25.10.2; and for the special case of Bernoulli filters, in Section 26.2.3. For the moment, however, let us set this issue aside until we have enough mathematical machinery to properly address it. 24.6.5

Mixed Expected-Value and PM Hedging

For sensor management using Kullback-Leibler objective function (or its approximations such as PENT and PENTI), it will prove convenient to employ a combination of hedging approaches (see Section 24.10.3, and Section 25.13.3): • Expected-value hedging for actuator-sensor measurements. • Single-sample PM hedging for sensor measurements. ∗

That is, let Ok+1 (u, z, z) be the original objective function. Eliminate z ∗ using PM hedging, but eliminate z by taking the expected value: ∗

M Ok+1 (u) = E[Ok+1 (u, zP k+1 , Zk+1 )].

24.7

(24.42)

SINGLE–SENSOR, SINGLE-TARGET CONTROL: OPTIMIZATION

The determination of the control u requires solution of the following optimization problem over an infinite solution space U of possible controls: uk = arg sup Ok+1 (u).

(24.43)

u

This will be computationally demanding in general. A commonly employed approximate approach is to limit the values of u to a small finite set U0 ⊆ U of representative control actions (“admissible controls”): uk = arg sup Ok+1 (u).

(24.44)

u∈U0

This type of approximation will be even more necessary for multisensormultitarget sensor management problems (see Section 25.11).

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24.8

SPECIAL CASE 1: IDEAL SENSOR DYNAMICS

A special case of interest occurs if we assume that: • The control space is the same as the sensor state space, ∗

(24.45)

U=X • The sensor Markov density has the form ∗

∗

∗

∗

∗

∗

∗

∗

f k+1|k (x|u, x′ ) = f k+1|k (x|u) = f k+1|k (x|xk )

(24.46)

∗

where now the control is u = xk . ∗

∗

∗

Here, f k+1|k (x|xk ) is interpreted as the probability (density) that the sensor ∗ ∗ state x is reachable at time tk+1 , given that the sensor has state xk at time tk . Suppose in particular that ∗

∗

∗

(24.47)

f k+1|k (x|u) = δu (x). ∗

That is, the sensor state xk+1 at time tk+1 is: ∗

• Completely decoupled from the sensor state x′

at time tk . ∗

• It is also the control variable at time tk : uk = xk+1 . This means that the sensor has ideal dynamical behavior. That is, between time tk and time tk+1 it can be redirected without any constraint from its previous state to any desired state. Selecting the control uk at time tk is the same as ∗ ∗ selecting the sensor state xk+1 at time tk+1 . In this case the variable x is deterministic, and as is shown in Section K.34, the predicted and measurementupdated joint distributions must have the form ∗

∗

fk+1|k (x, x|Z (k) , Z k , U k−1 , u) =

(24.48)

∗

δu (x) · fk+1|k (x|Z (k) , U k−1 )

and ∗

∗

fk+1|k+1 (x, x|Z (k+1) , Z k+1 , U k−1 , u) =

∗

δu (x) · fk+1|k+1 (x|Z

(k+1)

,U

k−1

, u).

(24.49)

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∗

∗

∗

The control-sequence U k−1 : u0 , ..., uk−1 becomes a sequence X k : x1 , ..., xk of sensor states. Thus the single-step control scheme reduces to the form single-target measurement-update ∗ ... → fk|k (x|Z (k) , X k ) optimal selection of next sensor state time-projection ∗

fk+1|k+1 (x|Z (k) , X k , ∗ z, x)

→

hedging

xk+1

↓ ∗ = arg supx∗ Ok+1 (x)

→

single-target time & measurement updates ∗ fk+1|k+1 (x|Z (k+1) , X k+1 )

∗

∗

xk+1 ,zk+1

→ ...

↗

.

Here, from (24.22) through (24.26), ∗

fk+1|k (x|Z (k) , X k )

=

∫

fk+1|k (x|x′ )

(24.50)

∗

∗

∗

fk+1|k+1 (x|Z (k) , X k , {z}, x)

·fk|k (x′ |Z (k) , X k )dx′ ( ) ∗ ∗ pD (x, x) · fk+1 (z|x, x) ∗ ·fk+1|k (x|Z (k) , X k ) =

∗

(24.51)

∗

fk+1 ({z}|Z k , X k , x) ∗

∗

fk+1 ({z}|Z k , X k , x)

=

∫

∗

∗

pD (x, x) · fk+1 (z|x, x)

(24.52)

∗

·fk+1|k (x|Z (k) , X k )dx. As a result, the Kullback-Leibler objective function of (24.28) becomes

∗

Ok+1 (x, z)

=

∫ (

∗

∗

fk+1|k+1 (x|Z (k) , X k , {z}, x) ∗

fk+1|k (x|Z (k) , X k ) ∗

∗

·fk+1|k+1 (x|Z (k) , X k , {z}, x)dx.

)

(24.53)

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24.9

SIMPLE EXAMPLE: LINEAR-GAUSSIAN CASE

The purpose of this section is to provide a concrete, analytically closed-form example to illustrate the sensor management concepts described in the previous sections. Consider the ideal-dynamics case, (24.50) through (24.52). Suppose that the sensor and motion models are linear-Gaussian: • Target Markov density: fk+1|k (x|x′ ) = fQk (x − Fk x′ ).

(24.54)

• Sensor probability of detection: ∗

pD (x, x) =

√

∗

∗

det 2πEk+1 · NEk+1 (Ak+1 x − Ak+1 x). ∗

(24.55)

∗

Here, Ak+1 x is the reference vector, Ak+1 x is the controlled vector, and Ek+1 specifies the spatial distribution of the sensor FoV (see Section 24.2.1). • Sensor likelihood function: ∗

∗

∗

fk+1 (z|x, x) = NVk+1 (z − Hk+1 x − J k+1 x).

(24.56)

∗

The sensor’s state x induces a translational bias on the measurement but does not affect sensor resolution, which is always the same. The following should be noted: • In (24.55), Ak+1 x could be the position of a target with state x, as in (24.5). ∗

∗

• In (24.55), Ak+1 x could be the center of the sensor’s field of view (FoV), as in (24.6). ∗

• pD (x, x) = 1 when the FoV is centered on the target’s position. The FoV is (hyper)ellipsoidal in shape, and its form is determined by the shapecovariance matrix Ek+1 . ∗

∗

• In (24.56), the term J k+1 x is not necessarily needed. Assume that the coordinate system is absolute (rather than, say, the sensor’s reference frame). Then regardless of how the sensor FoV is placed, a measurement collected ∗ from the target will remain the same. In this situation x has no influence on

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∗

fk+1 (z|x, x) and so we can set ∗

(24.57)

J k+1 = 0. Also assume that the prior distribution is linear-Gaussian: ∗

fk|k (x|Z (k) , X k ) = NPk|k (x − xk|k ).

(24.58)

Then in Section K.35 the following formulas will be demonstrated: • Time- and measurement-update equations: Equations (24.50) through (24.52) become ∗

fk+1|k (x|Z (k) , X k ) ∗

∗

fk+1|k+1 (x|Z (k) , X k , z, x)

∗

=

NPk+1|k (x − xk+1|k )

(24.59)

=

NQk +Fk Pk|k FkT (x − Fk xk|k )

(24.60)

=

NPk+1|k+1 (x − xk+1|k+1 (z, x))(24.61)

∗

fk+1 ({z}|Z (k) , X k , x) =

∗

√

(24.62)

det 2πEk+1

T ·NRk+1 +Hk+1 Pk+1|k Hk+1 (z − Hk+1 xk|k ) ( ) ∗ Ak+1 ck+1 (z, x) ∗ ·NEk+1 +Ak+1 Ck+1 ATk+1 ∗ −Ak+1 x

where −1 Pk+1|k+1

=

−1 Pk+1|k

(24.63)

−1 −1 T +Hk+1 Rk+1 Hk+1 + ATk+1 Ek+1 Ak+1 ∗

−1 Pk+1|k+1 xk+1|k+1 (z, x)

=

−1 Pk+1|k xk+1|k

(24.64) ∗

∗

−1 −1 T +Hk+1 Rk+1 z + ATk+1 Ek+1 Ak+1 x

and where −1 Ck+1 ∗ −1 Ck+1 ck+1 (z, x)

=

−1 −1 T Pk+1|k + Hk+1 Rk+1 Hk+1

−1 xk+1|k =Pk+1|k

+

−1 T Hk+1 Rk+1 (z

(24.65) ∗

− Jk+1 x).(24.66)

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• Kullback-Leibler sensor management objective function: (24.53) becomes ∗

(24.67)

2 · Ok+1 (x, z) =

−1 tr(Pk+1|k+1 Pk+1|k ) ∗

∗

−1 +(xk+1|k+1 (z, x) − xk+1|k )T Pk+1|k (xk+1|k+1 (z, x) − xk+1|k ) ( ) det Pk+1|k+1 − log −N det Pk+1|k

where N is the dimension of the target state space. • Optimal sensor management solution: For any given measurement z at time ∗ tk+1 , the objective function Ok+1 (x, z) is maximized if and only if the ∗ sensor state x at time tk+1 is chosen so that ( ) ∗ ∗ ∗ ∗ −1 ATk+1 Ek+1 Ak+1 xk+1|k − Ak+1 x − J k+1 x ( ) −1 T Hk+1 Rk+1 z − Hk+1 xk+1|k .

=

(24.68)

• Optimal sensor management solution with PM single-sample hedging: Suppose that z = Hk+1 xk+1|k is the predicted measurement as in (24.39). Then ∗ the objective function is maximized if and only if x is chosen so that ( ) ∗ ∗ ∗ −1 Ak+1 xk+1|k − (Ak+1 + J k+1 )x = 0. ATk+1 Ek+1

(24.69)

Thus single-sample PM optimization: – Directly causes sensor resolution (as defined by Rk+1 ) to have no ∗ influence over the choice of the sensor state x. ∗

∗

– Tends, if J k+1 x = 0 as in (24.57), to place the center of the sensor FoV at the predicted position of the target.

24.10

SPECIAL CASE 2: SIMPLIFIED NONIDEAL DYNAMICS

The purpose of this section is to introduce a simplified, approximate version of the general sensor management approach described in Section 24.4. It is a generalization of the ideal-sensor approach of Section 24.8, but one that allows

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sensor dynamics to be taken into account. It thus allows control of a nonideal sensor to be accomplished in the same, simplified setting as control of an ideal sensor. Strictly speaking, this approach is not computationally essential for the purpose of single-sensor, single-target control. However, its generalization to the multisensor-multitarget problem in Section 25.13 will be essential for computational reasons. Thus the approach is introduced here first to aid conceptual understanding. The section is organized as follows: 1. Section 24.10.1: dynamics.

Modeling assumptions for simplified nonideal sensor-

2. Section 24.10.2: The filtering equations for simplified nonideal sensordynamics. 3. Section 24.10.3: The optimization approach for simplified nonideal sensordynamics. 24.10.1

Simplified Nonideal Single-Sensor Dynamics: Modeling

Suppose that sensor dynamics are not ideal. That is, the sensor and/or the platform carrying it, are subject to temporal and/or spatial constraints on their ability to switch from one state to another. ∗ Because of Bayes’ rule, the joint sensor-target state (x, x) factors as described in Section 5.9, ∗

∗

∗

fk|k (x, x|Z (k) , Z k , U (k−1) )

=

∗

∗

f k|k (x|Z (k) , Z k , U (k−1) ) ∗

(24.70) ∗

·fk|k (x|Z k , Z (k) , U (k−1) , x). Assume that: 1. As in Section 24.8, the control space is the same as the sensor state space, ∗

(24.71)

U = X. ∗

2. The control at time tk is the sensor state uk = xk+1 at time tk+1 ; and ∗ ∗ ∗ thus the control-sequence has the form U k−1 = X k : x1 , ..., xk . 3. Rather than being selected by a control vector, the sensor Markov density is specified a priori: ∗

∗

∗

∗

∗

∗

f k+1|k (x|x′ , uk−1 ) = f k+1|k (x|x′ ).

(24.72)

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∗

∗

∗

That is, f k+1|k (x|x′ ) specifies an assumption about the reachability of the ∗ ∗ sensor-state x at time tk+1 , given that the sensor state at time tk is x′ . 4. The estimated sensor state depends only on the history of actuator-sensor measurements and controls: ∗

∗

∗

∗

∗

∗

∗

∗

f k|k (x|Z (k) , Z k , X k )

=

∗

∗

f k+1|k (x|Z (k) , Z k , X k )

=

∗

∗

f k|k (x|Z k ) ∗

(24.73)

∗

f k+1|k (x|Z k ).

(24.74)

That is, the sensor state does not depend on the history of the target-generated measurements. But in addition, it does not depend on the control history, either. This is because the sensor is treated as a noncooperative target. Its trajectory is to be estimated as though its control history is an unknown sequence of perturbations of that trajectory. 5. The estimated target state depends only on the history of target-generated measurements, and on the sensor-control sequence: ∗

∗

∗

∗

fk|k (x|Z (k) , Z k , X k , x) fk+1|k (x|Z

(k)

∗

k

∗

∗

k

fk|k (x|Z (k) , X k )

=

, Z , X , x)

=

fk+1|k (x|Z

(k)

∗

(24.75) k

, X ).

(24.76)

Equation (24.76) follows from the fact that the predicted target state cannot depend on the predicted sensor state. Equation (24.75) follows from the fact ∗ ∗ ∗ ∗ ∗ that fk|k (x|Z (k) , Z k , X k , x) cannot depend on x since xk has already been selected. ∗

6. The next control (sensor state) xk+1 is selected at time tk+1 , prior to application of the target measurement-update. Given this, the evolution of the sensor is described by a conventional Bayes filter ∗

... →

∗

∗

∗

f k|k (x|Z k )

∗

∗

∗

f k+1|k (x|Z k )

→

∗

∗

f k+1|k+1 (x|Z k+1 )

→

and the evolution of the target is described by a conventional Bayes filter ∗

... →

fk|k (x|Z (k) , X k ) ∗

→

fk+1|k (x|Z (k) , X k )

→

fk+1|k+1 (x|Z (k+1) , X k+1 )

∗

→ ...

→ ...

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24.10.2

885

Simplified Nonideal Single-Sensor Dynamics: Filtering Equations

Let: • Z (k) : Z1 , ..., Zk be the time sequence of target measurement sets at time tk , where either Zi = ∅ or Zi = {zi }. ∗

∗

∗

• Z k : z1 , ..., zk be the time sequence of actuator measurements at time tk . ∗

∗

∗

∗

• X k : x1 , ..., xk be a time sequence of sensor states at time tk , with xi being chosen at time ti , in a manner to be specified shortly. Then given the assumptions in the previous section, in Section K.36 it is shown that the single-filter control scheme in Section 24.4 can be equivalently replaced by the two-filter control scheme single-sensor/target measurement update ∗ ∗ ∗ ... → f k|k (x|Z k ) ∗

fk|k (x|Z (k) , X k )

... → →

optimal selection of next sensor state ∗ ∗ ∗ f k+1|k (x|Z k )

↓ time-projection ∗

fk+1|k+1 (x|Z (k) , X k , ∗ z, x)

→

hedging

↓ select next sensor state ∗

∗

xk+1 = arg supy∗ Ok+1 (y) ∗

zk+1

single-sensor/target time & measurement updates ∗ ∗ ∗ f k+1|k+1 (x|Z k+1 )

↗

→ ...

∗

fk+1|k+1 (x|Z (k+1) , X k+1 )

→ ∗

xk+1 ,zk+1 ∗

→ ...

↗

where xk+1 is to be determined as in the next section. The bottom filter indicated in each of the three boxes is a Bayes filter on the target state x; and the top filter is ∗ a Bayes filter on the sensor state x. The filtering equations for these filters are:

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• Time updates for the sensor and the target: ∗

∗

fk+1|k (x|Z

(k)

=

∫

f k+1 (x|x′ ) · f k|k (x′ |Z k )dx′ (24.77)

=

∫

fk+1|k (x|x′ )

∗

f k+1|k (x|Z k ) ∗

k

,X )

∗

∗

∗

∗

∗

∗

∗

(24.78)

∗

·fk|k (x′ |Z (k) , X k )dx′ . • Sensor measurement-update: ∗

∗

∗

∗

∗

∗

∗

∗

∗

f k+1|k+1 (x|Z k+1 ) ∝ f k+1 (zk+1 |x) · f k+1|k (x|Z k ).

(24.79)

• Target measurement-update for Zk+1 = {zk+1 }: ∗

fk+1|k+1 (x|Z (k+1) , X k+1 )

∗

∝

(24.80)

pD (x, xk+1 ) ∗

·fk+1 (zk+1 |x, xk+1 ) ∗

·fk+1|k (x|Z (k) , X k ). • Target measurement-update for Zk+1 = ∅: ∗

fk+1|k+1 (x|Z (k+1) , X k+1 )

∗

∝

(1 − pD (x, xk+1 ))

(24.81)

∗

·fk+1|k (x|Z (k) , X k ). 24.10.3

Simplified Nonideal Single-Sensor Dynamics: Optimization

A good choice of control should not result in a null target measurement set Zk+1 = ∅, since in this case no information about the target can be collected. Thus the future measurement set is Zk+1 = {z} for some z. From Bayes’ rule and given the assumptions in Section 24.10.1, the joint sensor-target measurement-update can be factored as ∗

∗

∗

∗

fk+1|k+1 (x, x|Z (k+1) , Z k , X k , {z}, z) ∗

=

∗

∗

∗

f k+1|k+1 (x|Z k , z) ∗

∗

·fk+1|k+1 (x|Z (k) , X k , {z}, x)

(24.82)

Single-Target Sensor Management

∗

∗

887

∗

where X k : x1 , ..., xk is the control-sequence at time tk and where ∗

∗

∗

∗

X k+1 : x1 , ..., xk , x ∗

∗

is the control-sequence at time tk+1 , with the selection x = xk+1 still to be determined. Apply the Kullback-Leibler cross-entropy to the target part of this distribution to get ∗ ∗ ∗ ∗ ∗ ∗ ∗ ˜ k+1 (x, Ok+1 (x, z, z) = f k+1|k+1 (x|Z k , z) · O z) where ∗ ˜ k+1 (x, O z)

∫

∗

∗

fk+1|k+1 (x|Z (k) , X k , {z}, x) ( ) ∗ ∗ fk+1|k+1 (x|Z (k) , X k , {z}, x) · log dx ∗ fk+1|k (x|Z (k) , X k )

=

(24.83)

is the objective function for the ideal-sensor case of (24.53). Now apply the mixed expected-value, PM single-sample hedging approach of Section 24.6.5: ∗

Ok+1 (x)

=

∫

∗

∗

=

∗

∗

∗

∗

∗

M k Ok+1 (x, zP k+1 , z) · f k+1 (z|Z )dz ∗

∗ ∗ M ˜ k+1 (x, f k+1|k (x|Z k ) · O zP k+1 ).

∗

It follows that to maximize Ok+1 (x): ∗ M ˜ k+1 (x, • Maximize the ideal-sensor objective function O zP k+1 ), subject to the constraint that ∗

• x must be reachable from time tk to time tk+1 . Thus the next sensor state is ∗

∗

xk+1 = arg sup Ok+1 (y). ∗

y

(24.84)

Chapter 25 Multitarget Sensor Management 25.1

INTRODUCTION

Having illustrated the basic sensor management concepts for the single-sensor, single-target case, let us now turn to the multisensor-multitarget case. The approach was summarized earlier in Section 23.3.1. The purpose of this chapter is to describe it in greater detail. The sensor management problem cannot be properly addressed unless it is first formulated in a mathematically complete manner. Thus much of the chapter will be devoted to definitions, notation, and specification of models. Sections 25.2-25.4 will be devoted to the description of state spaces and set integrals for targets and sensors; of joint target-sensor state spaces; of multisensor measurement spaces; of multisensor actuator-sensor spaces; and of multisensor control spaces. Likewise, Sections 25.5 and 25.6 will be devoted to specification of multisensor, multitarget motion models and multisensor-multitarget measurement models for multisensor-multitarget control. Since the notation will be fairly involved, a tabulated summary will be provided in Section 25.7. The remaining sections of the chapter will be devoted to the multisensormultitarget sensor management approach itself. Section 25.8 describes the approach for single-step look-ahead multisensor-multitarget control; and Sections 25.9 and 25.10 are devoted to the description of sensor management objective functions and hedging-optimization strategies. Sections 25.12 and 25.13 are devoted to simplifications of the approach necessary for computational tractability: an approach for sensors with ideal dynamical behavior, and a generalization of it to account for sensors with nonideal dynamics.

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The ultimate purpose of sensor management is not to collect information about all targets, but about targets of situational interest. This is the subject of Section 25.14, which shows how to statistically bias sensor collections in favor of targets of interest (ToIs). 25.1.1

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter: • Multisensor-multitarget sensor management requires a careful and detailed specification of state and measurement spaces, and of motion and measurement models for both targets and sensors. • In particular, the state space for a joint multisensor-multitarget system, containing a maximum of s sensors, is a disjoint union of the form ∗1

∗s

˘ = X ⊎ X ⊎ ... ⊎ X X

(25.1)

∗j

where X is the target state space and X is the state space for the jth sensor. ˘ The joint state of the system at any given time is, therefore, a finite subset X ˘ (Section 25.2.3). of X • Multisensor-multitarget sensor management is a direct generalization of single-sensor, single-target sensor management. • Information-theoretic objective functions can be defined for the multisensormultitarget case. However, these will almost always be computationally intractable (Section 25.9.1). • The PENT objective function, which was introduced in Section 23.2, is a computationally tractable and physically intuitive approximation of various information theoretic objective functions, including Kullback-Leibler crossentropy and R´enyi α-divergence (Section 25.9.4). • Since PHD filters are first-order in a point process sense, they must be used with a first-order objective function such as PENT—or perhaps the CauchySchwartz information functional. • A related and higher-order objective function—the cardinality variance (the variance of PENT)—is also computationally tractable and physically intuitive. It is potentially better-performing in certain circumstances, but at the

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current time it is not clear how to tractably incorporate sensor and platform dynamics into it (Section 25.9.3). • Hedging against unknown future measurements can be achieved for the multisensor-multitarget case, but the usual approaches will almost always be computationally intractable (Section 25.10). • Multisample hedging, which was introduced in Section 23.3.1 and Section 24.6.3 of, can also be applied to the multisensor-multitarget case. • Single-sample hedging was introduced in Section 23.3.1 and Section 24.6.4. The predicted ideal measurement set (PIMS) is a specific kind of singlesample hedging. The PIMS is the set of predicted measurements, mathematically adjusted to account for their collectability (Sections 25.10.1-25.10.3). • Computational complexity can be significantly reduced if one assumes that the sensors are dynamically ideal—that is, can be quickly redirected in any desired manner within a specified time-interval (Section 25.12). • The ideal-sensor approach can be modified to account for nonideal sensor dynamics (Section 25.13). • Sensor management can be statistically biased so as to promote the detection of situationally significant targets of interest, or ToIs (Section 25.14). – The primary means of doing so is the tactical importance function or TIF, which models the tactical priority of ToIs. – TIFs provide a mathematical basis for situational awareness (also known as “Levels 2 and 3 data fusion”). 25.1.2

Organization of the Chapter

The chapter is organized as follows: 1. Section 25.2: Target-state spaces and sensor-state spaces for multisensormultitarget control. 2. Section 25.3: Control spaces for multisensor-multitarget control. 3. Section 25.4: Measurement spaces for multisensor-multitarget control. 4. Section 25.5: Motion models for multisensor-multitarget control. 5. Section 25.6: Measurement models for multisensor-multitarget control.

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6. Section 25.7: A summary of notation. 7. Section 25.8: Single-step look-ahead multisensor-multitarget control. 8. Section 25.9: Objective functions for multisensor-multitarget sensor management. 9. Section 25.10: Hedging for multisensor-multitarget sensor management. 10. Section 25.11: Optimization for multisensor-multitarget sensor management. 11. Section 25.12: A special case: ideal sensor dynamics. 12. Section 25.13: A generalization of the ideal-sensor approach that accounts for nonideal sensor dynamics. 13. Section 25.14: Target prioritization—that is, the statistical biasing of sensor management in order to preferentially focus on situationally important targets.

25.2

MULTITARGET CONTROL: TARGET AND SENSOR STATE SPACES

The purpose of this section is to specify those aspects of the approach that involve target states and sensor states. The material follows that of [169], pp. 245-249, and is organized as follows: 1. Section 25.2.1: Target state spaces. 2. Section 25.2.2: Sensor state spaces. 3. Section 25.2.3: Joint multisensor-multitarget state spaces. 4. Section 25.2.4: Integrals and set integrals on joint sensor-target state spaces 5. Section 25.2.5: Probability generating functionals (p.g.fl.’s) on joint sensortarget state spaces. 25.2.1

Target State Spaces

As usual, the single-target state space is denoted X, and the multitarget state space is the hyperspace X∞ of all finite subsets of X.

Multitarget Sensor Management

25.2.2

893

Sensor State Spaces

Assume the following: • Each sensor has a “sensor tag” j = 1, ..., s that uniquely identifies it, its controls, and its measurements. ∗j

The state space of the jth sensor is X, with individual sensor states denoted ∗j

∗j

as x ∈ X. Assume that there are a maximum of s distinct sensors with respective ∗1

∗s

state spaces X, ..., X. Then the joint state space for all sensors is the disjoint union (“topological sum”) ∗

∗1

∗s

(25.2)

X=X ⊎ ... ⊎ X. ∗

˘ Thus a The state of a sensor with an unspecified sensor tag is denoted as x ∈ X. multisensor system will have a state set of the form ∗

∗

∗

(25.3)

X = {x1 , ..., xn∗ } ∗

where n is the current number of sensors in the system. The hyperspace of all ∗ such multisensor states is denoted by X∞ . ∗ ∗ ∗j ∗j A finite subset X is physically unrealizable if there exist x, y ∈ X such ∗j ∗j that x ̸= y. That is, the sensor with tag j cannot have two different physical ∗ states. Thus any physically realizable X must have the form ∗

∗j1

∗je

(25.4)

X = { x , ..., x }

∗

where 1 ≤ j1 ̸= ... ̸= je ≤ s are distinct indices. Thus multisensor state sets X are inherently labeled in the sense of Section 15.2.2. 25.2.3

Joint Multisensor-Multitarget State Space

Taken together, the sensors and the targets form a single, joint stochastic system. In general, there will be not only a variable number of targets, but also a variable number of sensors. Thus a state of this joint system is a finite subset whose elements are target-states or sensor-states. The notation ∗

∗ ∗ ˘ = {x1 , ..., xn , x ∗} = X ∪ X X 1 , ..., xn

(25.5)

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∗

∗

with |X| = n and |X| = n indicates that the scene contains n = 0, 1, ... distinct ∗ targets and n = 0, 1, ... distinct sensors. A joint multisensor-multitarget state is a finite subset of ∗ ∗1 ∗s ˘ = X ⊎ X = X ⊎ X ⊎ ... ⊎ X. X (25.6) ˘ ∞ . An arbitrary element The hyperspace of all such finite subsets is denoted as X ˘ will be denoted as x ˘ so that a finite subset of X ˘ has the form of X ˘ ∈ X, ˘ = {˘ X x1 , ..., x ˘n˘ }

(25.7)

∗

with n ˘ = n + n. 25.2.4

Integrals and Set Integrals on State Spaces

˘ of sensor-states and target-states Functions with arguments in the joint space X have the form { ˘ h(x) if x ˘=x ˘ x) = h(˘ . (25.8) ∗j ˘ ∗j h( x) if x ˘=x ∫ ˘ is defined as The integral ·d˘ x on X ∫ ∫ ∫ ∗ ∗ ˘ x)d˘ ˘ ˘ x)d h(˘ x= h(x)dx + ∗ h( x (25.9) X

∫

X

∗

∗

where the integral ·dx on X is defined as ∫ ∫ ∫ ∗ ∗ ∗ ∗1 ∗ ∗s ∗ ∗1 ∗s h(x)dx = ∗1 h( x)d x + ... + ∗s h( x)d x. X

(25.10)

X

˘ is Thus the set integral on X ∫ ∑ 1 ∫ ˘ X ˘ = f˘(X)δ f˘({˘ x1 , ..., x ˘n˘ })d˘ x1 · · · d˘ xn˘ . n ˘!

(25.11)

n ˘ ≥0

Equation (25.11) can be written in a simpler and more intuitive form. Abbreviate ∗

∗

˘ = f˘(X ∪ X) abbr. f˘(X) = f (X, X).

(25.12) ˘ Then because of (3.53), the single set integral on the joint space X can be written as a double set integral on the target and sensor subspaces: ∫ ∫ ∗ ∗ ˘ ˘ ˘ f (X)δ X = f (X, X)δXδ X. (25.13)

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p.g.fl.’s on Target/Sensor State Spaces

˘ on the joint target/sensor The p.g.fl. of a multitarget probability distribution f˘(X) space is, by definition, ∫ ˘ = h ˘ X˘ · f˘(X)δ ˘ h] ˘ X. ˘ G[ (25.14) Define the two-variate p.g.fl. ∗

G[h, h] =

∫

∗ ∗

∗

∗

hX · hX · f (X, X)δXδ X.

(25.15)

∗

∗ ∗ ˘ ˘ x) ˘ x), then If we define the restrictions h(x) = h(x) and h(x) = h( of h(˘ ∗

˘ = G[h, h]. ˘ h] G[

(25.16)

That is: ˘ can be equivalently replaced by the two˘ h] • The single-variate p.g.fl. G[ ∗ variate p.g.fl. G[h, h]. To prove (25.16), note that: ∫ ∫ ∗ ∗ ∗ ˘ = ˘ X˘ · f˘(X)δ ˘ X⊎X · f (X, X)δXδ X ˘ h] ˘ X ˘ = h G[ h ∫ ∗ ∗ ∗ ˘X · h ˘ X · f (X, X)δXδ X = h ∫ ∗ ∗ ∗ ∗ = hX · hX · f (X, X)δXδ X

(25.17) (25.18) (25.19)

∗

=

25.3

(25.20)

G[h, h].

MULTITARGET CONTROL: CONTROL SPACES j

j j

Corresponding to the jth sensor there is a space U of controls. Each u ∈ U is associated with the tag j of its corresponding sensor. The hyperspace of finite j

j

subsets of U is denoted as U∞ . The space of all controls for all sensors is 1

s

U =U ⊎ ... ⊎ U and the hyperspace of all finite subsets of U is U∞ .

(25.21)

896

25.4

Advances in Statistical Multisource-Multitarget Information Fusion

MULTITARGET CONTROL: MEASUREMENT SPACES

The purpose of this section is to specify those concepts of the approach that involve measurements. The material follows that of [169], pp. 245-249, and is organized as follows: 1. Section 25.4.1: Spaces of target-generated measurements. 2. Section 25.4.2: Spaces of actuator-sensor measurements. 3. Section 25.4.3: Joint measurement and actuator-measurement spaces. 4. Section 25.4.4: Integrals and set integrals on measurement spaces. 5. Section 25.4.5: Probability generating functionals (p.g.fl.’s) on measurement spaces. 25.4.1

Sensor Measurements j

The measurement space for the jth sensor is Z, with individual measurements j j

denoted as z ∈ Z. Thus any measurement collected by a sensor has that sensor’s tag attached to it. The space of all measurements from all sensors is 1

s

Z = Z ⊎ ... ⊎ Z.

(25.22)

The measurement set collected by any sensors that are present is a finite subset of Z of the form 1 s Z = Z ∪ ... ∪ Z (25.23) j

where—and this is important to note—each Z can be empty. The set of all measurement sets from all sensors—the hyperspace of all finite subsets of Z—is denoted as Z∞ . A measurement with an unspecified sensor tag is denoted as z ∈ Z. A finite subset of such measurements has the form Z = {z1 , ..., zm } j1

(25.24)

je

where m = m + ... + m; where j1 , ..., je are the tags of the e distinct sensors that ji are present; and where m is the number of measurements collected by the sensor with tag ji .

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897

Actuator-Sensor Measurements

As in the single-sensor, single-target case of Chapter 24, assume that the states of the sensors are observed by internal actuator sensors. The measurement space of the ∗j

actuator sensor(s) for the jth sensor is Z, with individual actuator-measurements ∗j

∗j

denoted as z ∈ Z. The space of all actuator-sensor measurements is ∗

∗1

∗s

(25.25)

Z = Z ⊎ ... ⊎ Z.

∗

∗

An actuator-sensor measurement with an unspecified sensor tag is denoted z ∈ Z; and a finite subset of such measurements is denoted as ∗

∗

∗

(25.26)

Z = {z1 , ..., zm }. ∗

The hyperspace of all such measurement sets is denoted as Z∞ . 25.4.3

Joint Multisensor-Multitarget Measurements

In general, the total measurement provided by the jth sensor will have the form ∗j

j

j

∗j

( z , Z) where z is the actuator-sensor measurement and Z is the measurement set collected by the sensor itself. Thus the measurements provided by all sensors ∗1

1

∗s

s

have the rather complicated form {( z , Z), ..., ( z , Z)}. However, we can apply the same reasoning that was used in Sections 21.8.2 and 21.8.3, in regard to the “natural” and “simplified” state representations for multiple level-1 group targets. That is, any measurement collected from the joint multisensor-multitarget system will be represented as a finite subset of the form ∗

∗ ∗ ∗ } = Z ∪ Z. Z˘ = {z1 , ..., zm , z1 , ..., zm

(25.27)

This indicates that m ≥ 0 measurements have been collected from the targets by ∗ the sensors; and m ≥ 0 measurements from the sensors by the actuator sensors. ∗ For a given z ∈ Z˘ with sensor tag j, the set of target-generated measurements ∗ corresponding to z is the set of all z ∈ Z˘ that have the same tag. If there are no such z, then the jth sensor did not collect any measurements at all. Thus a joint multisensor-multitarget measurement is a finite subset of ∗

1

s

∗1

∗s

˘ = Z ⊎ Z = Z ⊎ ... ⊎ Z ⊎ Z ⊎ ... ⊎ Z. Z

(25.28)

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Any measurement, whether a sensor measurement or an actuator-sensor measure˘ so that ment, will be notated as ˘ z ∈ Z, Z˘ = {˘ z1 , ..., ˘ zm ˘}

(25.29)

∗

with m ˘ = m+ m. The hyperspace of all such joint observation-sets will be denoted ˘∞. Z 25.4.4

Integrals and Set Integrals on Measurement Spaces

˘ of sensor-states and target-states have the form Functions on the joint space Z

g˘(˘ z) =

The integral

∫

{

j

j

if ˘ z=z . ∗j if ˘ z= z

g˘(z) ∗j g˘( z )

(25.30)

˘ is defined as ·d˘ z on the joint measurement space Z ∫

g˘(˘ z)d˘ z=

∫

g˘(z)dz +

∫

∗

∗

(25.31)

g˘(z)dz ∗

Z

where the integrals ∫ ∫

∫

·dz and

g(z)dz ∗

∗

∗

g(z)dz

= =

∫

Z ∗

∗

·dz on Z and Z are defined by

∫

1

1

g(z)dz + ... +

∫

s

s

g(z)dz ∫ ∫ ∗1 ∗s ∗ ∗1 ∗ ∗s g( z )d z + ... + g( z )d z . ∗1 ∗s 1

s

Z

Z

Z

(25.32) (25.33)

Z

˘ is The set integral on Z ∫ ∑ 1 ∫ ˘ Z˘ = f˘(Z)δ f˘({˘ z1 , ..., ˘ zm z1 · · · d˘ zm ˘ })d˘ ˘. m! ˘

(25.34)

m≥0 ˘

Suppose that there are s sensors. Then as in (25.13) and because of (3.53), we can write ˘ f˘(Z)

∗

=

1

=

∗

f˘(Z ∪ Z) = f (Z, Z) ∗1

s

(25.35)

∗s

f (Z, Z, ..., Z, Z)

(25.36)

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and thus ∫

25.4.5

˘ Z˘ f˘(Z)δ

=

∫

f (Z, Z)δZδ Z

=

∫

f (Z, Z, ..., Z, Z)δ Zδ Z · · · δ Zδ Z.

∗

1

∗

∗1

s

(25.37) ∗s

∗1

1

s

∗s

(25.38)

p.g.fl.’s on Measurement Spaces

From (25.37) it follows, by analogy with (25.17), that the single-variate p.g.fl. of the multisensor-multitarget likelihood function ∗

˘ k+1 [˘ G g |X, X] =

∫

∗

˘

˘ g˘Z · f˘k+1 (Z|X, X)δ Z˘

(25.39)

is equivalent to an s-variate p.g.fl. as described in Section 4.2.5.2: ∗

∗

1 s ˘ k+1 [˘ G g |X, X] = Gk+1 [g, ..., g|X, X]

(25.40)

where j

g = g˘| j

(25.41)

∗j

Z⊎ Z ∗j

j

denotes the restriction of g˘ to Z ⊎ Z. To see why, first note that, because of (25.37), =

∫

˘ g˘Z · f˘k+1 (Z|X, X)δ Z˘

=

∫

g˘Z g˘ Z · · · g Z g˘ Z

∗

Gk+1 [˘ g |X, X]

∗

˘

∗1

1

1

(25.42)

∗s

s

∗1

s

(25.43) ∗s

∗

·fk+1 (Z, Z, ..., Z, Z|X, X) 1 ∗1

s

∗s

·δ Zδ Z · · · δ Z δ Z. Then note that j

j

∗j

∗

j

j

∗j

∗j

f k+1 (Z, Z|X, X) = f k+1 (Z, Z|X, x)

(25.44)

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j

∗j

because Z, Z can have no dependence on sensors other than the jth one. We then get, because of conditional independence, ∗

=

˘ k+1 [˘ G g |X, X] ∫ 1 ∗1 s ∗s 1 1 ∗1 ∗1 g˘Z g˘ Z · · · g Z g˘ Z · f k+1 (Z, Z|X, x)

=

· · · f k+1 (Z, Z|X, x)δ Zδ Z · · · δ Z δ Z ∫ 1 ∗1 1 1 ∗1 ∗1 g˘Z g˘ Z · f k+1 (Z, Z|X, x)

s

∗s

s

s

∗s

1 ∗1

∗s

s

s

∗s

(25.45)

∗s

s

(25.46)

1 ∗1

∗s

s

∗s

=

· · · g Z g˘ Z · f k+1 (Z, Z|X, x)δ Zδ Z · · · δ Z δ Z (∫ 1 ∗1 ) 1 1 ∗1 1 ∗1 ∗1 Z Z g˘ g˘ · f k+1 (Z, Z|X, x)δ Zδ Z (∫ s ∗s ) s s ∗s s ∗s ∗s Z Z ··· g g˘ · f k+1 (Z, Z|X, x)δ Z δ Z

=

Gk+1 [˘ g |X, x] · · · Gk+1 [˘ g |X, x]

1

where j

s

∗1

∗j

Gk+1 [˘ g |X, x] =

∫

j

(25.47)

∗s

∗j

j

j

(25.48)

∗j

j

∗j

∗j

g˘Z g˘ Z · f k+1 (Z, Z|X, x)δ Zδ Z.

(25.49)

Thus, as claimed, ∗

˘ k+1 [˘ G g |X, X]

1

=

Gk+1 [˘ g| 1

Z⊎ Z

1 1

=

1

25.5

∗1

s

∗

s

s

∗s

∗s

Z⊎ Z

s

∗s

s

|X, x] (25.50) (25.51)

∗

Gk+1 [g|X, X] · · · Gk+1 [g|X, X] 1

=

|X, x] · · · Gk+1 [˘ g| s

Gk+1 [g|X, x] · · · Gk+1 [g|X, x] 1

=

s

∗1

∗1

(25.52)

∗

Gk+1 [g, ..., g|X, X].

(25.53)

MULTITARGET CONTROL: MOTION MODELS

This section specifies the dynamical models for targets and sensors. Assume the following: 1. All targets and sensors dynamically evolve independently of each other.

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2. Targets can appear or disappear, and this fact is reflected by target appearance and target-disappearance models. 3. The motion of every sensor at a given time-step is directed by a control that was selected at the previous time-step. 4. While sensors can in actuality appear and disappear, this fact will not be reflected in the motion modeling. That is, in the modeling the number of sensors does not change from one time to the next. (This assumption will allow us to avoid, for sensor state sets, the labeled RFS formalism of Chapter 15.) The section is organized as follows: 1. Section 25.5.1: Single-target motion models and multitarget motion models. 2. Section 25.5.2: Single-sensor motion models and multisensor motion models, with controls. 3. Section 25.5.3: Motion models for the joint multisensor-multitarget system. 25.5.1

Single-Target and Multitarget Motion Models

As usual, the evolution of single targets is governed by a single-target Markov transition density fk+1|k (x|x′ ). Since target states can contain a discrete identity variable, it is possible to assign different motion models to targets of different types. The disappearance of single targets is governed by a probability of survival pS (x′ ) abbr. = pS,k+1|k (x′ ). Also as usual, the evolution of multiple targets is governed by a multitarget Markov transition density fk+1|k (X|X ′ ). 25.5.2

Single-Sensor Motion and Multisensor Motion with Sensor Controls

The evolution of single sensors is governed by Markov densities for each sensor j = 1, ..., s: ∗j

∗j ∗j

j

f k+1|k ( x| x ′ , u)

j

where u is the control for the jth sensor at time tk . As in Section 24.8, it will later be assumed that these Markov models have the nonlinear-additive form ∗j

∗j ∗j

∗j

j

∗j

∗j

j

f k+1|k ( x| x ′ , u) = f ∗j ( x − φ k ( x ′ , u)). Wk

(25.54)

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Advances in Statistical Multisource-Multitarget Information Fusion

∗j

∗j

∗j

j

Here Wk is a zero-mean noise vector, φ k ( x ′ , uk ) is the state transition function, j and the control uk selects among these functions. The Markov transition density for the entire multisensor system has the form ∗ ∗ ∗ f k+1|k (X|X ′ , U ), where U is the set of controls for all sensors. Since each control j

j

uk ∈ U contains the tag j for the sensor it controls, it is always clear that uk ∗ ∗j can be applied only to the sensor with state x ′ ∈ X ′ .1 25.5.3

Joint Multisensor-Multitarget Motion

In what follows, it will be assumed that: 1. Target dynamics and sensor dynamics are independent. Thus the Markov ∗ ˘ = X ∪ X factors as density for joint multisensor-multitarget states X follows: ∗

˘ X ˘ ′, U ) f˘k+1|k (X|

=

∗

fk+1|k (X ⊎ X|X ′ ⊎ X ′ , U ) ∗

= =

(25.55)

∗

fk+1|k (X, X|X ′ , X ′ , U ) ∗

′

∗

(25.56) ∗

′

fk+1|k (X|X ) · f k+1|k (X|X , U ). (25.57)

2. fk+1|k (X|X ′ ) is the multitarget Markov density for the standard multitarget motion model of Section 7.4. 3. Every sensor has a control, and is controlled by it and no other. That is, if ∗ j is the tag of a sensor in X ′ , then it is also the tag of a control in U . Furthermore, let U ′ be the subset of U of those controls whose tags are ∗ tags of sensors in X ′ . (That is, the controls in U − U ′ do not correspond to ∗ any sensor in X ′ .) Then ∗

∗

∗

∗

∗

∗

f k+1|k (X|X ′ , U ) = f k+1|k (X|X ′ , U ′ ).

(25.58)

4. Neither sensor appearance nor sensor disappearance are modeled; and ∗ sensors transition independently to their next states. That is, let X = ∗ ∗ ∗ ∗j1 ∗j1 ∗je ∗je { x , ..., x } with |X| = e; let X ′ = { x ′ , ..., x ′ } with |X ′ | = e, 1

The approach described in Remark 1 of [169], in which sensor states are to be paired with controls, is actually unnecessary.

Multitarget Sensor Management

j1

903

je

and let U = { u, ..., u} with |U | = e. Then ∗

∗

∗

f k+1|k (X|X ′ , U )

=

δ

∗

∗

|X|,|X ′ |

∑ ∗j1 ∗j1 ∗jπ1 jπ1 f k+1|k ( x | x ′ , u ) (25.59) π

∗je

··· f

k+1|k (

∗je ∗jπe ′ jπe

x| x , u)

where the summation is taken over all permutations π on 1, ..., e. Or, expressed in vector notation, ∗

∗j1

∗je ∗j1

∗je

j1

je

f k+1|k ( x , ..., x | x ′ , ..., x ′ , u, ..., u) ∗j1

=

f

k+1|k (

∗j1 ∗j1 ′ j1

∗je

x | x , u) · · · f

k+1|k (

(25.60)

∗je ∗je ′ je

x | x , u).

Given these assumptions, the following two degenerate control situations must be addressed: ∗

• U does not contain a control for one of the sensors in X ′ . In this case, it ∗ ∗ is impossible for the state set X ′ to evolve to the state set X since the jth sensor cannot evolve in the absence of an explicitly specified control. Thus if ∗ ∗ there are no birth targets, X ′ can evolve only to the empty state X = ∅: ∗

∗

∗

f k+1|k (X|X ′ , U ) = δ

∗

(25.61)

.

0,|X|

If there are birth targets, it can evolve only the target-birth distribution. ∗

• U contains a control for a sensor that is not in X ′ . This could be, for example, because the sensor has temporarily disappeared from the scene. In this case this control will have no influence on the evolution of its sensor during this cycle.

25.6

MULTITARGET CONTROL: MEASUREMENT MODELS

This section specifies the measurement models for targets and sensors. It is organized as follows: 1. Section 25.6.1: Basic assumptions about measurements. 2. Section 25.6.2: Sensor-noise models.

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3. Section 25.6.3: Models of sensor fields of view (FoVs) and clutter. 4. Section 25.6.4: Models for actuator sensors, and for communication-transmission failures. 5. Section 25.6.5: Multitarget likelihood functions. 6. Section 25.6.6: Joint multisensor-multitarget likelihood functions. 25.6.1

Measurements: Assumptions

The following assumptions are made about measurements collected by the sensors and actuator sensors: 1. Without loss of generality, each platform is assumed to carry exactly one sensor. (Mathematically speaking, multiple sensors on the same platform can be modeled as single sensors on multiple copies of the platform.) 2. For each sensor, each target generates at most one measurement; and no measurement is generated by more than one target. 3. For each sensor, each measurement collected from a target is contaminated by sensor noise. 4. For each sensor, measurements from different targets are conditionally independent of target state. 5. For each sensor, any multitarget measurement is contaminated by a clutter and/or false alarm process that is independent of the target-generated measurement process. Typically, this process is functionally dependent on the sensor state. 6. For each sensor, the state of that sensor is observed by an internal actuator sensor. 7. For each sensor, the measurement collected by its actuator sensor is contaminated by sensor noise. 8. For each sensor, its measurements and the actuator sensor’s measurement are transmitted to an information fusion site. This transmission may not be successful because of communications channel drop-outs (due to terrain or weather obscuration), communication latencies, and other effects.

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9. Measurements from different sensors are conditionally independent of target state. In the following subsections, these assumptions will be elaborated in greater detail. 25.6.2

Measurements: Sensor Noise

The noise characteristics of the sensors are modeled using sensor likelihood functions: j j j j ∗j j ∗j Lj (x) abbr. = Lj ∗j (x) abbr. = Lj (x, x) abbr. = f k (z|x, x) (25.62) z

z, x

z

for j = 1, ..., s. In what follows, it will be assumed that these likelihood functions have the nonlinear-additive form j

∗j

j

∗j

j

f k (z|x, x) = f j (z − ηk (x, x))

(25.63)

Vk j

where Vk is a zero-mean noise vector and ∗j

j

η k (x) abbr. = ηk (x, x)

(25.64)

is the deterministic measurement function. 25.6.3

Measurements: Fields of View (FoVs) and Clutter

The FoVs of the sensors at time tk are modeled as state-dependent probabilities of detection ∗j ∗j j pD (x) abbr. = pD (x, x) abbr. = pD,k (x, x). (25.65) That is, the probability that the jth sensor will collect an observation from a target ∗j ∗j with state x at time tk is pD (x, x) if the state the sensor at that time-step is x. Also, for each sensor, target-generated measurements will typically be corrupted by a clutter process specified by a multiobject probability distribution j j

j

∗j

κk+1 (Z) abbr. = κk+1 (Z| x).

(25.66)

As indicated by the notation, clutter measurements will typically depend on the sensor state. For example, clutter is often heavier at the edges of a sensor FoV (where probability of detection is smallest) than in its center(s). For the sake of notational simplicity, in what follows the dependence of the clutter process ∗ ∗ κk+1 (Z|x) on the sensor state x will usually be suppressed.

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25.6.4

Measurements: Actuator Sensors and Transmission Failure

The actuator sensors are modeled using actuator-sensor likelihood functions ∗j

∗j

∗j

∗j ∗j

L ∗j ( x) abbr. = f k ( z | x).

(25.67)

z

Actuator sensors are assumed to have unity probability of detection and no clutter. ∗ ∗j1 ∗je Thus suppose that the states of the sensors are given by X = { x , ..., x } with ∗ |X| = e; and that the corresponding actuator-sensor measurements are given by ∗ ∗j1 ∗je Z = { z , ..., z }. Then the joint likelihood function for the actuator sensors is given by ∗

∗

∗

f k+1 (Z|X) = δ

∗

∗

|Z|,|X|

∑ ∗j1 ∗je ∗j1 ∗jπ1 ∗jn ∗jπe f k+1 ( z | x ) · · · f k+1 ( z | x )

(25.68)

π

where the summation is taken over all permutations π on 1, ..., n. In vector notation, ∗

∗j1

∗jn ∗j1

∗j1

∗je

f k+1|k ( z , ..., z | x , ..., x ) = f

∗j1 ∗j1

k+1 (

∗je

z | x )··· f

∗je ∗je

k+1 (

z | x ).

(25.69)

It may not be the case that a sensor’s measurements and actuator-sensor measurement will be successfully transmitted to an information fusion site. This could, for example, be because of transmission drop-outs due to atmospheric interference, terrain blockage, latency, and so on Suppose that there are s sensors, and that each sensor has access to some catalog of current possible transmission paths, along with their associated latencies; ∗j and that the state x of the jth sensor contains parameters specifying which path choice to make. In other words, a “sensor” in the most abstract sense is a specific sensing mode of a specific physical sensor, located at a specific site or on a specific mobile platform, and communicating to a ground station via a specific transmission path. Given this, transmission disruptions can be modeled using actuator-sensor probabilities of detection ∗j ∗j ∗j ∗j p D ( x) abbr. = p D,k ( x) (25.70) ∗j

∗j

for j = 1, ..., s. That is, p D ( x) is a mathematical convenience, and not the probability that an actuator-sensor measurement will be detected. Rather, it is the ∗j probability that a measurement from the jth sensor with state x will actually be

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received at the fusion site. A consequence of this model is that the jth actuator measurement will now have the fictitious but mathematically convenient form { ∗j ∅ if transmission was unsuccessful Z= . (25.71) ∗j { z } if otherwise Similarly, communication latencies can be modeled using the sensor Markov ∗j

∗j ∗j

∗j

transition densities f k+1|k ( x| x ′ ). The reason is that the state-vector x ′ of the jth sensor at time-step k contains, by assumption, parameters that specify the ∗j

∗j ∗j

transmission path. So, f k+1|k ( x| x ′ ) is the probability that the jth sensor can ∗j

∗j

reach state x at time-step k + 1 if it had state x ′ at time-step k + 1. If the ∗j transmission path specified in x ′ has large latency, then it may not be possible for the sensor’s information to traverse that particular path in the time-interval between time-step k and time-step k + 1. 25.6.5

Measurements: Multitarget Likelihood Functions

The multitarget likelihood function for the jth sensor at time tk+1 will have the form j j ∗j f k+1 (Z|X, x) (25.72) for j = 1, ..., s. Given our independence assumptions, if X = {x1 , ..., xn } with |X| = n then in general these will have the form j

j

∑

∗j

f k+1 (Z|X, x)

= j

j

j j

(25.73)

κk+1 (W 0 ) j

j

W 0 ⊎W 1 ⊎...⊎W n =Z j

j

j

∗j

j

∗j

·f k+1 (W 1 |x1 , x) · · · f k+1 (W n |xn , x) where j

j

j

∗j

κk+1 (Z) abbr. = κk+1 (Z| x) is the distribution of the clutter process for the jth  ∗j   1 − pD (x, x)  j j ∗j j ∗j ∗j f k+1 (Z|x, x) = pD (x, x) · Lj (x, x)  z   0 2

(25.74)

sensor;2 and where j

if

Z=∅ j

if if

j Z = {z} . otherwise

Caution: Eq. (63) in [169] is a typo. Equation (25.72) replaces it and Eqs. (64,65).

(25.75)

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More generally, the joint likelihood function for the jth sensor and its actuator sensor is  ∗j ∗j ∗j  1 − p D ( x) if Z=∅   j j ∗j j ∗j j ∗j ∗j ∗j ∗j ∗j ∗j ∗j f k+1 (Z,Z|X, x) = p D ( x) · L ∗j ( x) · f k+1 (Z|X, x) if Z = { z } .  z  ∗j  0 if |Z| ≥ 2 (25.76) 25.6.6

Measurements: Joint Multitarget Likelihood Functions

Finally, suppose that the sensors with distinct tags j1 , ..., je collect measurements ∗ ∗j1 ∗je and let X = { x , ..., x } be their state set. Then because of conditional independence, the joint likelihood function for these sensors is j1 ∗j1

je ∗je

j1 ∗j1

j

∗

j

∗j1

je ∗je

∗je

fk+1 (Z, Z , ..., Z, Z |X, X) = f k+1 (Z, Z |X, x ) · · · f k+1 (Z, Z |X, x ). (25.77) Because of (25.37), we can write j1

∗j1

je

∗je

j1

∗

fk+1 (Z, Z , ..., Z, Z |X, X)

je

∗j1

∗je

∗

=

fk+1 (Z, ..., Z, Z , ..., Z |X, X)(25.78)

=

fk+1 (Z, Z|X, X)

∗

∗

(25.79)

where j1

je

Z = Z ⊎ ... ⊎ Z,

25.7

∗

∗j1

∗je

Z = Z ⊎ ... ⊎ Z .

(25.80)

MULTITARGET CONTROL: SUMMARY OF NOTATION

Because the notation in the previous sections is rather involved, for ease of reference it is compiled here in a single place. 25.7.1

Notation for Spaces of Interest

• x ∈ X: states and state space for targets. • X ⊆ X: state sets for targets. ∗j

∗j

• x ∈ X: states and state space for the jth sensor.

Multitarget Sensor Management

∗

∗

909

∗

∗1

∗s

• x ∈ X: states and state space for all sensors, with X = X ⊎ ... ⊎ X. ∗

∗

• X ⊆ X: state sets for all sensors. ∗

∗

∗

• X (k) : X 1 , ..., X k : time sequence of sensor state sets. ∗

˘ joint states and state space for targets and sensors, with X ˘ = X⊎X. • x ˘ ∈ X: ˘ joint state sets for targets and sensors. ˘ ⊆ X: • X j j

• z ∈ Z: measurements and measurement space for the jth sensor. j

j

• Z ⊆ Z: measurement sets for the jth sensor. j

j

j

• Z (k) : Z 1 , ..., Z k : time sequence of measurement sets for the jth sensor. • z ∈ Z: target-generated measurements and total measurement space for all 1 s sensors, with Z = Z ⊎ ...⊎Z. • Z ⊆ Z: measurement sets for all sensors. • Z (k) : Z1 , ..., Zk : time sequence of measurement sets for all sensors. ∗j

∗j

• z ∈ Z: actuator-sensor measurements and actuator measurement space for the jth sensor. ∗j

∗j

∗j

• Z ⊆ Z: actuator-sensor measurement set for the jth sensor, with Z = ∅ ∗j

∗j

or Z = { z }. ∗j

∗j

∗j

• Z (k) : Z 1 , ..., Z k : time sequence of actuator-sensor measurement sets for the jth sensor. ∗

∗

• z ∈ Z: actuator-sensor measurements and actuator measurement space for ∗

∗1

∗s

all sensors, with Z = Z ⊎ ... ⊎ Z. • ˘ z ∈ with

˘ sensor or actuator-sensor measurements and measurement space, Z: ∗ ˘ = Z⊎Z. Z

˘ joint measurement sets for all sensors. • Z˘ ⊆ Z: j j

• u ∈ U: controls and control space for the jth sensor.

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j

j

j

• U (k) : u0 , ..., uk : time sequence of controls (control policy) for the jth sensor. 1

s

• u ∈ U: controls and control space for all sensors, with U = U ⊎ ...⊎U. • U ⊆ U: control sets for all sensors. • U (k) : U0 , ...Uk : time sequence of control sets for all sensors (multisensor control policy). 25.7.2

Notation for Motion Models

• fk+1|k (x|x′ ): single-target Markov transition density. • fk+1|k (X|X ′ ): multitarget Markov transition density. ∗

∗

∗j

∗j ∗j

∗

• f k+1|k (x|x′ , u): single-sensor Markov transition density. ∗j

j

∗j

∗j

j

• f k+1|k ( y| x ′ , uk ) = f ∗j ( y − φ k ( x ′ , uk )): additive version, with state Wk ∗j ′ j

∗j

transition function φ k ( x , uk ). ∗

∗

∗

∗

∗

˘ X ′ , U ) = fk+1|k (X|X ′ ) · f k+1|k (X|X ′ , U ): joint multi• fk+1|k (X, X|X, sensor, multitarget Markov transition density. 25.7.3

Notation for Measurement Models j

j

∗j

j

j

∗j

• f k+1 (z|x, x) = Lj (x, x) = Lj (x): likelihood function for the jth sensor. z

j

z

∗j

j

∗j

j

(z − ηk+1 (x, x)): additive version, with measure-

• f k+1 (z|x, x) = f ∗j Wk+1

∗j

ment function ηk+1 (x, x). ∗j

∗j

j

• pD,k+1 (x, x) = pD (x, x) = pD (x): probability of detection (field of view, FoV) for the jth sensor. ∗j

∗j

∗j ∗j

∗j

• f k+1 ( z | x) = L ∗j ( x): actuator-sensor likelihood function for the jth z sensor. ∗j

∗j

∗j

∗j

• p D,k+1 ( x) = p D ( x): probability that measurements from the jth sensor and actuator sensor will be successfully transmitted.

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j j

• κk+1 (Z): probability distribution of the clutter process for the jth sensor. j

j

∗j

• f k+1 (Z|x, x): single-target likelihood function for the jth sensor, with

j

j

∗j

f k+1 (Z|x, x) =

j

j

   

j

∗j

if

1 − pD (x, x) j

∗j

Z=∅ j

∗j

j Z = {z} . otherwise

if if

pD (x, x) · Lj (x, x)  z   0

(25.81)

∗j

• f k+1 (Z|X, x): multitarget likelihood function for the jth sensor, with j

j

∑

∗j

f k+1 (Z|X, x)

= j

j

j j

∗j

(25.82)

κk+1 (W 0 | x) j

j

W 0 ⊎W 1 ⊎...⊎W n =Z j

j

j

∗j

j

∗j

·f k+1 (W 1 |x1 , x) · · · f k+1 (W n |xn , x). j

j ∗j

∗j

• f k+1 (Z,Z|X, x): joint sensor/actuator-sensor multitarget likelihood function for the jth sensor, with

j

j ∗j

∗j

f k+1 (Z,Z|X, x) =

∗

   

∗j

∗j

∗j

if

1 − p D ( x) ∗j

∗j

∗j

∗j

j

j

∗j

p D ( x) · L ∗J ( x) · f k+1 (Z|X, x)  z   0

Z=∅ ∗j

if

∗j Z = {z} . ∗j

if

|Z| ≥ 2 (25.83)

∗

• fk+1 (Z, Z|X, X): joint sensor/actuator-sensor multitarget likelihood function for all sensors, with ∗

∗

j

j1 ∗j1

∗j1

j

je ∗je

∗je

fk+1 (Z, Z|X, X) = f k+1 (Z, Z |X, x ) · · · f k+1 (Z, Z |X, x ).

25.8

(25.84)

MULTITARGET CONTROL: SINGLE STEP

The material in this section was summarized in Section 23.3.1. Let: • Z (k) : Z1 , ..., Zk be the time sequence of target-generated measurement sets for all sensors at time tk .

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∗

∗

∗

• Z (k) : Z 1 , ..., Z k be the time sequence of actuator measurements for all ∗ sensors at time tk , with |Z j | ≤ 1 for each j. • U (k) : U0 , ..., Uk be the time sequence of controls (multisensor control policy) for all sensors at time tk , with Ui being chosen at time ti . Given this, the multisensor-multitarget Bayes filter with controls is a direct generalization of the single-sensor, single-target Bayes filter with controls of (24.22) through (24.25): ∗

∗

∗

∗

fk|k (X, X|Z (k) , Z (k) , U (k−1) )

... → →

fk+1|k (X, X|Z (k) , Z (k) , U (k) )

→

fk+1|k+1 (X, X|Z (k+1) , Z (k+1) , U (k) )

∗

∗

→ ...

where the filtering equations are as follows: • Time update: ∗

=

∗

fk+1|k (X, X|Z (k) , Z (k) , U (k) ) ∫ ∗ ∗ ∗ fk+1|k (X|X ′ ) · f k+1|k (X|X ′ , Uk ) ∗

(25.85)

∗

∗

·fk|k (X ′ , X ′ |Z (k) , Z (k) , U (k−1) )δX ′ δ X ′ • Measurement update: ∗

∗

fk+1|k+1 (X, X|Z (k+1) , Z (k+1) , U (k) ) ( ) ∗ ∗ ∗ fk+1 (Zk+1 , |X, X) · fk+1 (Z k+1 |X) ∗ ∗ ·fk+1|k (X, X|Z (k) , Z (k) , U (k) )

=

∗

(25.86)

∗

fk+1 (Zk+1 , Z k+1 |Z (k) , Z (k) , U (k) ) where ∗

=

∗

fk+1 (Zk+1 , Z k+1 |Z (k) , Z (k) , U (k) ) ∫ ∗ ∗ ∗ fk+1 (Zk+1 |X, X) · fk+1 (Z k+1 |X) ∗

∗

∗

·fk+1|k (X, X|Z (k) , Z (k) , U (k) )δXδ X.

(25.87)

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Now consider single-step look-ahead control. The previous multisensor control policy U (k−1) has been determined, and we are to determine the next multisensor control set Uk . It should be chosen so that the placements of the sensor FoVs are as efficacious as possible, given the resolutions of the sensors. In this case, the control solution is a closed-loop version of the open-loop filter:

multisensor & multitarget measurement-update ∗ ∗

fk|k (X, X|Z (k) , Z (k) , U (k−1) )

... −→

time-projection ∗ ∗

−→

fk+1|k+1 (X, X|Z (k) , Z (k) , U (k−1) , ∗ Z, Z, U ) hedging

↓ select next control set

Uk = arg supU Ok+1 (U ) multisensor & multitarget time & measurement-update ∗ ∗

fk+1|k+1 (X, X|Z (k+1) , Z (k+1) , U (k) )

−→ ∗

Uk Zk+1 ,Z k+1

−→ ...

↗

This has a more detailed structure, to be described in the subsections that follow.

25.9

MULTITARGET CONTROL: OBJECTIVE FUNCTIONS

This section discusses possible sensor management objective functions. organized as follows:

It is

1. Section 25.9.1: Information-theoretic objective functions. 2. Section 25.9.2: The posterior expected number of targets (PENT) objective function. 3. Section 25.9.3: The cardinality-variance (variance of PENT) objective function. 4. Section 25.9.4: PENT as an approximate information-theoretic objective function.

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Information-Theoretic Objective Functions

By analogy with Section 24.5, one could consider the multitarget versions of Kullback-Leibler cross-entropy and, more generally, of Csisz´ar discrimination: ∗

=

Ok+1 (U, Z, Z) ∫ ∗ ∗ fk+1|k+1 (X|Z k , Z (k) , U (k−1) , Z, Z, U ) ( ) ∗ ∗ fk+1|k+1 (X|Z (k) , Z (k) , U (k−1) , Z, Z, U ) · log δX ∗ fk+1|k (X|Z (k) , Z (k) , U (k−1) , U )

(25.88)

and ∗

=

Ok+1 (U, Z, Z) ( ) ∗ ∗ ∫ fk+1|k+1 (X|Z (k) , Z (k) , U (k−1) , Z, Z, U ) ck+1 ∗ fk+1|k (X|Z (k) , Z (k) , U (k−1) , U )

(25.89)

∗

·fk+1|k (X|Z (k) , Z (k) , U (k−1) , U )δX where the marginal distributions are given by ∗

=

∗

fk+1|k+1 (X|Z (k) , Z (k) , U (k−1) , Z, Z, U ) ∫ ∗ ∗ ∗ ∗ fk+1|k+1 (X, X|Z (k) , Z (k) , U (k−1) , Z, Z, U )δ X

(25.90)

and ∗

=

fk+1|k (X|Z (k) , Z (k) , U (k−1) , U ) ∫ ∗ ∗ ∗ fk+1|k (X, X|Z (k) , Z (k) , U (k−1) , U )δ X.

(25.91)

Both objective functions will almost always be computationally intractable. Drastic but principled approximations are therefore required. One such approximation has been suggested by Ristic and Vo [259]. Suppose that the underlying multitarget filter is a sequential Monte Carlo (SMC) implementation of the multitarget Bayes filter. Also suppose that the objective function is R´enyi α-divergence (see Section 6.3). In this single-sensor case and assuming the

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ideal-sensor approach of Section 25.12, Ristic and Vo proposed the following SMC approximation of this objective function. In an SMC implementation, the predicted multitarget distribution is approximated as a multitarget Dirac mixture νk+1|k

∑

fk+1|k (X|Z (k) ) ∼ =

k+1|k

wi

· δX k+1|k (X),

(25.92)

i

i=1

where δX ′ (X) is the multitarget Dirac distribution concentrated at the multitarget state X ′ (see (4.15)). Given this, R´enyi divergence can be approximated as ([259], Eq. (15)): 1 Rα (Z|u) = log 1−α

=

∫

fk+1|k+1 (X|Z (k) , U k , Z, u)

·fk+1|k (X|Z (k) , U k )1−α δX ∫ fk+1 (Z|X, u)α · fk+1|k (X|Z (k) , U k−1 )α 1 log 1−α fk+1 (Z|Z (k) , U k )α

(25.93)

(25.94)

·fk+1|k (X|Z (k) , U k )1−α δX

=

∼ =

=

∫ fk+1 (Z|X, u)α · fk+1|k (X|Z (k) , U k )δX 1 )α log (∫ 1−α fk+1 (Z|Y, u) · fk+1|k (Y |Z (k) , U k )δY ∑νk+1|k k+1|k k+1|k 1 wi · fk+1 (Z|Xi , u)α )α log (∑i=1 ν k+1|k k+1|k k+1|k 1−α w · f (Z|X , u) k+1 i=1 i i 1 γα (Z|u) log 1−α γ1 (Z|u)α

(25.95)

(25.96)

(25.97)

where νk+1|k

γα (Z|u) =

∑

k+1|k

wi

k+1|k

· fk+1 (Z|Xi

, u)α .

(25.98)

i=1

One other potentially useful objective function should be mentioned: the Cauchy-Schwartz divergence. When the predicted and posterior multitarget distributions are Poisson, (6.86) tells us that it reduces to the square-L2 norm [108]:

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∗

=

Ok+1 (U, Z, Z) (25.99) )2 ∗ ∗ ∫ ( c Dk+1|k+1 (x|Z (k) , Z (k) , U (k−1) , Z, Z, U ) dx.(25.100) ∗ 2 −Dk+1|k (x|Z (k) , Z (k) , U (k−1) , U )

This norm can be computed in exact closed form when the PHDs are approximated as Gaussian mixtures. Initial research indicates that it produces good sensor management behavior—see [108] and Section 26.6.4.3. 25.9.2

The PENT Objective Function

Suppose that the multitarget filter is to be approximated using a PHD filter. Then the objective function must be definable entirely in terms of first-order information. Specifically, it must be entirely definable in terms of the PHD Dk|k (x|Z (k) ). Such an objective function was described at a more intuitive level in Section 23.2. It is the posterior expected number of targets (PENT): ∫ ∗ ∗ ∗ Nk+1|k+1 (U, Z, Z) = |X| · fk+1|k+1 (X|Z k , Z (k) , U (k−1) , Z, Z, U )δX. (25.101) That is, it is the expected number of targets, given the unknowable future measure∗ ment sets Z, Z and the to-be-determined control set U . PENT can be expressed in two different ways: • As an expected value of the posterior cardinality distribution: ∑ ∗ ∗ ∗ Nk+1|k+1 (U, Z, Z) = n · pk+1|k+1 (n|Z k , Z (k) , U (k−1) , Z, Z, U ). n≥0

(25.102) • As the integral of the posterior PHD: ∫ ∗ ∗ ∗ Nk+1|k+1 (U, Z, Z) = Dk+1|k+1 (x|Z k , Z (k) , U (k−1) , Z, Z, U )dx. (25.103) 25.9.3

The Cardinality-Variance Objective Function

The PENT objective function is computationally tractable, but—being statistically first-order in a point process sense—is oriented towards use with PHD filters.

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Suppose that, in addition to the PHD, the cardinality distribution is available (as is the case with CPHD and CBMeMBer filters). Then, in principle, it should be possible to devise objective functions that are still computationally tractable and physically intuitive while being potentially more effective than PENT. In [108], Hung Gia Huong and B.-T. Vo proposed such an objective function: the cardinality variance—that is, the variance of the posterior expected number of targets (variance of PENT). Let the posterior cardinality distribution be ∫ ∗ ∗ ∗ pk+1|k+1 (n|U, Z, Z) = fk+1|k+1 (X|Z k , Z (k) , U (k−1) , Z, Z, U )δX. |X|=n

(25.104) Then the variance of this distribution is ∗

2 σk+1|k+1 (U, Z, Z)

∗

=

−Nk+1|k+1 (U, Z, Z)2 ∑ ∗ + n2 · pk+1|k+1 (n|Z, Z, U ).

(25.105)

n≥0

Huong and Vo proposed that sensor management be based on the minimization of this objective function. They demonstrated that, when used with a high-resolution sensor, it performs well in comparison to the R´enyi information functional (see [108] and Section 26.6.4.2). 25.9.4

PENT as an Approximate Information-Theoretic Objective Function

PENT can be regarded as an approximation of the multitarget Kullback-Leibler cross-entropy and other Csisz´ar discrimination functionals. Suppose, for example, that the predicted and posterior multitarget distributions are Poisson (Section 4.3.1): ∗

fk+1|k (X|Z (k) , Z (k) , U (k−1) , U ) ∗

X =e−Nk+1|k · Dk+1|k (25.106)

∗

fk+1|k+1 (X|Z (k) , Z (k) , U (k−1) , Z, Z, U )

=

e−Nk+1|k+1

(25.107)

X ·Dk+1|k+1

where for notational clarity the dependence of Nk+1|k+1 and Dk+1|k+1 (x) on ∗

Z, Z, U is neglected. Also, abbreviate c(x) = ck+1 (x) = 1 − x + x log x,

(25.108)

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which is strictly increasing for x ≥ 1. Then from (6.79) we know that the KullbackLeibler cross-entropy is given by ∗

Ok+1 (U, Z, Z) Nk+1|k

=

) Nk+1|k+1 (25.109) Nk+1|k ( ) ∫ Nk+1|k+1 sk+1|k+1 (x) + sk+1|k (x) · c dx. Nk+1|k sk+1|k (x) c

(

Thus when Nk+1|k+1 ≥ Nk+1|k , we will maximize the cross-entropy if we independently maximize both Nk+1|k+1 and the cross-entropy ( ) ∫ sk+1|k+1 (x) Ic (sk+1|k+1 ; sk+1|k ) = sk+1|k (x) · c dx (25.110) sk+1|k (x) of the normalized PHDs. Suppose that we wish to avoid the computationally challenging task of maximizing (25.110). Then: • PENT as an approximation of Kullback-Leibler cross-entropy: Neglecting the term of (25.109) that involves the right side of (25.110), we write ( ) ∗ Nk+1|k+1 ∼ Ok+1 (U, Z, Z) = Nk+1|k · c . (25.111) Nk+1|k Thus when Nk+1|k+1 ≥ Nk+1|k , maximizing Nk+1|k+1 amounts to the ∗

same thing as approximately maximizing Ok+1 (U, Z, Z). ∗

Similar reasoning can be applied if fk+1|k (X|Z (k) , Z (k) , U (k−1) , U ) and ∗

∗

fk+1|k+1 (X|Z (k) , Z (k) , U (k−1) , Z, Z, U ) are i.i.d.c. (Section 4.3.2). In this case, by (6.88) the cross-entropy is Ic (fk+1|k+1 ; fk+1|k )

=

Ic (pk+1|k+1 ; pk+1|k ) (25.112) +Nk+1|k+1 · Ic (sk+1|k+1 ; sk+1|k ).

Thus the cross-entropy is maximized if Nk+1|k+1 , Ic (pk+1|k+1 ; pk+1|k ), and Ic (sk+1|k+1 ; sk+1|k ) are independently maximized. If we neglect the latter two items for computational reasons, the cross-entropy is maximized if Nk+1|k+1 is maximized. PENT is an approximation of at least two other Csisz´ar discrimination functionals:

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• PENT as an approximation of chi-square discrimination: From (6.81) we know that for Poisson processes, the chi-square discrimination is (

( ) ) Nk+1|k c Nk+1|k+1 Nk+1|k · Ic (fk+1|k+1 ; fk+1|k ) = · +Ic (sk+1|k+1 ; sk+1|k ) (25.113) where c(x) = (x − 1)2 . If we neglect the term involving the normalized PHDs we get 2 Nk+1|k+1

Ic (fk+1|k+1 ; fk+1|k )

∼ = =

2 Nk+1|k+1

(

Nk+1|k Nk+1|k Nk+1|k+1 ( )2 Nk+1|k+1 − Nk+1|k . ·c

)

(25.114) (25.115)

So, by maximizing PENT we approximately maximize the chi-square discrimination. • PENT as an approximation of R´enyi α-divergence: From (6.85), the R´enyi α-divergence for Poisson processes is Rα (fk+1|k+1 ; fk+1|k )

=

α · Nk+1|k · c

(

Nk+1|k+1 Nk+1|k

)

(25.116)

1−α α +α · Nk+1|k+1 Nk+1|k · Ic (sk+1|k+1 ; sk+1|k ) −1

where c(x) = α−1 (1 − α) · (αx + 1 − α − xα ). Again neglecting the term involving the normalized PHDs, we get Rα (fk+1|k+1 ; fk+1|k ) ∼ = α · N0 · c

(

Nk+1|k+1 Nk+1|k

)

(25.117)

By maximizing PENT we approximately maximize the α-divergence.

25.10

MULTISENSOR-MULTITARGET CONTROL: HEDGING

As in the single-sensor, single-target case (Section 24.6), we must address the fact ∗ that the objective function Ok+1 (U, Z, Z) depends on the future measurement sets

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∗

Z, Z. The obvious multitarget analogs of expected-value hedging and minimumvalue hedging are, respectively: ∫ ∗ ∗ ∗ ∗ Ok+1 (U ) = Ok+1 (U, Z, Z) · fk+1 (Z, Z|Z (k) , Z (k) , U (k−1) , U )δZδ Z (25.118) and ∗

Ok+1 (U ) = inf∗ Ok+1 (U, Z, Z).

(25.119)

Z,Z

Both of these possibilities will almost always be computationally intractable. Thus we must find a more computationally attractive alternative. One approach is the multiobject version of multiple-sample approximation, described earlier in (23.4). A computationally still simpler approach is singlesample approximation, in which a single “most representative” measurement set is chosen.3 One single-sample approach was introduced in [169], pp. 269-270, and generalizes the single-sensor, single-target PM approach of (24.6.4). Called “max-PIMS hedging” or just “PIMS hedging,” it is based on the concept of the predicted ideal measurement set (PIMS). This is described in the remainder of this section, which is organized as follows: 1. Section 25.10.1: The predicted measurement set (PMS) and the difficulties associated with using it for hedging. 2. Section 25.10.2: A general approach for hedging, using the predicted ideal measurement set (PIMS). 3. Section 25.10.3: Important special cases of max-PIMS hedging: for CPHD filters, PHD filters, Bernoulli filters, and CBMeMBer filters. 4. Section 25.10.4: Derivation of the PIMS single-sample hedging approach. 25.10.1

Hedging Using Predicted Measurement Set (PMS)?

An obvious approach to hedging would be to generalize the PM approach of Section 24.6.4 to the multisensor-multitarget case.4 Assume that at time tk+1 : 3

4

An earlier and ill-considered approach, first proposed in [162], was to use the empty set Z = ∅ as the single sample (on the mistaken belief that it was in some sense noninformative). This approach resulted in poor sensor management performance. This approach has been employed by several authors as a simplified version of the PIMS approach. It seems to have performed reasonably well under the particular conditions assumed.

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• We have n predicted target tracks at time tk+1 with estimated states Xk+1|k = {x1 , ..., xn } with |Xk+1|k | = n. ∗j

• x for j = 1, ..., e denotes the states of the sensors that are present. • The single-sensor likelihood functions have the additive form j

∗j

j j

∗j

j

z

∗j

(z − ηk+1 (x, x)).

Lj (x, x) = f k+1 (z|x, x) = f j

(25.120)

Vk+1

Then for the jth sensor, the “ideal” noise- and clutter-free single-sensor measurement set at time tk+1 would be the set of predicted target-measurements, given the predicted states of the sensors: j

∗j

j

∗j

def. MS ZP k+1 = η k+1 (Xk+1|k ) = {ηk+1 (x1 , x), ..., ηk+1 (xn , x)}.

(25.121)

If clutter is not too dense and sensor resolution is good, this would be the measurement set most likely to be collected (given that the x1 , ..., xn are actually detectable). The ideal noise- and clutter-free measurement set for a subset of sensors with tags j1 , ..., je would be j1

je

MS P MS P MS ZP k+1 = Z k+1 ⊎ ... ⊎ Z k+1 .

(25.122)

For the moment, ignore the actuator-sensor measurements; and assume that we have an objective function Ok+1 (U, Z). By analogy with (24.6.4), we could hedge as follows: P MS Ok+1 (U ) = Ok+1 (U, Zk+1 ). (25.123) However, this approach is problematic since, as will be noted more fully in Remark 96 in Section 26.3.1.4, it does not always produce intuitively correct sensor management behavior. The reason why will become clear because of the following reasoning. Suppose that at time tk+1 , the sensor fields of view are “cookie clutters”: ∗j

pD,k+1 (x, x) = 1 j (x)

(25.124)

S j

j

for some subsets S ⊆ X. If xi ∈ / S then, as was noted in Section 24.6.4 , it is impossible to collect any measurement from xi . In particular, it is impossible ∗j to collect the predicted measurement ηk+1 (xi , x). This presents an inherent difficulty:

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• It would clearly be inadvisable to force the sensors to try to collect measurements that are impossible to collect. • It would be better to induce the sensors to try to collect measurements that can be most probably collected. This difficulty can be addressed as follows. The set of collectable predicted measurements for the jth sensor is ∪

j

IM S ZP = k+1

∗j

j

{η k+1 (xi , x)}.

(25.125)

j

i:xi ∈S

This is the predicted ideal measurement set (PIMS), given the probability of detection 1 j (x). If there are e sensors with tags j1 , ..., je , then the PIMS S

for all sensors is: j1

je

P IM S IM S IM S Zk+1 = ZP ⊎ ... ⊎ Z P k+1 k+1 .

25.10.2

(25.126)

Predicted Ideal Measurement Set (PIMS): A General Approach

In general, sensor FoVs are not cookie cutters. So how can the concept of a PIMS be generalized to arbitrary FoVs? The purpose of this section is to describe a general approach. It is more general than the approach originally proposed in [169], pp. 268-271 (which is applicable only if the predicted multitarget distribution is Poisson). However, it also has the following limitation: • It implicitly depends on the assumption that there is at least some clutter (see Remark 92 at the end of the section). For the sake of conceptual clarity, consider the single-sensor case. Let us be given the set ˆ abbr. X = Xk+1|k = {x1 , ..., xn } (25.127) of estimated predicted tracks, and let ˆ abbr. ˆ Zˆ = η(X) = ηk+1 (X)

(25.128)

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923

be the predicted measurement set (PMS) as defined in (25.121). Also, for any subset I ⊆ {1, ..., n} define ˆI X

∪

=

(25.129)

{xi }

i∈I

ZˆI

ˆI ) = η(X

=

∪

{ηk+1 (xi )}.

(25.130)

i∈I

ˆ define the joint probability of detection of V Next, for any subset V ⊆ X, to be ∏ pD (V ) = pVD = pD (x). (25.131) x∈V

ˆ will be detected. That is, pD (V ) is the probability that all xi ∈ V ⊆ X Given this, the PIMS hedging approach is defined in terms of a “hedged” posterior p.g.fl., defined as follows. From Section 5.10.3 we know that, for the standard multitarget measurement model, the actual posterior p.g.fl. is

Gk+1|k+1 [h] =

δF δZk+1 [0, h] δF δZk+1 [0, 1]

(25.132)

where δF [g, h] δZk+1

=

∑ W ⊆Zk+1

Lg (x)

=

δGκk+1 [g] δ(Z − W )

δ · Gk+1|k [h(1 + pD Lg−1 )] ∫δW g(z) · fk+1 (z|x)dz

(25.133)

(25.134)

and where Gκk+1 [g] is the p.g.fl. of the clutter process. Define the single-sample hedged posterior p.g.fl. to be (see Section 25.10.4)

IM S GP k+1|k+1 [h]

=

δF P IM S δZk+1 [0, h] δF P IM S δZk+1 [0, 1]

(25.135)

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where δF P IM S [g, h] δ Zˆ

∑

=

I⊆{1,...,n}

·

δGκk+1 ˆI ) [g] · pD (X δ(Zˆ − ZˆI )

(25.136)

δ Gk+1|k [h(1 − pD + pD Lg )]. δ ZˆI

The reasoning that led to this equation will be explained in Section 25.10.4. A slight adjustment is required if (25.136) is to be mathematically well ˆ I ) ̸= 0. defined. Note that there is a largest Imax ⊆ {1, ..., n} such that pD (X max ∗ ∗ ˆ − Imax . That is, pD (x, x) ̸= 0 for all x ∈ Imax but pD (x, x) = 0 for all x ∈ X Thus (25.136) reduces to the final form δF P IM S [g, h] δ Zˆ

∑

=

I⊆Imax

·

δGκk+1 ˆI ) [g] · pD (X δ(Zˆ − ZˆI )

(25.137)

δ Gk+1|k [h(1 − pD + pD Lg )]. δ ZˆI

Remark 91 In the situations that we will consider, this equation will have the effect of substituting pD (xi ) · Lηk+1 (xi ) for Lηk+1 (xi ) , wherever Lηk+1 (xi ) occurs in formulas. From the hedged p.g.fl. one can derive various items of interest, such as: • The hedged posterior PHD: P IM S Dk+1|k+1 (x) =

δF P IM S [1]. δx

(25.138)

• The hedged posterior expected number of targets (hedged PENT): P IM S Nk+1|k+1 =

[

d P IM S G [x] dx k+1|k+1

]

.

(25.139)

x=1

• The hedged posterior p.g.f.: IM S P IM S GP k+1|k+1 (x) = Gk+1|k+1 [x].

(25.140)

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• The hedged cardinality distribution: IM S pP k+1|k+1 (n) =

[

1 dn P IM S G (x) n! dxn k+1|k+1

]

.

(25.141)

x=0

• The hedged variance: 2,P IM S P IM S σk+1|k+1 = −(Nk+1|k+1 )2 +

∑

IM S n 2 · pP k+1|k+1 (n).

(25.142)

n≥0 P IM S For sensor management, Nk+1|k+1 will functionally depend on a control set U . We will therefore write P IM S Nk+1|k+1 (U ) = Nk+1|k+1 (U, Zk+1 )

(25.143)

as an abbreviation to indicate that PIMS single-sample hedging is being used. Remark 92 (PIMS-hedging requires some clutter) The hedging approach described in this section depends on the following implicit assumption: there is at least some clutter. Suppose to the contrary that there is no clutter: κk+1 (Z) = δ0,|Z| . Then in (25.137) δGκk+1 [g] = 0 (25.144) δ(Zˆ − ZˆI ) ˆ unless ZˆI = Z—that is, unless I = {1, ..., n}. Thus the only term in the sum in (25.137) that will survive is the term corresponding to I = {1, ..., n}. But according to this equation, this is not possible unless {1, ..., n} ⊆ Imax —that is, ˆ ̸= ∅. Consequently, (25.136) reduces unless Imax = {1, ..., n} and thus pD (X) to δF P IM S ˆ · δ Gk+1|k [h(1 − pD + pD Lg )] [g, h] = pD (X) (25.145) ˆ δZ δ Zˆ

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and thus, from (25.132), IM S GP k+1|k+1 [h]

=

δF P IM S [0, h] ˆ δZ δF P IM S [0, 1] ˆ δZ

] − pD + pD Lg )] [ ] g=0 (25.147) = δ ˆ pD (X) · δZˆ Gk+1|k [1 − pD + pD Lg ] g=0 [ ] δ ˆ Gk+1|k [h(1 − pD + pD Lg )] δZ [ ] g=0 . = (25.148) δ G [1 − p + p L ] D D g k+1|k ˆ δZ ˆ · pD (X)

[

(25.146)

δ ˆ Gk+1|k [h(1 δZ

g=0

Thus when there is no clutter, the PIMS-hedged posterior p.g.fl. will not have any ˆ dependence upon pD (X)—in which case there is no PIMS hedging. Thus for this approach to be effective, one must assume that there is at least some clutter. Under certain circumstances, this limitation can be sidestepped—see Remark 93 in Section 26.3.1.3. 25.10.3

Predicted Ideal Measurement Set (PIMS): Special Cases

The purpose of this section is to describe four important special cases of (25.135) and (25.136). Later in the chapter, these special cases will be required to use PIMS single-sample hedging for CPHD, PHD, Bernoulli, and CBMeMBer filters. In Section K.37, the following are verified: • PIMS single-sample hedging for CPHD filters: Assume that Gκk+1 [g] and Gk+1|k [h] are i.i.d.c. in the sense of Section 4.3.2: Gκk+1 [g]

=

Gκk+1 (ck+1 [g])

(25.149)

Gk+1|k [h]

=

Gk+1|k (sk+1|k [h]).

(25.150)

Then the PIMS-hedged posterior p.g.fl. is ( ∑n ) κ l=0 (n − l)! · pk+1 (n − l) (l) P IM S ·Gk+1|k (ϕk [h]) · σn,l [h] P IM S ) Gk+1|k+1 [h] = ( ∑n κ i=0 (n − i)! · pk+1 (n − i) (i) P IM S ·Gk+1|k (ϕk [1]) · σn,i [1]

(25.151)

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where ϕk [h] P IM S σn,l [h]

= =

sk+1|k [h(1 − pD )]  σn,l 

pD (x1 )·sk+1|k [hpD Lηk+1 (x1 ) ] ck+1 (ηk+1 (x1 )) pD (xn )·sk+1|k [hpD Lηk+1 (xn ) ] , ..., ck+1 (ηk+1 (xn ))

(25.152)  (25.153)

and where σn,l (x1 , ..., xn ) is the elementary homogeneous symmetric function of degree l in n variables. • PIMS single-sample hedging for PHD filters: Assume that Gκk+1 [g] and Gk+1|k [h] are Poisson in the sense of Section 4.3.1: Gκk+1 [g] Gk+1|k [h]

= =

eκk+1 [g−1] e

Dk+1|k [h−1]

(25.154) .

(25.155)

Then the PIMS-hedged posterior p.g.fl. is IM S τ0 [h−1] GP k+1|k+1 [h] = e

n ∏ κk+1 (ηk+1 (xi )) + pD (xi ) · τi [h] κ (η (x )) + pD (xi ) · τi [1] i=1 k+1 k+1 i

(25.156)

where, if i = 1, ..., n, ∫ τ0 [h] = h(x) · (1 − pD (x)) · Dk+1|k (x)dx (25.157) ∫ τi [h] = h(x) · pD (x) · Lηk+1 (xi ) (x) · Dk+1|k (x)dx.(25.158) • PIMS single-sample hedging for Bernoulli filters: Assume that Gk+1|k [h] is Bernoulli in the sense of Section 4.3.3: Gk+1|k [h] = 1 − pk+1|k + pk+1|k · sk+1|k [h].

(25.159)

Then the PIMS-hedged posterior p.g.fl. is IM S GP k+1|k+1 [h]

(25.160)

1 − pk+1|k + pk+1|k · τ0 [h] ∑n κk+1 (Z−{η ˆ k+1 (xi )}) +pk+1|k i=1 · pD (xi ) · τi [h] ˆ κ (Z) k+1

=

1 − pk+1|k + pk+1|k · τ0 [1] ∑n ˆ k+1 (xl )}) +pk+1|k l=1 κk+1 (Z−{η · pD (xl ) · τl [1] ˆ κ (Z) k+1

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where, if i = 1, ..., n, ∫ τ0 [h] = h(x) · (1 − pD (x)) · sk+1|k (x)dx (25.161) ∫ τi [h] = h(x) · pD (x) · Lηk+1 (xi ) (x) · sk+1|k (x)dx. (25.162) • PIMS single-sample hedging for CBMeMBer filters: Assume that Gk+1|k [h] is multi-Bernoulli in the sense of Section 4.3.4, and that the CBMeMBer approximation of Section 13.4 has been employed. Then the hedged posterior p.g.fl. has the form ( n ) ∏ P IM S L L L Gk+1|k+1 [h] = (1 − qi + qi · si [h]) (25.163) i=1



·

n ∏ j=1



 (1 − q˜jU + q˜jU · s˜U j [h])

where, by (13.52) through (13.56), 1 − sik+1|k [pD ]

qiL

=

i qk+1|k ·

sL i (x)

=

sik+1|k (x) ·

i 1 − qk+1|k · sik+1|k [pD ]

1 − pD (x) 1 − sik+1|k [pD ]

(25.164) (25.165)

and where q˜jU ( · =

∑νk+1|k

pD (xj ) i i qk+1|k (1−qk+1|k )·sik+1|k [pD Lηk+1 (xj ) ] i (1−qk+1|k ·pD (xi )·sik+1|k [pD ])2

i=1

(

κk+1 (ηk+1 (xj )) +pD (xj )

(25.166) )

∑

i i νk+1|k qk+1|k ·sk+1|k [pD Lηk+1 (xj ) ] i i=1 1−qk+1|k ·sik+1|k [pD ]

s˜U j (x)

)

(25.167) ∑νk+1|k i=1

=

i qk+1|k i 1−qk+1|k

∑νk+1|k i=1

· sik+1|k (x) · pD (x) · Lηk+1 (xj ) (x)

i qk+1|k i 1−qk+1|k

. · sik+1|k [pD Lηk+1 (xj ) ]

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25.10.4

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Predicted Ideal Measurement Set (PIMS): Derivation of General Approach

The purpose of this section is to describe the reasoning behind, and derivation of, (25.136). Let the sensor FoV be a cookie cutter: (25.168)

pD (x) = 1S (x).

In (25.133), set Zk+1 to the predicted measurement set (PMS), Zk+1 = Zˆ = {ηk+1 (x1 ), ..., ηk+1 (xn )}. Then we get

=

δF [g, h] δ Zˆ ∑ δGκk+1 δ [g] · Gk+1|k [h(1 + 1S Lg−1 )] ˆ δW δ(Z − W )

(25.169)

ˆ W ⊆Z

∑

=

I⊆{1,...,n}

·

δGκk+1 [g] δ(Zˆ − ZˆI )

(25.170)

δ Gk+1|k [h(1 + 1S Lg−1 )]. δ ZˆI

The terms in the summation in (25.170) correspond to different hypotheses about which measurements are clutter and which are target-generated. For each I, the subset Zˆ − ZˆI is hypothesized to consist of clutter measurements and ZˆI is hypothesized to consist of target-generated measurements. However, if there exists a x ∈ ZˆI such that x ∈ / S, then ZˆI cannot be a valid set of target-generated measurements (since x could not possibly have been collected). Thus any term involving such a ZˆI must be eliminated from the summation. This will be accomplished if we write δF P IM S [g, h] δ Zˆ

∑

=

I⊆{1,...,n}

· where ˆI ) = 1S (X

δGκk+1 ˆI ) [g] · 1S (X δ(Zˆ − ZˆI )

(25.171)

δ Gk+1|k [h(1 + 1S Lg−1 )] δ ZˆI

∏ ˆI x∈X

1S (x) =

∏ i∈I

1S (xi ).

(25.172)

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We are to generalize this to arbitrary FoVs pD (x). Intuitively speaking, ˆ containing xi should be when pD (xi ) is very small then any subset of X correspondingly unlikely to be collected. If all elements of the subset have small pD , that subset should be even less likely to be collected. Thus it is intuitively reasonable to generalize (25.171) to δF P IM S [g, h] δ Zˆ

∑

=

I⊆{1,...,n}

·

δGκk+1 ˆI ) [g] · pD (X δ(Zˆ − ZˆI )

(25.173)

δ Gk+1|k [h(1 + pD Lg−1 )], δ ZˆI

which is (25.136).

25.11

MULTISENSOR-MULTITARGET CONTROL: OPTIMIZATION

The determination of the multisensor control set U at time tk requires solution of the following optimization problem over an infinite solution space: Uk = arg sup Ok+1 (U ).

(25.174)

U

This will be computationally intractable in general. As in the single-sensor, singletarget case (Section 24.7), a commonly employed approximate approach is to limit j the values of Uk to a small finite number of “admissible” possibilities ui (i = 1, ..., aj ) for each of the sensors that are present (j = 1, ..., e). In this case, we must maximize an objective function of the form Ok+1 (S) over all possible subsets S of a finite set. Even this approach may be computationally intensive in general. Witkoskie, Kuklinski, Stein, Theophanis, Otero, and Winters [277], [324], [276] have proposed the following additional approximation. Take the inverse M¨obius transform of Ok+1 (S): ∑ ˜ k+1 (T ) = O (−1)|T −S| · Ok+1 (S). (25.175) S⊆T

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Then truncate the transform using some threshold τ : { ˜ k+1 (T ) if |T | ≤ τ O τ ˜ k+1 O (T ) = . 0 if otherwise

(25.176)

Finally, apply the M¨obius transform: ∑

τ Ok+1 (S) =

τ ˜ k+1 O (T ).

(25.177)

T ⊆S τ Then Ok+1 (S) = 0 for |S| ≤ τ , limiting the number of subsets S that must be considered.

25.12

SENSOR MANAGEMENT WITH IDEAL SENSOR DYNAMICS

This section addresses the multisensor analog of the approach in Section 24.8. As in that section, assume that the control spaces are the same as the sensor state spaces, ∗j

j

(25.178)

U = X,

and that, as in (24.47), the sensor Markov densities completely decouple the sensor dynamics: ∗j

∗j j

∗j

∗j

f k+1|k ( x|u, x ′ ) = δ j ( x).

(25.179)

u

Then given the assumptions in Section 25.5.3, it follows that ∗

∗

∗

∗

f k+1|k (X|X ′ , U ) = δU (X)

(25.180)

∗

where δU (X) is the multitarget Dirac delta density of (4.15). For, from (25.59), ∗

∗

∗

f k+1|k (X|X ′ , U )

=

δ

∗

∗

|X|,|X ′ |

∑ ∗j1 ∗j1 ∗jπ1 jπ1 f k+1|k ( x | x ′ , u )

(25.181)

π ∗je

··· f =

δ

∗

k+1|k ( ∗

|X|,|X ′ |

∑

∗je ∗jπe ′ jπe

x| x , u) ∗j1

∗je

δjπ1 ( x ) · · · δjπe ( x ) u

(25.182)

u

π ∗

=

δU (X)

(25.183)

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where the summation is taken over all permutations π on 1, ..., e. As a consequence, it is easily shown (by direct analogy with the derivation in Section K.34) that ∗

∗

fk+1|k (X, X|Z (k) , Z (k) , U (k−1) , U )

(25.184)

∗

δU (X) · fk+1|k (X|Z (k) , U (k) )

= and

∗

∗

fk+1|k+1 (X, X|Z (k+1) , Z (k+1) , U (k−1) , U )

(25.185)

∗

δU (X) · fk+1|k+1 (X|Z (k+1) , U (k−1) , U ).

=

Selecting the control set Uk at time tk therefore amounts to the same thing ∗ as selecting the sensor state set X k+1 at time tk+1 . Also, the control-sequence ∗ ∗ ∗ U (k−1) : U0 , ..., Uk−1 becomes a sequence X (k) : X 1 , ..., X k of sensor state sets. In this case the single-step multisensor-multitarget control scheme reduces to the form multisensor & multitarget measurement-update ∗

fk|k (X|Z (k) , X (k) )

... →

time-projection ∗

fk+1|k+1 (X|Z (k) , X (k) , ∗ Z, X)

→

hedging ∗

X k+1

↓ ∗ = arg sup ∗ Ok+1 (X) X multisensor & multitarget time & measurement-update ∗

fk+1|k+1 (X|Z (k+1) , X (k+1) )

→ ∗

X k+1 ,Zk+1

→ ...

↗

where ∗

fk+1|k (X|Z (k) , X (k) )

=

∫

fk+1|k (X|X ′ ) ∗

·fk|k (X ′ |Z (k) , X (k) )δX ′

(25.186)

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and ∗

∗

fk+1|k+1 (X|Z (k+1) , X (k) , Z, X) ∗

=

fk+1 (Z|X, X) · fk+1|k (X|Z ∗

(k)

(25.187) ∗

,X

(k)

)

∗

fk+1 (Zk+1 |Z k , X k , X (k) ) and where ∗

∗

f (Z |Z (k) , X (k) , X (k) ) ∫k+1 k+1 ∗ ∗ fk+1 (Zk+1 |X, X) · fk+1|k (X|Z (k) , X (k) )δX.

=

(25.188)

Thus the multitarget analog of (24.53), the Kullback-Leibler objective function with ideal sensor dynamics, is ∗

Ok+1 (X, Z)

=

∫ (

∗

∗

fk+1|k+1 (X|Z (k) , X (k) , Z, X)

)

∗

(25.189)

fk+1|k (X|Z (k) , X (k) ) ∗

∗

·fk+1|k+1 (X|Z (k+1) , X (k) , Z, X)δX. The posterior expected number of targets (PENT) is ∫ ∗ ∗ ∗ Nk+1|k+1 (X, Z) = |X| · fk+1|k+1 (X|Z (k+1) , X (k) , Z, X)δX.

(25.190)

From (25.105), the cardinality variance is ∗

2 σk+1|k+1 (X, Z)

=

−Nk+1|k+1 (U, Z)2 (25.191) ∫ ∗ + |X|2 · fk+1|k+1 (X|Z k , Z (k) , U (k−1) , Z, U )δX.

If we use PIMS single-sample hedging for Z, as in Section 25.10.2, then the hedged objective function is ∗

∗

P IM S Nk+1|k+1 (X) = Nk+1|k+1 (X, Zk+1 )

(25.192)

or, alternatively, ∗

∗

2 2 P IM S σk+1|k+1 (X) = σk+1|k+1 (X, Zk+1 ).

(25.193)

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Sensor management is then accomplished by solving the optimization problems ∗

∗

(25.194)

X k+1|k+1 = arg sup Nk+1|k+1 (X) ∗

X

or ∗

∗

2 X k+1|k+1 = arg inf σk+1|k+1 (X). ∗

(25.195)

X

25.13

SIMPLIFIED NONIDEAL MULTISENSOR DYNAMICS

This section addresses the multitarget generalization of the material in Section 24.10. It is organized as follows: 1. Section 25.13.1: The assumptions underlying the simplified nonideal-sensor approach. 2. Section 25.13.2: Multitarget filtering equations for the simplified nonidealsensor approach. 3. Section 25.13.3: Optimization-hedging approach for simplified management of nonideal sensors. 25.13.1

Simplified Nonideal Multisensor Dynamics: Assumptions

The assumptions are analogous to those in Section 24.10.1. Because of Bayes’ rule, ∗ the joint multisensor-multitarget state (X, X) factors in the manner described in Section 5.9: ∗

∗

fk|k (X, X|Z (k) , Z k , U (k−1) )

∗

=

∗

fk|k (X|Z (k) , Z k , U (k−1) ) ∗

(25.196) ∗

·fk|k (X|Z k , Z (k) , U (k−1) , X). Assume that: 1. As in Section 24.8, the sensor control spaces are the same as the sensor state spaces, j

∗j

(25.197)

U = X. j

∗j

2. The control for the jth sensor is that sensor’s state uk = x k+1 at time tk+1 . Thus:

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∗

(a) The multisensor control sets have the form Uk = X k+1 ; and ∗

∗

∗

(b) The control-sequences have the form U (k−1) : X (k) : X 1 , ..., X k . 3. The sensor Markov densities for the sensors are specified a priori: ∗j

∗j ∗j

∗

j

∗j ∗j

f k+1|k ( x| x ′ , uk−1 ) = f k+1|k ( x| x ′ ). ∗j

(25.198)

∗ j ∗j

That is, at time tk we have a model f k+1|k ( x| x ′ ) of how reachable the ∗j

∗j

sensor-state x at time tk+1 is, given that the sensor state at time tk is x ′ . 4. The sensor distributions depend only on actuator-sensor measurements: ∗j

∗j

∗j

∗j

∗j

f k|k ( x|Z (k) , Z k , X (k) ) ∗j

∗j

∗j

=

∗j ∗j

f k|k ( x|Z k ) ∗j

∗j

f k+1|k ( x|Z (k) , Z k , X (k) )

=

(25.199)

∗j ∗j k

(25.200)

f k+1|k ( x|Z ).

5. The multitarget distributions depend only on target-generated measurements and the multisensor control sequence: ∗

∗

∗

fk|k (X|Z (k) , Z (k) , X (k) , X) ∗

∗

∗

=

fk|k (X|Z (k) , X (k) ) (25.201)

=

fk+1|k (X|Z (k) , X (k) ).(25.202)

∗

fk+1|k (X|Z (k) , Z (k) , X (k) , X)

∗

Equation (25.202) follows from the fact that the predicted multitarget state cannot depend on the predicted sensor states. Equation (25.201) follows from ∗ ∗ ∗ ∗ the fact that fk|k (X|Z (k) , Z k , X k , X) cannot depend on X since, at time ∗

∗

tk , the set X = X k of sensor states has already been selected. ∗

6. The next control set (set of sensor states) X k+1 is selected at time tk+1 , prior to application of the multitarget measurement-update. Given these assumptions, if there are s sensors with given tags, then the multisensor distributions factor as ∗

∗

∗

f k|k (X|Z (k) )

∗

=

|X|! · δ

∗

s,|X|

∏ ∗j

∗j

j ∗j ∗

f k|k ( x|Z k )

(25.203)

∗

x ∈X

∗

∗

∗

f k+1|k (X|Z (k+1) )

∗

=

|X|! · δ

∗

s,|X|

∏ ∗j

∗

x ∈X

∗j

j ∗j ∗

f k+1|k ( x|Z k ).

(25.204)

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Thus the joint multisensor filter separates into parallel, single-object Bayes filters: ∗j

... →

∗j

∗j ∗j

f k|k ( x|Z k )

∗j

∗j ∗j

f k+1|k ( x|Z k )

→

→

∗j ∗j

f k+1|k+1 ( x|Z k+1 )

→ ...

Similarly, the evolution of the targets is described by a conventional multitarget Bayes filter of the form ∗

fk|k (X|Z (k) , X (k) )

... →

∗

→

fk+1|k (X|Z (k) , X (k) )

→

fk+1|k+1 (X|Z (k+1) , X (k+1) )

∗

25.13.2

→ ...

Simplified Nonideal Multisensor Dynamics: Filtering Equations

Given these assumptions, let: • Z (k) : Z1 , ..., Zk be the time sequence of target measurement sets at time tk . ∗

∗

∗

• Z (k) : Z 1 , ..., Z k be the time sequence of multi-actuator measurement sets at time tk . ∗

∗

∗

• X (k) : X 1 , ..., X k be a time sequence of sensor state sets at time tk , with ∗ X i to be selected at time ti (as indicated shortly). Also, we choose the sensor management objective function to be the PENT or PENTI objective function of Sections 25.9.2 and 25.9.4. (The cardinality-variance objective function of Section 25.9.3 cannot be employed, since at this time it is not known how to incorporate sensor and platform dynamics into it using the approach described in this section.) Then as in the single-sensor case, the single-filter control

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scheme in Section 25.8 can be equivalently replaced by the control scheme multisensor & multitarget measurement-update ∗ ∗ ∗

... →

f k|k (X|Z (k) )

... →

fk|k (X|Z (k) , X (k) )

∗

time-projectiion ∗ ∗

∗

f k+1|k (X|Z (k) ) ↓ ∗ fk+1|k+1 (X|Z (k) , X (k) , ∗ Z, X)

−→

−→

hedging

↓ select next multisensor state set nonideal

∗

∗

X k+1 = arg sup ∗ Nk+1|k+1 (Y ) Y multisensor & multtitarget time & measurement-update ∗ ∗ ∗

f k+1|k+1 (X|Z (k+1) )

↗

∗

Z k+1

−→ ...

∗

fk+1|k+1 (X|Z (k+1) , X (k+1) )

→ ∗

X k+1 ,Zk+1

−→ ...

↗

with time-update ∗

∗

∗

∫

∗

∗

∗

∗

∗

∗

∗

f k+1 (X|X ′ ) · f k|k (X ′ |Z (k) )δ X ′ (25.205) ∫ ∗ (k) (k) fk+1|k (X|Z , X ) = fk+1|k (X|X ′ ) (25.206) f k+1|k (X|Z (k) ) =

∗

·fk|k (X ′ |Z (k) , X (k) )δX ′ and measurement-update ∗

∗

∗

∗

f k+1|k+1 (X|Z (k+1) )

∝

∗

∗

(25.207)

f k+1 (Z k+1 |X) ∗

∗

∗

·f k+1|k (X|Z (k) ) ∗

fk+1|k+1 (X|Z (k+1) , X (k+1) )

∗

∝

fk+1 (Zk+1 |X, X k+1 ) ∗

·fk+1|k (X|Z (k) , X (k) ).

(25.208)

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25.13.3

Advances in Statistical Multisource-Multitarget Information Fusion

Simplified Nonideal Single-Sensor Dynamics: Hedging and Optimization

The purpose of this section is to show how a combination of expected-value and PIMS hedging can be applied to the Kullback-Leibler objective function and thus, in turn, to the PENT objective function. The discussion generalizes that of Section 24.6.5. Because of Bayes’ rule, the joint multisensor-multitarget distribution can be factored as ∗

∗

∗

∗

fk+1|k+1 (X, X|Z (k+1) , Z (k) , X (k) , Z, Z) ∗

=

∗

∗

(25.209)

∗

f k+1|k+1 (X|Z (k) , Z) ∗

∗

·fk+1|k+1 (X|Z (k) , X (k) , Z, X) ∗

where where

∗

∗

X (k) : X 1 , ..., X k is the multisensor control-sequence at time tk and ∗

∗

∗

∗

X (k+1) : X 1 , ..., X k , X ∗

∗

is the control-sequence at time tk+1 , with the selection X = X k+1 still to be determined. Apply the multitarget version of Kullback-Leibler cross-entropy to the target part of this distribution to get ∗

∗

∗

∗

∗

∗

˜ k+1 (X, Z) Ok+1 (X, Z, Z) = f k+1|k+1 (X|Z (k) , Z) · O where ∫

∗

˜ k+1 (X, Z) O

∗

∗

fk+1|k+1 (X|Z (k) , X (k) , Z, X) (25.210) ) ( ∗ ∗ fk+1|k+1 (X|Z (k) , X (k) , Z, X) · log δX ∗ fk+1|k (X|Z (k) , X (k) )

=

is the objective function for the ideal-sensor case. Now apply the mixed expectedvalue, PIMS single-sample hedging approach as in analogy with Section 24.6.5:

=

∫

=

P IM S ˜ k+1 (X, Zk+1 f k+1|k (X|Z (k) ) · O ).

∗

Ok+1 (X)

∗

∗

∗

∗

∗

∗

∗

P IM S Ok+1 (X, Zk+1 , Z) · f k+1 (Z|Z (k) )δ Z ∗

∗

∗

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939

Thus maximizing the objective function for nonideal sensors will be the same thing as maximizing the objective function for ideal sensors, but given that the sensor states are reachable by the next time-step. The same reasoning applies to PENT (which is, after all, an approximation of the Kullback-Leibler objective function): nonideal

N

=

∫

=

P IM S ˜k+1|k+1 (X, Zk+1 f k+1|k (X|Z (k) ) · N )

=

f k+1|k (X|Z (k) ) · N k+1|k+1 (X)

∗

k+1|k+1 (X)

∗

∗

∗

∗

∗

∗

P IM S Nk+1|k+1 (X, Zk+1 , Z) · f k+1 (Z|Z (k) )δ Z

∗

∗

∗

∗

∗

∗

∗

ideal

∗

where ideal

∗

∗

P IM S ˜k+1|k+1 (X, Zk+1 N k+1|k+1 (X) = N )

(25.211)

is the PENT objective function for the ideal-sensor case of Section 25.12. ∗ ∗ ∗j1 ∗je If X = { x , ..., x } with |X| = e, then the formula for PENT becomes ∗

Nk+1|k+1 (X, Z)

=

˜k+1|k+1 ({∗jx1 , ..., ∗jxe }, Z) N ∗j1

·f

k+1|k (

∗j1

∗ j1

(25.212)

∗je

x | Z (k) ) · · · f

k+1|k (

∗ je

∗je

x | Z (k) ).

Finally, make use of PIMS single-sample hedging for Z. Then nonideal

N

ideal

∗

k+1|k+1 (X)

=

∗j1

∗je

(25.213)

N k+1|k+1 ({ x , ..., x }) ∗j1

·f

k+1|k (

∗ j1

∗j1

∗je

x | Z (k) ) · · · f

k+1|k (

∗je ∗je (k)

x |Z

).

That is: nonideal

• Maximizing

N

∗

k+1|k+1 (X)

ideal

is the same as maximizing the ideal-sensor

∗

PENT N k+1|k+1 (X), subject to the constraint that all of the sensor states ∗j1

∗je

x , ..., x

must be reachable.

Thus the next multisensor state is nonideal

∗

X k+1 = arg sup N ∗

Y

∗

k+1|k+1 (Y

).

(25.214)

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Advances in Statistical Multisource-Multitarget Information Fusion

25.14

TARGET PRIORITIZATION

The goal of sensor management is usually not to direct sensors and platforms to any and all possible targets. Rather, it is to direct them to situationally significant targets of interest (ToIs): tanks in preference to trucks, say, or missile launchers in preference to tanks. In principle, one could address situational significance by simply waiting until accumulated information strongly suggests that particular targets have high situational interest, and then bias sensor management towards these targets. Unfortunately, deterministic ad hoc techniques of this kind have inherent weaknesses. For example: • Information about target type accumulates incrementally, not suddenly. Preferential biasing of sensors should likewise be accomplished incrementally, and only to the degree supported by accumulated evidence. • Deterministic decisions to ignore some tracks may be ill-conceived, since information about target type may be erroneous and reversed by later, better data. • Consequently, it would be better to have a theoretically principled way of integrally incorporating tactical significance into the fundamental statistical representation of multisensor-multiplatform-multitarget problems. This is the purpose of this section, which is based on concepts introduced in [179], pp. 531-536. The section is organized as follows: 1. Section 25.14.1: The concept of situational significance. 2. Section 25.14.2: The mathematical modeling of situational significance using tactical importance functions (TIFs). 3. Section 25.14.3: Characteristics and examples of TIFs: mission-oriented TIFs, Time evolution of TIFs, and prediction of TIFs. 4. Section 25.14.4: The multitarget statistics of TIFs. 5. Section 25.14.5: The PENTI objective function—that is, the PENT objective function, biased towards ToIs using a TIF. 6. Section 25.14.6: The ToI-biased cardinality-variance objective function— that is, the cardinality-variance objective function, biased towards ToIs using a TIF.

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25.14.1

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The Concept of Tactical Significance

The situational significance of a target depends on multiple factors, some of which are intrinsic (a target is significant in and of itself) and some of which are relational (a target is significant because of its relationships to friendly assets or other targets). One factor is target type or class c. Any target that is capable of causing massive damage or disruption, such as a tank or mobile missile launcher, is of inherent situational interest regardless of situational context. Another factor is the threat state w (for example, fire-control radar is on). A third factor is position p. Any target of given type may have greater or lesser situational significance depending on its proximity to to friendly assets and on the ranges of its weapons. The same is true in regard to speed and heading v. Any target of undetermined type that is heading towards a friendly asset at high speed is of inherent situational interest. As another example, targets that are moving in concert as a group are potentially of greater interest than an isolated target. There may be other factors as well, such as a commander’s evolving preferences and priorities, or an operator’s assessment of the current situation. Finally, since the scene is dynamically changing, the definition of situational significance must also be dynamic. For example, the significance of a target will increase as it approaches an asset. The current situational significance of a target is thus a function of its current state-vector x = (p, v, c, w, ...). However, its degree of significance will also depend on the states of the other objects, fixed or mobile, in the scene. 25.14.2

Tactical Importance Functions (TIFs) and Higher-Level Fusion

Target importance at a given time tk can be expressed as a tactical importance function (TIF). A TIF can be regarded as: • A mathematical representation of the tactical situation at any given time. • The mathematical basis for situational awareness (“Levels 2 and 3 data fusion”). • The mathematical portal through which operator contextual information can be supplied.

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Figure 25.1 The concept of a tactical importance function (TIF) and targets of interest (ToIs). In the “crisp” or “hard” TIF on the left, targets are either of no tactical interest at all (non-ToIs), or of crucial tactical interest (ToIs). In the “fuzzy” or “soft” TIF on the right, targets can be assigned intermediate degrees of tactical importance.

A TIF is a function ιk|k (x) of target state x that is always between zero and one. If ιk|k (x) = 0, then at time tk a target with state x has no situational significance whatsoever. If ιk|k (x) = 1, however, it has the greatest possible significance. TIFs can be updated semi-automatically to reflect changing priorities, or to reflect operator contextual information. Both “crisp” and “fuzzy” TIFs are illustrated in Figure 25.1. In general, a TIF will have a much more complex structure than is indicated there. It will usually have multiple and irregularly-shaped peaks, with each peak corresponding to a tactically “hot” part of the scenario. The section is organized as follows:

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1. Section 25.14.2.1: A general mathematical representation for TIFs. 2. Section 25.14.2.2: Proximity-only TIFs. 3. Section 25.14.2.3: Velocity-only TIFs. 25.14.2.1

A General Mathematical Representation for TIFs

Suppose that, in a given scenario and at a particular time tk , we have a list of νk|k νk|k static or moving assets of interest (AoIs) with state-vectors ˚ x1k|k , ...,˚ xk|k . An AoI will generally be a friendly or neutral target, but can also be an unfriendly one. Given this, a TIF for the scenario will have the general form νk|k νk|k 0 1 ιk|k (x) = ˚ Ik|k ·˚ ι0k|k (x) + ˚ Ik|k ·˚ ι1k|k (x) + ... + ˚ Ik|k ·˚ ιk|k (x)

(25.215)

where: • ˚ ι0k|k (x) indicates an AoI-independent TIF,5 and • ˚ ιik|k (x) for i ≥ 1 is a TIF specific to the ith AoI. i i The weights ˚ Ik|k satisfy 0 ≤ ˚ Ik|k ≤ 1 and νk|k

∑

i ˚ Ik|k = 1,

(25.216)

i=0

and indicate the relative degrees of importance attached to the different AoIs. The following subsections provide some examples of AoI-specific TIFs. 25.14.2.2

TIF Example: Proximity-Only

Suppose that ˚ ι1k|k (x) is a TIF oriented towards proximity only—that is, the closer that a target is to the first asset, the greater its significance. A simple AoI-specific TIF of this kind might have the form 1 ˆ ˚1,pos (A ˚1,pos x − B ˚1,pos˚ ˚ ι1,pos k|k (x) = NE k|k k|k xk|k )

(25.217)

k|k

5

For example, a mobile missile launcher may be regarded as significant regardless of its relationship to any asset or any other target.

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Advances in Statistical Multisource-Multitarget Information Fusion

˚1,pos , B ˚1,pos project target-state x to target position p; where where the matrices A k|k k|k ˚1,pos determines the (ellipsoidal) shape and extent of the the covariance matrix E k|k “zone of significance” surrounding the first asset; and where: √ ˚1,pos · N ˆ ˚1,pos (p) ˆ ˚1,pos (p) = det 2π E N (25.218) E E k|k k|k

k|k

is a normalized Gaussian distribution with mean 0. 25.14.2.3

TIF Example: Velocity-Only

Suppose that ˚ ι1k|k (x) is oriented towards velocity only—that is, the faster that a target is approaching the first asset’s position, the greater its significance. A TIF of this kind might have the form ˚ ι1,pos,vel (x) k|k =

ˆ ˚1,vel sup N E y

k|k

(25.219) (

1 ˚1,pos y − B ˚1,pos˚ ˚1,vel x A B k|k k|k k|k xk|k − 1 ∥ ˚1,pos y − B ˚1,pos˚ ˚1,vel x∥ ∥A x ∥ B k|k k|k k|k k|k

)

·1s≥sthresh (˚ sk|k ) ˚1,vel projects to velocity; where sthresh is a speed beyond which an where B k|k ˚1,vel determines approaching target is deemed potentially threatening; and where E k|k the (ellipsoidal) “zone of velocities” surrounding the first asset. 25.14.3

Characteristics of TIFs

The following subsections highlight some important points about TIFs and their use. 25.14.3.1

Mission-Specific TIFs

Situational importance as a concept is primarily definitional and deterministic rather than physics-based or statistical. On the one hand, determinations of the intent and degree of threat of particular targets must be inferred statistically from accumulating evidence. But on the other hand, such inference depends on specific, missiondependent definitions of what “tactical importance” means. As just the most obvious example, target types that are of central importance in one mission may be of little significance in another. Or significance in a specific

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mission may depend on a particular commander’s preferences and priorities. Or, moment-to-moment significance may be determined on the basis of an operator’s contextual understanding of a scene. TIFs can be constructed to reflect any of these influences. In particular, a TIF can be altered to reflect the preferences of a system operator. 25.14.3.2

Time-Evolution of TIFs

Since situational importance changes with time, TIFs must be updated continuously as necessary. In many instances, such updates can be performed automatically. For example, if a target has moved even closer to an AoI than previously then the proximity part of the AoI’s TIF does not have to be updated, since significance due to proximity is already built into the defining formula for the TIF. On the other hand, many TIF updates may require human mediation. For example, if a commander and/or an operator might designate an AoI as being more crucial than was previously the case. In such situations, the formula for the TIF must be modified accordingly. 25.14.3.3

Time-Prediction of TIFs

For purposes of inferring future intent it may be necessary to predict a TIF ιk|k (x) to a TIF ιk+k′ |k (x) at some future time tk+k′ , in order to assess what the future threat situation might look like. This can be accomplished using conventional kinematic prediction methods. For example, suppose that the TIF ιk|k (x) reflects current tactical interest, given that the targets have certain positions and velocities. By applying suitable motion models to the assets and the targets, we can time-extrapolate ιk|k (x) to ιk+k′ |k (x). 25.14.4

The Multitarget Statistics of TIFs

TIFs cannot be employed unless they are correctly incorporated into the statistics of a multitarget system. The purpose of this section is to show how this is accomplished. The basic concepts were originally proposed in Section 14.9.1 of [179]. Suppose that Gk|k [h] abbr. = Gk|k [h|Z (k) ] (25.220)

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Advances in Statistical Multisource-Multitarget Information Fusion

is the p.g.fl. of the multitarget distribution fk|k (X) abbr. = fk|k (X|Z (k) ) at time tk . For conceptual clarity, begin with a special case: ιk|k (x) = 1S (x) is “crisp” in the sense of Figure 25.1. That is, if the target’s state is x then either ιk|k (x) = 1 (the target has highest possible tactical interest) or ιk|k (x) = 0 (it is of no interest at all). Let Ξk|k be the multitarget RFS. Then the “censored” RFS Ξ→ι k|k = Ξk|k ∩ S

(25.221)

is the RFS of only the tactically important targets. The p.g.fl. of Ξ→ι k|k —that is, the p.g.fl. that has been biased to preferentially emphasize ToIs—is, from (14.295) of [179] or (4.135) in Chapter 4: G→ι k|k [h]

=

GΞ→ι [h] = GΞk|k [1 − 1S + 1S · h] k|k

(25.222)

=

Gk|k [1 − 1S + 1S · h].

(25.223)

Now consider a general G→ι [h] is given by k|k

ιk|k (x). From (14.296) of [179] we know that

G→ι k|k [h] = Gk|k [1 − ιk|k + ιk|k · h].

(25.224)

It is also easily shown that the corresponding ToI-biased PHD is given by →ι Dk|k (x) = ιk|k (x) · Dk|k (x)

(25.225)

and therefore that the ToI-biased expected number of targets is →ι Nk|k =

∫

ιk|k (x) · Dk|k (x)dx.

(25.226)

Similarly, let Gk|k [h] = Gk|k (sk|k [h]) be i.i.d.c. Then its cardinality variance is (2) →ι 2 (σk|k )

Gk|k (1) = (1)

Gk|k (1)

· Dk|k [ιk|k ]2 − Dk|k [ιk|k ]2 + Dk|k [ιk|k ].

(25.227)

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947

To see why (25.225) is true, note that from the fourth chain rule, (3.84) with T [h] = 1 − ιk|k + ιk|k h, δG→ι k|k [h]

=

δx = =

∫

δGk|k δT [h](w) · [1 − ιk|k + ιk|k h]dw (25.228) δx δw ∫ δGk|k ιk|k (w) · δx (w) · [1 − ιk|k + ιk|k h]dw (25.229) δw δGk|k ιk|k (x) · [1 − ιk|k + ιk|k h] (25.230) δx

and so by (4.75), →ι Dk|k (x) =

δG→ι k|k [1] = ιk|k (x) ·

δx

δGk|k [1] = ιk|k (x) · Dk|k (x). δw

(25.231)

Equation (25.227) follows immediately from (4.70) and d →ι G (x) dx k|k 2 d G→ι (x) dx2 k|k 25.14.5

(1)

(25.232)

=

Gk|k (sk|k [1 − ιk|k + ιk|k · x]) · sk|k [ιk|k ]

=

Gk|k (sk|k [1 − ιk|k + ιk|k · x]) · sk|k [ιk|k ]2 . (25.233)

(2)

Posterior Expected Number of Targets of Interest (PENTI)

For the purposes of sensor management, we need to incorporate the TIF ιk|k (x) into the PENT objective function. Once this is accomplished, the new objective function will tend to bias sensors to preferentially collect measurements from more important targets (based on their relative degree of importance). The purpose of this section is to describe how this is accomplished. ∗ ∗ ∗1 ∗s Let X = { x, ..., x} be the set of undetermined sensor states with |X| = s and let ∗

∗

∗

abbr. IM S P IM S (k) P IM S GP , X (k) , Zk+1 , X] k+1|k+1 [h|X] = Gk+1|k+1 [h|Z

(25.234)

be the PIMS-hedged p.g.fl. as defined in Section 25.10.2. Then the corresponding ToI-biased p.g.fl. is P IM S

G

∗

→ι k+1|k+1 [h|X]

∗

IM S = GP k+1|k+1 [1 − ιk|k + ιk|k h|X].

(25.235)

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Thus from (25.225), the ToI-biased, PIMS-hedged PHD is ∗

P IM S

D

→ι k+1|k+1 (x|X)

∗

P IM S = ιk|k (x) · Dk+1|k+1 (x|X)

(25.236)

where ∗

P IM S Dk+1|k+1 (x|X) =

IM S δGP k+1|k+1

∗

(25.237) δx is the PIMS-hedged PHD. Thus from (25.226), the ToI-biased, PIMS-hedged posterior expected number of targets—the posterior expected number of targets of interest (PENTI) is ideal

∗

∗

P IM S

def. →ι N →ι k+1|k+1 (X) = N k+1|k+1 (X) ∫ ∗ P IM S ιk|k (x) · Dk+1|k+1 (x|X)dx.

= 25.14.6

[1|X]

(25.238)

Biasing the Cardinality Variance to Targets of Interest (ToIs)

From (4.67), the ToI-biased, PIMS-hedged cardinality distribution is ∗

P IM S

G

→ι k+1|k+1 (x|X)

∗

IM S = GP k+1|k+1 [1 − ιk|k + x · ιk|k |X].

(25.239)

Thus we can bias the cardinality-variance objective function towards preferential collection from targets of interest. From (4.70) and (25.239) we know that the PIMS-hedged cardinality variance is ideal 2

ideal

∗

σ k+1|k+1 (X)

=

∗

ideal

∗

→ι 2 N →ι k+1|k+1 (X) − N k+1|k+1 (X)

(25.240)

P IM S

→ι ∗ k+1|k+1 (1|X). 2 dx

d2 G +

(25.241)

Chapter 26 Approximate Sensor Management 26.1

INTRODUCTION

A general RFS approach to sensor and platform management was introduced in Chapter 25. However, because this approach is based on the multitarget Bayes filter, it will be computationally intractable in general. The purpose of this chapter is to describe potentially tractable approximate sensor management algorithms based on approximate RFS filters: Bernoulli, PHD, CPHD, and CBMeMBer filters. 26.1.1

Summary of Major Lessons Learned

The following are the major concepts, results, and formulas that the reader will learn in this chapter: • The ideal-sensor and approximate nonideal-sensor approximations of Chapter 25 can be extended for use with Bernoulli, PHD, CPHD, and CBMeMBer filters. • The single-sample PIMS approximate hedging strategy (Section 25.10.2) can be extended for use with Bernoulli, PHD, CPHD, and CBMeMBer filters. • In the case of PHD filters, the PENT objective function with PIMS hedging exhibits intuitively reasonable behavior (Sections 26.3.1.4 and 26.3.2.5). with PMS hedging, however, it does not necessarily do so (Section 26.3.1.4). • The “pseudosensor” approximation allows management of multiple sensors to be used more tractably with PHD, CPHD, and CBMeMBer filters.

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• Explicit formulas for the PENT, PENTI, and cardinality-variance objective functions can be derived for use with Bernoulli, PHD, CPHD, and CBMeMBer filters. 26.1.2

Organization of the Chapter

The chapter is organized as follows: 1. Section 26.2: Sensor management using Bernoulli filters. 2. Section 26.3: Approximate sensor management using PHD filters. 3. Section 26.4: Approximate sensor management using CPHD filters. 4. Section 26.5: Approximate sensor management using CBMeMBer filters. 5. Section 26.6: Implementations of RFS sensor management algorithms.

26.2

SENSOR MANAGEMENT WITH BERNOULLI FILTERS

The Bernoulli filter of Section 13.2 is the optimal approach for detecting and tracking at most a single target, observed by a single sensor. The purpose of this section is to describe single-sensor, single-target sensor management using this filter, an approach first investigated by Ristic and Arulampalam in 2012 [251]. The material in this section is a generalization, to arbitrary clutter processes, of the single-target sensor management approach described in Chapter 24. Single-step look-ahead sensor management using the Bernoulli filter has the same form as in Section 24.4: ... →

multitarget measurement-update ∗ ∗

fk|k (X, x|Z (k) , Z k , U k−1 ) time-projection ∗

∗

→

fk+1|k+1 (X, x|Z (k) , Z (k) , U k−1 , ∗ Z, Z, u) hedging

↓ select next control

uk = arg supu Ok+1 (u) multitarget time-update & measurement-update ∗

∗

uk ,Zk+1 ,zk+1

∗

fk+1|k+1 (X, x|Z (k+1) , Z k+1 , U k )

→ ↗

→ ...

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Here, however, the target state is a finite set X with X = ∅ (no target) or X = {x} (single target with state x). Also, a single sensor collects a sequence Z (k) : Z1 , ..., Zk of measurement sets rather than single measurements, since measurements can be clutter-generated as well as generated by the target (if it is present). In what follows, the simplified nonideal-sensor approach of Section 25.13 will be employed. Let: • Z (k) : Z1 , ..., Zk be the time sequence of target measurement sets at time tk ; ∗

∗

∗

• Z k : z1 , ..., zk the time sequence of actuator measurements at time tk ; and ∗

∗

∗

∗

• X k : x1 , ..., xk a time sequence of sensor state sets at time tk , with xi to be selected at time ti . In this case, the control scheme has the two-filter form shown in Figure 26.1. That is, it consists of the following sequence of steps: ∗

∗

∗

∗

• Sensor time-update: Time-extrapolate the current x—Z k ) to f k+1|k (x|Z k ) using the prediction integral. • Target time-update: Time-extrapolate ∗

pk|k (Z (k) , X k ),

∗

sk|k (x|Z (k) , X k )

to ∗

pk+1|k (Z (k) , X k ),

∗

sk+1|k (x|Z (k) , X k )

using the Bernoulli filter prediction equations. • Time-projection: Measurement update ∗

pk+1|k (Z (k) , X k ),

∗

sk+1|k (X|Z (k) , X k )

to ∗

∗

pk+1|k+1 (Z (k) , X k , Z, x),

∗

∗

sk+1|k+1 (x|Z (k) , X k , Z, x) ∗

using the Bernoulli filter corrector equations—given the sensor state x at time tk+1 (yet to be determined) and the unknowable target measurement set Z at time tk+1 .

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Figure 26.1 sensor case.

The sensor management control scheme for the simplified nonideal-

Approximate Sensor Management

∗

• Objectivization: Given x ∗ Ok+1 (x, Z).

953

and Z, determine the objective function

• Hedging: use multisample hedging or PIMS single-sample hedging to eliminate Z: ideal ∗ ∗ P IM S O k+1 (x) = Nk+1|k+1 (x, Zk+1 ). (26.1) • Dynamicization: Modify PENT to account for sensor dynamics (that is, for the reachable sensor states at time tk+1 ): nonideal

N

∗

k+1|k+1 (x)

ideal

∗

∗

∗

∗

= N k+1|k+1 (x) · f k+1|k (x|Z k ).

(26.2)

• Optimization: Determine the best next sensor state: nonideal

∗

xk+1 = arg sup N

∗

(26.3)

k+1|k+1 (x).

∗

x ∗

∗

∗

∗

∗

∗

• Sensor measurement-update: Update f k+1|k (x|Z k ) to f k+1|k+1 (x|Z k+1 ) using Bayes’ rule. ∗

• Target measurement-update: Employing xk+1 as the sensor state at time tk+1 , update ∗

pk+1|k (Z (k) , X k ),

∗

sk+1|k (x|Z (k) , X k )

to ∗

pk+1|k+1 (Z (k+1) , X k+1 ),

∗

sk+1|k+1 (x|Z (k+1) , X k+1 )

using the Bernoulli filter corrector equations. • Repeat. The section is organized as follows: 1. Section 26.2.1: The filtering equations for single-sensor, single-target, singlestep control using Bernoulli filters. 2. Section 26.2.2: Objective functions for single-sensor, single-target, singlestep control using Bernoulli filters. 3. Section 26.2.3: Hedging using predicted ideal measurement set (PIMS) single-sample hedging and the PENT objective function or its variants. 4. Section 26.2.4: Multisensor, single-step look-ahead using Bernoulli filters.

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26.2.1

Sensor Management with Bernoulli Filters: Filtering Equations

From (13.3) through (13.8), the following are the filtering equations for the approach: • Sensor time-update: ∗

∗

∗

f k+1|k (x|Z k ) =

∫

∗

∗

∗

∗

∗

∗

∗

f k+1|k (x|x′ ) · f k|k (x′ |Z k )dx′ .

(26.4)

• Target time-update: ∗

pk+1|k (Z (k) , X k )

=

∗

sk+1|k (x|Z (k) , X k )

=

pB · (1 − pk|k ) + pk|k · sk|k [pS ] (26.5) ( ) pB · (1 − pk|k ) · ˆbk+1|k (x) +sk|k [pS Mx ] (26.6) pk+1|k

where sk|k [pS ] sk|k [pS Mx ]

=

∫

pS (x′ ) · sk|k (x′ |Z (k) , X k )dx′

(26.7)

=

∫

pS (x′ ) · fk+1|k (x|x′ )

(26.8)

∗

∗

·sk|k (x′ |Z (k) , X k )dx′ . • Sensor measurement-update: ∗

∗

∗

∗

∗

∗

∗

∗

∗

f k+1|k+1 (x|Z k+1 ) ∝ f k+1 (zk+1 |x) · f k+1|k (x|Z k ).

(26.9)

• Target measurement-update: ∗

=

∗

pk+1|k+1 (Z (k+1) , X k , x) (26.10) ( ) ∗ 1 − sk+1|k [pD |x] ∑ ∗ (Zk+1 −{z}) + z∈Zk+1 sk+1|k [pD Lz |x] · κk+1 κk+1 (Zk+1 ) ( ) ∗ ∗ pk+1|k (Z (k) , X k )−1 − sk+1|k [pD |x] ∑ ∗ (Zk+1 −{z}) + z∈Zk+1 sk+1|k [pD Lz |x] · κk+1 κk+1 (Zk+1 )

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and ∗

=

∗

sk+1|k+1 (x|Z (k+1) , X k , x) (26.11) ( ) ∗ 1 − pD (x, x) ∗ ∑ (Zk+1 −{z}) +pD (x, x) z∈Zk+1 Lz (x) · κk+1 κk+1 (Zk+1 ) ( ) ∗ 1 − sk+1|k [pD |x] ∑ ∗ (Zk+1 −{z}) + z∈Zk+1 sk+1|k [pD Lz |x] · κk+1 κk+1 (Zk+1 ) ∗

·sk+1|k (x|Z (k) , X k ) where ∗

sk+1|k [pD |x] ∗

sk+1|k [pD Lz |x]

=

∫

pD (x, x) · sk+1|k (x|Z (k) , X k )dx (26.12)

=

∫

pD (x, x) · Lz (x, x)

∗

∗

∗

∗

(26.13)

∗

·sk+1|k (x|Z (k) , X k )dx and where by convention the summations vanish if Zk+1 = ∅. 26.2.2

Sensor Management with Bernoulli Filters: Objective Functions

As per the discussion of Section 25.9, any Csisz´ar information functional could be employed as an objective function for sensor management. In this section, only the following are considered: The R´enyi α-divergence (Section 26.2.2.1) and its approximation, the posterior expected number of targets or PENT, and its cardinality-variance variant (Section 26.2.2.2). 26.2.2.1

Sensor Management with Bernoulli Filters: R´enyi α-Divergence

This objective function has been considered by Ristic and Arulampalam [251]. By (6.71), it is given by ∗

Rα (x, Z)

=

1 log α−1

∫

∗

∗

fk+1|k+1 (X|Z (k) , X k , Z, x)α ∗

·fk+1|k (X|Z (k) , X k )1−α δX

(26.14)

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or, alternatively, by ([251], Eq. (19)) ∗

e(α−1)·Rα (x,Z) =

(26.15)

(k)

∗

∗

k

α

fk+1|k+1 (∅|Z , X , Z, x) · fk+1|k (∅|Z ∫ ∗ ∗ + fk+1|k+1 ({x}|Z (k) , X k , Z, x)α

(k)

∗

k 1−α

,X )

∗

·fk+1|k ({x}|Z (k) , X k )1−α dx ∗

=

∗

(1 − pk+1|k+1 (Z (k) , X k , Z, x))α

(26.16)

∗

·(1 − pk+1|k (Z (k) , X k ))1−α ∗

∗

∗

+pk+1|k+1 (Z (k) , X k , Z, x)α · pk+1|k (Z (k) , X k )1−α ∫ ∗ ∗ ∗ · sk+1|k+1 (x|Z (k) , X k , Z, x)α · sk+1|k (x|Z (k) , X k )1−α dx. The following heuristic derivation1 shows that, for the Bernoulli filter, R´enyi divergence is particularly easy to approximate using sequential Monte Carlo (SMC) methods. Let sk+1|k (x) ∼ =

ν ∑

sk+1|k+1 (x) ∼ =

ui · δxi (x),

∫

vi · δxi (x).

(26.17)

i=1

i=1

Then

ν ∑

sk+1|k+1 (x)α · sk+1|k (x)1−α dx =

ν ∑

1−α uα . i · vi

(26.18)

i=1

This is because ∫

=

sk+1|k+1 (x)α · sk+1|k (x)1−α dx

∫ (∑ ν

ui · δxi (x)

)α (

i=1

and so =

∫ (∑ ν i=1

1

uα i

· δxi (x)

ν ∑

vi · δxi (x)

(26.19) )1−α

dx

)

dx

i=1

α

)(

ν ∑

wi1−α

· δxi (x)

1−α

i=1

This derivation is heuristic, since fractional powers of Dirac delta functions appear to be undefined.

Approximate Sensor Management

∫ (∑ ν

∼ =

sk+1|k (x)

uα i

·

vi1−α

957

)

(26.20)

· δxi (x) dx

i=1 ν ∑

=

1−α uα . i · vi

(26.21)

i=1

26.2.2.2

Sensor Management with Bernoulli Filters: PENT and Cardinality Variance

Equation (26.16) will be computationally demanding in general. Thus as per the reasoning in Section 25.9.2, we approximate it by the posterior expected number of targets: ∗

Nk+1|k+1 (x, Z)

=

∫

=

pk+1|k+1 (Z (k) , X k , Z, x).

∗

∗

|X| · fk+1|k+1 (X|Z (k+1) , X k , Z, x)δX (26.22) ∗

∗

(26.23)

∗

That is, for a Bernoulli filter Nk+1|k+1 (x, Z) is just the posterior probability of ∗ existence of the target, hereafter abbreviated as pk+1|k+1 (Z, x). Sensor control consists of trying to maximize this probability. Similarly, the cardinality-variance objective function of Section 23.2.4 is ( ) ∗ ∗ ∗ 2 σk+1|k+1 (Z, x) = 1 − pk+1|k+1 (Z, x) · pk+1|k+1 (Z, x).

(26.24)

In this case, sensor control is achieved by trying to minimize this variance. This, in ∗ turn, amounts to either maximizing or minimizing pk+1|k+1 (Z, x). 26.2.3

Bernoulli Filter Control: Hedging

Equations (26.16) through (26.22) both depend on the unknowable future measurement set Z. The typical strategy—averaging over Z—would be computationally troublesome. The predicted ideal measurement set (PIMS) single-sample hedging approach of Section 25.10.2 can be used instead.

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From (25.160), we know that the PIMS-hedged p.g.fl. for a Bernoulli targetprocess is given by IM S GP k+1|k+1 [h] (

=

(

(26.25)

1 − pk+1|k + pk+1|k · τ0 [h] ∑n κk+1 (Z−{η ˆ k+1 (xi )}) +pk+1|k i=1 · pD (xi ) · τi [h] ˆ κ (Z) k+1

1 − pk+1|k + pk+1|k · τ0 [1] ∑n ˆ k+1 (xl )}) +pk+1|k l=1 κk+1 (Z−{η · pD (xl ) · τl [1] ˆ κ (Z)

) )

k+1

where τ0 [h] τi [h]

=

∫

h(x) · (1 − pD (x)) · sk+1|k (x)dx

(26.26)

=

∫

h(x) · pD (x) · Lηk+1 (xi ) (x) · sk+1|k (x)dx.

(26.27)

The corresponding p.g.f. is IM S GP k+1|k+1 (x) (

=

(26.28) )

1 − pk+1|k + x · pk+1|k · τ0 [1] ∑n ˆ k+1 (xi )}) +x · pk+1|k i=1 κk+1 (Z−{η · pD (xi ) · τi [1] ˆ κk+1 (Z) ( ) 1 − pk+1|k + pk+1|k · τ0 [1] ∑n ˆ k+1 (xl )}) +pk+1|k l=1 κk+1 (Z−{η · pD (xl ) · τl [1] ˆ κ (Z) k+1

and so the PIMS-hedged PENT is P IM S

P IM S Nk+1|k+1 = G

( +pk+1|k = (

∑n

i=1

(1) k+1|k+1 (1)

)

pk+1|k · τ0 [1] ˆ κk+1 (Z−{η k+1 (xi )}) ˆ κk+1 (Z)

(26.29)

· pD (xi ) · τi [1]

1 − pk+1|k + pk+1|k · τ0 [1] ∑n ˆ k+1 (xl )}) +pk+1|k l=1 κk+1 (Z−{η · pD (xl ) · τl [1] ˆ κ (Z)

)

(26.30)

k+1

1 − pk+1|k ). =1− ( 1 − pk+1|k + pk+1|k · τ0 [1] ∑n κk+1 (Z−{η ˆ k+1 (xl )}) +pk+1|k l=1 · pD (xl ) · τl [1] ˆ κ (Z) k+1

(26.31)

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This is maximized if and only if the following “alternative PIMS” formula (the denominator of the fraction in (26.31)) is maximized: AP IM S Nk+1|k+1

=

(26.32)

1 − pk+1|k + pk+1|k · τ0 [1] n ∑ κk+1 (Zˆ − {ηk+1 (xl )}) +pk+1|k · pD (xl ) · τl [1]. ˆ κk+1 (Z) l=1

Thus in our current notation, we get: • Single-step look-ahead alternative-PENT for a single sensor with ideal sensor dynamics (Bernoulli filter): ideal

∗

(26.33)

N k+1|k+1 (x) ∗

=

∗

1 − pk+1|k (Z (k) , X k ) · τ˜0 (x) n ∑ ∗ κk+1 (Zˆ − {ηk+1 (xl )}) +pk+1|k (Z (k) , X k ) ˆ κk+1 (Z) l=1 ∗

∗

·pD (xl , x) · τ˜l (x) where ∗

τ˜0 (x) ∗

τ˜l (x)

=

∫

pD (x, x) · sk+1|k (x|Z (k) , X k )dx

=

∫

pD (x, x) · Lηk+1 (xl ) (x, x)

∗

∗

∗

∗

(26.34) (26.35)

∗

·sk+1|k (x|Z (k) , X k )dx. Now consider the case of nonideal sensor dynamics. From (25.212) we have: • Single-step look-ahead Alternative-PENT for a single sensor with nonideal sensor dynamics (Bernoulli filter): nonideal

N 26.2.4

∗

k+1|k+1 (x)

ideal

∗

∗

∗

∗

= N k+1|k+1 (x) · f k+1|k (x|Z k ).

(26.36)

Bernoulli Filter Control: Multisensor

The multisensor Bernoulli filter was described in Section 13.3. It consists of iterating the Bernoulli filter measurement-update equations, once for each sensor.

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Single-step look-ahead control for the Bernoulli filter is achieved in the same general manner as described in Chapter 24. It will not be further considered here.

26.3

SENSOR MANAGEMENT WITH PHD FILTERS

The following summarizes the major topics addressed thus far in Part V: • A general approach for single-step look-ahead multisensor-multitarget sensor management (Section 25.8). • A special case: ideal sensor dynamics (Section 25.12). • A simplified approach for nonideal sensor dynamics that is also a generalization of the ideal-sensor formulation (Section 25.13). All of these approaches will be computationally intractable in general, largely because they rely on the general multisensor-multitarget Bayes filter. The purpose of this and the following sections is to develop approaches to sensor management using approximate multitarget filters. This section is devoted to sensor management using the PHD filter. Sections 26.4 and 26.5 will be devoted to sensor management using the CPHD and CBMeMBer filters. The PHD and CPHD filter sensor management approaches will require the following assumptions: • The simplified approach to sensors with nonideal dynamics (25.13). • The predicted ideal measurement set (PIMS) approximate single-sample hedging approach (Section 25.10.2). The section is organized as follows: 1. Section 26.3.1: Single-sensor, single-step look-ahead using PHD filters. 2. Section 26.3.2: Multisensor, single-step look-ahead using PHD filters. 26.3.1

Single-Sensor, Single-Step PHD Filter Control

The goal of this section is to modify the simplified approach for nonideal sensors (Section 25.13) for use with PHD filters. We begin with the single-sensor, singlestep look-ahead case, with the multisensor, single-step look-ahead case addressed in Section 26.3.2. Let:

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• Z (k) : Z1 , ..., Zk be the time sequence of target measurement sets at time tk ; ∗

∗

∗

• Z k : z1 , ..., zk tk ; and ∗

∗

be the time sequence of actuator measurement sets at time

∗

∗

• X k : x1 , ..., xk be a time sequence of sensor states at time tk , with xi to be selected at time ti . Then the PHD filter approximation of the simplified nonideal-sensor case in Section 25.13 has the structure shown in Figure 26.2. That is, it consists of the following sequence of steps: ∗

∗

∗

• Sensor time-update: Extrapolate the current sensor-state distribution f k|k (x|Z k ) ∗

∗

∗

to f k+1|k (x|Z k ) using the prediction integral. ∗

• Target time-update: Extrapolate the current PHD Dk|k (x|Z (k) , X k ) to ∗

Dk+1|k (x|Z (k) , X k ) using the PHD filter prediction equations. ∗

• Time-projection: Measurement update Dk+1|k (x|Z (k) , X k ) to ∗

∗

Dk+1|k+1 (x|Z (k) , X k , Z, x) using the PHD filter corrector equations, given the yet-to-be-determined ∗ sensor state x at time tk+1 and the unknowable target measurement set Z at time tk+1 . ∗

• Objectivization: Given x and Z, determine the posterior expected number of targets (PENT) ∫ ∗ ∗ ∗ Nk+1|k+1 (x, Z) = Dk+1|k+1 (x|Z (k) , X k , Z, x)dx. (26.37) • Hedging: Use multisample hedging or single-sample PIMS hedging to eliminate Z: ideal

∗

∗

P IM S N k+1|k+1 (x) = Nk+1|k+1 (x, Zk+1 ).

(26.38)

• Dynamicization: Modify PENT to account for sensor dynamics (that is, for the reachability of the sensor states at time tk+1 ): nonideal

N

∗

k+1|k+1 (x)

ideal

∗

∗

∗

∗

= N k+1|k+1 (x) · f k+1|k (x|Z k ).

(26.39)

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Figure 26.2 The structure of the PHD filter approximation for the simplified single nonideal-sensor case.

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• Optimization: Determine the best next sensor state: nonideal

∗

xk+1 = arg sup N

∗

(26.40)

k+1|k+1 (x).

∗

x ∗

∗

∗

∗

∗

∗

• Sensor measurement-update: Update f k+1|k (x|Z k ) to f k+1|k+1 (x|Z k+1 ) using Bayes’ rule. ∗

• Target measurement-update: Employing xk+1 as the sensor state at time ∗ ∗ tk+1 , update Dk+1|k (x|Z (k) , X k ) to Dk+1|k+1 (x|Z (k+1) , X k+1 ) using the PHD filter corrector equations. • Repeat. The section is organized as follows: 1. Section 26.3.1.1: The filtering equations for single-sensor, single-step control using PHD filters. 2. Section 26.3.1.2: Hedging using multisample hedging or single-sample PIMS hedging and the PENT objective function. 3. Section 26.3.1.3: Formulas for ideal and nonideal sensor dynamics for the single-sensor, single-step PENT objective function for PHD filters. 4. Section 26.3.1.4: A simple example using a sensor with a “cookie cutter” FoV and no clutter. 5. Section 26.3.1.5: A simple example using a sensor with a cookie cutter FoV, clutter, and perfect sensor resolution. 6. Section 26.3.1.6: The formulas for the PENTI objective function for singlesensor, single-step control using PHD filters. 26.3.1.1

Single-Sensor, Single-Step PHD Filter Control: Filtering Equations

From (8.15) and (8.50), the following are the filtering equations for the approach: • Sensor time-update: ∗

∗

∗

k

f k+1|k (x|Z ) =

∫

∗

∗

∗

∗

∗

∗

∗

f k+1 (x|x′ ) · f k|k (x′ |Z k )dx′ .

(26.41)

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• Target time-update (neglecting target-spawning): ∗

Dk+1|k (x|Z (k) , X k )

=

bk+1|k (x) ∫ + pS (x′ ) · fk+1|k (x|x′ )

(26.42)

∗

·Dk|k (x′ |Z (k) , X k )dx′ . • Sensor measurement-update: ∗

∗

∗

∗

∗

∗

∗

∗

∗

f k+1|k+1 (x|Z k+1 ) ∝ f k+1 (zk+1 |x) · f k+1|k (x|Z k ). • Target measurement-update: ∗

∗

Dk+1|k+1 (x|Z (k) , X k , Zk+1 , xk+1 )

(26.43)

∗

∗

LZk+1 (x, xk+1 ) · Dk+1|k (x|Z (k) , X k )

= where ∗

LZk+1 (x, xk+1 )

∗

τk+1 (z|xk+1 )

∗

1 − pD (x, xk+1 ) (26.44) ∗ ∗ ∑ pD (x, xk+1 ) · Lz (x, xk+1 ) + ∗ ∗ κ (z|xk+1 ) + τk+1 (z|xk+1 ) z∈Zk+1 k+1 ∫ ∗ ∗ pD (x, xk+1 ) · Lz (x, xk+1 ) (26.45)

=

=

∗

·Dk+1|k (x|Z (k) , X k )dx and where ∗

∗

Lz (x, xk+1 ) abbr. = fk+1 (z|x, xk+1 ). 26.3.1.2

(26.46)

Single-Sensor, Single-Step PHD Filter Control: Hedging ∗

∗

Integration of Dk+1|k+1 (x|Z (k) , X k , Zk+1 , xk+1 ) with respect to x yields the posterior expected number of targets ∗

∗

Nk+1|k+1 (x, Z) = Nk+1|k (x) +

∑ z∈Z

∗

τk+1 (z|x) ∗ ∗ κk+1 (z|x) + τk+1 (z|x)

(26.47)

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where ∗

Nk+1|k (x) =

∫

∗

∗

(1 − pD (x, x)) · Dk+1|k (x|Z (k) , X k )dx.

(26.48)

Equation (26.47) must be hedged against the unknowable future measurement set Z. Multisample hedging can be employed as described in (23.4). Here, however, the PIMS single-sample hedging approach of Section 25.10.2 is adapted to current assumptions as follows. ˆ = {x1 , ..., xn } be the set of predicted target states. Then we must Let X IM S determine the PIMS-hedged posterior p.g.fl. GP k+1|k+1 [h] as defined in (25.135) and (25.136). From (25.156) we know that this is IM S τ0 [h−1] GP k+1|k+1 [h] = e

n ∏ κk+1 (ηk+1 (xi )) + pD (xi ) · τi [h] κ (η (x )) + pD (xi ) · τi [1] i=1 k+1 k+1 i

(26.49)

where τ0 [h]

=

∫

h(x) · (1 − pD (x)) · Dk+1|k (x)dx

(26.50)

τi [h]

=

∫

h(x) · pD (x) · Lηk+1 (xi ) (x) · Dk+1|k (x)dx.

(26.51)

The corresponding p.g.f. is IM S GP k+1|k+1 (x)

=e

(x−1)·τ0 [1]

n ∏ κk+1 (ηk+1 (xi )) + x · pD (xi ) · τi [1] κk+1 (ηk+1 (xi )) + pD (xi ) · τi [1] i=1

(26.52)

and so the PIMS-hedged posterior expected number of targets (PENT) is the expected value P IM S

P IM S Nk+1|k+1

= =

G

(1) k+1|k+1 (1) n ∑

τ0 [1] +

i=1

26.3.1.3

pD (xi ) · τi [1] . κk+1 (ηk+1 (xi )) + pD (xi ) · τi [1]

(26.53) (26.54)

Single-Sensor, Single-Step PHD Filter Control: PENT

Expressed in our current notation, the predicted measurement ηk+1 (xi ) actually ∗ has the form ηk+1 (xi |x). Thus (26.54) becomes:

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• Single-step look-ahead PENT for a single sensor with ideal sensor dynamics (PHD filter): ideal

∗

∗

N k+1|k+1 (x) = Nk+1|k (x) +

n ∑ i=1

∗

∗

pD (xi , x) · τi (x) ∗ ∗ κi (x) + pD (xi , x) · τi (x) ∗

(26.55)

where ∗

Nk+1|k (x) ∗

τi (x)

=

∫

=

∫

∗

∗

(1 − pD (x, x)) · Dk+1|k (x|Z (k) , X k )dx ∗

∗

(26.56) (26.57)

∗ (x, x) pD (x, x) · Lηk+1 (xi ,x) ∗

·Dk+1|k (x|Z (k) , X k )dx ∗

κi (x)

=

∗

∗

(26.58)

κk+1 (ηk+1 (xi , x)|x).

As is indicated by the notation, (26.55) is the formula for PENT assuming ideal sensor dynamics. From (25.212), the corresponding formula for nonideal dynamics is • Single-step look-ahead PENT for a single sensor with nonideal sensor dynamics (PHD filter): nonideal

N

∗

k+1|k+1 (x)

ideal

∗

∗

∗

∗

= N k+1|k+1 (x) · f k+1|k (x|Z k ). nonideal

(26.59)

∗

That is, optimization consists of maximizing N k+1|k+1 (x), subject to the ∗ constraint that the next sensor state x must be reachable. Remark 93 (PENT with a “cookie cutter” FoV in no clutter) Suppose that the ∗ sensor FoV is a cookie clutter, pD (xi , x) = 1S (x), for some S ⊆ X. Also suppose that there is almost no clutter—that is, κk+1 (z) > 0 but κk+1 (z) ∼ =0 identically. Then—despite the conclusions of Remark 92— (26.55) can be written as ∑ ∗ ∗ P IM S Nk+1|k+1 (x) = Nk+1|k (x) + 1S (xi ). (26.60) i=1

For, suppose that ∗

pD (xi , x) = 1S (xi ) · (1 − ε) + (1 − 1S (xi )) · ε

(26.61)

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967

∗

for some arbitrarily small number ε. Then if xi ∈ / S then pD (xi , x) = ε and as ε → 0, ∗

∗

∗

pD (xi , x) · τi (x) ε · τi (x) ∼ ∼ = = 0 = 1S (xi ). ∗ ∗ ∗ ∗ κi (x) + pD (xi , x) · τi (x) κi (x)

(26.62)

Otherwise, if xi ∈ S then as ε → 0, ∗

∗

∗

∗

pD (xi , x) · τi (x) pD (xi , x) · τi (x) = = 1 = 1S (xi ). ∗ ∗ ∗ ∗ κi (x) + pD (xi , x) · τi (x) pD (xi , x) · τi (x) ∗

(26.63)

Remark 94 (New versus old formulas for PENT) Equation (26.55) should be compared to the expression that was derived in [169] (the unnumbered equation at bottom of p. 277), using a different approach: ∗

∗

P IM S Nk+1|k+1 (x) = Nk+1|k (x) +

n ∗ ∗ ∑ pD (xi , x) · τi (x) ∗

∗

.

(26.64)

κi (x) + τi (x)

i=1

Equation (26.55), which follows from the general analysis of Section 25.10.2, ∗ ∗ (x) is a cookie appears to be less heuristic than (26.64). If pD (xi , x) = 1S(x) cutter, however, then the two formulas are identical. Remark 95 (PENT and information theory) This point was previously noted in Section 25.9.4, but is important enough to be emphasized again. For c(x) = 1 − x + x log x (Kullback-Leibler cross-entropy), we know from (6.79) that Ic (fk+1|k+1 ; fk+1|k )

=

Nk+1|k  ·

∼ =

c

(

Nk+1|k+1 Nk+1|k

)

( )  · Ic sk+1|k+1 ; sk+1|k ( ) Nk+1|k+1 Nk+1|k · c . (26.66) Nk+1|k +

Nk+1|k+1 Nk+1|k

Similarly, from (6.85) we know that the R´enyi α-divergence has the form ( ) Nk+1|k+1 Rα (fk+1|k+1 ; fk+1|k ) = αNk+1|k · c Nk+1|k

∼ =

(26.65) 

(26.67)

1−α α +αNk+1|k+1 Nk+1|k · Ic (sk+1|k+1 ; sk+1|k ) ( ) Nk+1|k+1 α · Nk+1|k · c (26.68) Nk+1|k

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where c(x) = α−1 (1 − α) 26.3.1.4

−1

· (αx + 1 − α − xα ).

Single-Sensor, Single-Step PHD Filter Control Using PENT: Simple Example

This example, adapted from [169], pp. 278-279, demonstrates that PENT with PIMS single-sample hedging behaves in an intuitively reasonable manner. Suppose that: • Measurements are states (Z = X) and that the measurement function is η(x) = x. • There is almost no clutter (κk+1 (z) ∼ = 0 identically). • The sensor FoV is a cookie cutter ∗

(26.69)

pD (x, x) = 1S (x)

where S ⊆ X is a (hyper)spherical subset of state space that can be translated to any location. • The predicted PHD has the form Dk+1|k (x)

=

n ∑

(26.70)

ωi · si (x)

i=1

Nk+1|k

=

∫

Dk+1|k (x)dx =

n ∑

ωi def. =ω

(26.71)

i=1

where si (x) is the spatial density of the ith track, where 0 ≤ ωi ≤ 1 is its existence probability (its “firmness”), and where the MAP estimate of si (x) is xi . • The tracks are sufficiently well-localized that they can be completely encompassed ∫ by the FoV—that is, for all i = 1, ..., n, there is a choice of S such that S si (x)dx = 1. First note that ∫ ∫ ∗ pD (x, x) · si (x)dx = 1S (x) · si (x)dx = pi (S)

(26.72)

is the amount of probability mass of the ith predicted track that is contained in S. That is, pi (S) is a measure of how well the track xi has been located by the sensor FoV. It is 1 if it has been well-located, and small if otherwise.

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Second, note that (26.56) becomes ∗

Nk+1|k (S) abbr. = Nk+1|k (x) = ω −

n ∑

ωi · pi (S).

(26.73)

i=1

Third, note that (26.55) becomes, given the discussion in Remark 93 of Section 26.3.1.3, (26.74)

Nk+1|k+1 =

∗

Nk+1|k (x) +

n ∑ i=1

=

=

ω−

n ∑

ω−

i=1 n ∑

∗

∗

pD (xi , x) · τk+1 (xi |x) ∗ ∗ κk+1 (xi ) + pD (xi , x) · τk+1 (xi |x)

ωi · pi (S) +

n ∑ i=1

1S (xi ) · τ (xi |S) κk+1 (xi ) + 1S (xi ) · τ (xi |S)

(26.75)

(26.76)

ωi · pi (S)

i=1

+

n ∑

1S (xi ) ·

i=1

=

ω−

n ∑

τ (xi |S) κk+1 (xi ) + τ (xi |S)

ωi · pi (S) +

i=1

n ∑

1S (xi ).

(26.77)

i=1

Now consider the following possible placements of the field of view S: • S is in free space—that is, xi ∈ / S for all i. Then pi (S) = 0 for all i = 1, ..., n and so the corresponding value of PENT is Nk+1|k+1 = ω.

(26.78)

• S is placed over x1 and no other track. Then 1S (xi ) = 0 and pi (S) = 0 for all i ̸= 1, whereas 1S (x1 ) = 1. In this case PENT has the form Nk+1|k+1

= ≥

ω − ω1 · p1 (S) + 1 ω.

(26.79) (26.80)

Thus Nk+1|k+1 is maximized by placing S over some track rather than somewhere over free space. Furthermore, Nk+1|k+1 is maximized by choosing that track such that the product ω1 · p1 (S) is minimized. In other words:

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– More firm and more well-localized tracks (tracks with larger ω1 and larger p1 (S)) should be ignored in favor of less firm and less localized tracks. – This is what one would hope to see, since total information is not increased by placing the FoV over tracks for which one already possesses sufficient information. – However, the choice depends on a balance between the track’s firmness ω1 and its degree of localization p1 (S). The FoV may be placed over a relatively firm track (ω1 ∼ = 1) if it is sufficiently poorly localized (p1 (S) ∼ = 0), and vice versa. Yet, broadly speaking, the FoV will be placed over the track that is simultaneously least firm and least localized. • S is placed over two tracks x1 and x2 but no others. In this case PENT becomes Nk+1|k+1

ω − ω1 · p1 (S) − ω2 · p2 (S) + 2 ω − ω1 · p1 (S) + 1.

= ≥

(26.81) (26.82)

That is, Nk+1|k+1 is maximized by placing it over two tracks rather than a single track. More generally, it will be maximized by placing it over as many tracks as possible. Remark 96 (PIMS hedging versus PMS hedging) Compare these results with those obtained if, rather than PIMS-based hedging, we instead employed hedging using the predicted measurement set (PMS) of Section 25.10.1. In this case PENT would have the form Nk+1|k+1

=

Dk+1|k [1 − pD ] +

n ∑

Dk+1|k [1 − pD ] +

n ∑ τk+1 (xi |S)

i=1

=

i=1

=

ω−

n ∑

τk+1 (xi |S) (26.83) κk+1 (xi ) + τk+1 (xi |S) (26.84)

τk+1 (xi |S)

ωi · pi (S) + n.

(26.85)

i=1

Thus: if

S is in free space we get Nk+1|k+1 = ω + n

(26.86)

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whereas if S is placed over x1 we get a smaller—rather than the desired larger— value: Nk+1|k+1 = ω − ω1 · p1 (S) + n ≤ ω + n. (26.87) Likewise, if S is placed over x1 and x2 then we again get a smaller value: Nk+1|k+1 = ω − ω1 · p1 (S) − ω2 · p2 (S) + n ≤ ω − ω1 · p1 (S) + n.

(26.88)

Thus unlike PIMS-hedged PENT, PMS-hedged PENT does not exhibit good sensormanagement behavior. However, this comment is no longer true if the sensor is high-resolution—see the continuation of the example in the next section. 26.3.1.5

Single-Sensor, Single-Step PHD Filter Control Using PENT: Simple Example, Continued

Now, change the example by assuming: • There is significant clutter, with an intensity function κk+1 (x|S) = κk+1 (x) that is independent of the sensor state (recall that z = x). • The likelihood function has the form Lz (x) = δz (x)—that is, the sensor has perfect resolution. In this case, (26.77) becomes Nk+1|k+1 = ω −

n ∑

ωi · pi (S) +

i=1

n ∑

1S (xi ) ·

i=1

τ (xi |S) κk+1 (xi ) + τ (xi |S)

(26.89)

where τ (xi |S)

= =

∫

pD (x) · Lxi (x) · Dk+1|k (x)dx

n ∑

ωl

∫

1S (x) · δxi (x) · sl (x)dx

(26.90) (26.91)

l=1

=

1S (xi )

n ∑

ωi · sl (xi ) = 1S (xi ) · Dk+1|k (xi ).

(26.92)

l=1

Let ρi =

Dk+1|k (xi ) , κk+1 (xi )

(26.93)

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which measures the degree to which the measurement xi is target-like rather than clutter-like. Then Nk+1|k+1 = ω −

n ∑

ωi · pi (S) +

i=1

n ∑

1S (xi ) ·

ρi . 1 + ρi

(26.94)

i=1

If S is placed over free space, then as in the no-clutter case we get Nk+1|k+1 = ω. Now, place S over x1 and no other track. Then Nk+1|k+1 = ω − ω1 · p1 (S) +

ρ1 . 1 + ρ1

(26.95)

This will exceed ω only if ρ1 > ω1 · p1 (S). 1 + ρ1

(26.96)

Thus: • Nk+1|k+1 will be maximized by placing S over a track that is: – Simultaneously not firm or well-localized (ω1 · p1 (S) is small). – In a low-clutter region (ρi is large). – High-clutter regions tend to be treated as low-information regions. • If all tracks are too firm, too well-localized, and in higher-clutter regions, then Nk+1|k+1 will be maximized by placing S over free space. 26.3.1.6

Single-Sensor, Single-Step PHD Filter Control: PENTI

Let ιk+1|k+1 (x) be a tactical significance function (TIF) at time tk+1 , as defined in Section 25.14.2. For a given TIF, the general formula for the posterior expected number of targets of interest (PENTI) objective function was given in (25.238). The purpose of this section is to show that, for sensor management, the PHD-filter formula for PENTI is ideal

∗

∗

→ι N →ι k+1|k+1 (x) = Nk+1|k (x) +

n ∑ i=1

∗

∗

pD (xi , x) · τi→ (x) ∗ ∗ κi (x) + pD (xi , x) · τi (x) ∗

(26.97)

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where ∗

→ι Nk+1|k (x)

=

∫

=

·Dk+1|k (x|Z (k) , X k )dx ∫ ∗ ∗ ∗ (x, x) ιk+1|k+1 (x) · pD (x, x) · Lηk+1 (xi ,x)

∗

(26.98)

ιk+1|k+1 (x) · (1 − pD (x, x)) ∗

∗

τi→ι (x)

(26.99)

∗

·Dk+1|k (x|Z (k) , X k )dx and ∗

τi (x)

=

∫

∗

∗

∗ (x, x) pD (x, x) · Lηk+1 (xi ,x)

(26.100)

∗

·Dk+1|k (x|Z (k) , X k )dx ∗

κi (x)

=

∗

∗

κk+1 (ηk+1 (xi , x)|x).

For, from (25.238), PENTI is given by ∫ ideal ∗ ∗ P IM S N →ι ( x) = ιk|k (x) · Dk+1|k+1 (x|x)dx k+1|k+1

(26.101)

(26.102)

where, by (4.75), ∗

P IM S Dk+1|k+1 (x|x) =

IM S δ log GP k+1|k+1

[1]. (26.103) δx Thus by (3.31), the integral of the functional derivative can be written as a Gˆateaux derivative: ∫ IM S δ log GP ideal k+1|k+1 ∗ →ι N k+1|k+1 (x) = ιk|k (x) · [1]dx (26.104) δx IM S ∂ log GP k+1|k+1 = [1]. (26.105) ∂ιk|k IM S But GP k+1|k+1 [h] was given in (26.49), which leads to IM S log GP k+1|k+1 [h]

=

τ0 [h − 1] +

n ∑ i=1

log

(26.106) κk+1 (ηk+1 (xi )) + pD (xi ) · τi [h] . κk+1 (ηk+1 (xi )) + pD (xi ) · τi [1]

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Thus from (26.50) and (26.51), IM S ∂ log GP k+1|k+1

[h] = τ0 [ιk|k ] + ∂ιk|k

n ∑ i=1

pD (xi ) · τi [ιk|k ] (26.107) κk+1 (ηk+1 (xi )) + pD (xi ) · τi [h]

and so ideal

∗

N →ι k+1|k+1 (x) = τ0 [ιk|k ] +

n ∑ i=1

pD (xi ) · τi [ιk|k ] κk+1 (ηk+1 (xi )) + pD (xi ) · τi [1]

(26.108)

from which (26.97) follows. 26.3.2

PHD Filter Sensor Management: Multisensor Single-Step

This section addresses the situation in which multiple sensors are used to achieve single-step look-ahead control. The approach generalizes and refines the one proposed in [169], pp. 279-280. It is based on the idea of approximating multiple sensors as a single “pseudosensor.” Assume the simplified approach for nonideal sensors described in Section 25.13. Let: • Z (k) : Z1 , ..., Zk be the time sequence of measurement sets at time tk . ∗j

∗j

∗j

∗j

∗j

• Z k : z 1 , ..., z k be the time sequence of actuator measurement sets for the jth sensor at time tk . ∗j

• X k : x 1 , ..., x k be a time sequence of sensor state sets for the jth sensor ∗j at time tk , with x i to be selected at time ti . ∗

∗

∗

• X (k) : X 1 , ..., X k be the time sequence of multisensor state sets for all ∗ ∗1 ∗s sensors at time tk , where X i = { x i , ..., x i }.

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Then the PHD filter approximation of the simplified nonideal-sensor case in Section 25.13 has the following structure: sensor & multitarget measurement-updates ∗j ∗j ∗j

... →

f k|k ( x|Z k )

... →

Dk|k (x|Z (k) , X (k) )

∗

∗j

∗j ∗j

f k+1|k ( x|Z k ) ↓

→

time-projection ∗

Dk+1|k+1 (x|Z (k) , X (k) , ∗ Z, X)

→

hedging

↓ select next multisensor state set nonideal

∗

∗

X k+1 = arg sup ∗ Nk+1|k+1 (X) X sensor & multitarget time- & measurement-updates ∗j ∗j ∗j

f k+1|k+1 ( x|Z k+1 )

↗

∗j

z k+1

→ ...

∗

Dk+1|k+1 (x|Z (k+1) , X (k+1) )

→ ∗

X k+1 ,Zk+1

→ ...

↗

That is, it consists of the following sequence of steps: ∗j

∗j ∗j

• Sensor time-update: Extrapolate the sensor-state distributions f k|k ( x|Z k ) ∗j

∗j ∗j

to f k+1|k ( x|Z k ) using the respective prediction integrals. ∗

• Target time-update: Extrapolate the current PHD Dk|k (x|Z (k) , X k ) Dk+1|k (x|Z

(k)

∗

to

k

, X ) using the PHD filter prediction equations. ∗

• Pseudosensor approximation: For purposes of choosing the next set X k+1 of sensor states, approximate the currently available sensors as a single “pseudosensor” (see Sections 26.3.2.2 and 26.3.2.3).

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∗

• Time-projection: Given this approximation, update Dk+1|k (x|Z (k) , X (k) ) ∗

∗

to Dk+1|k+1 (x|Z (k) , X (k) , Z, X) using the PHD filter corrector equations— ∗

given the yet-to-be-determined multisensor state set X at time tk+1 and the unknowable target measurement set Z at time tk+1 . ∗

• Objectivization: Given X and Z, determine the posterior expected number of targets (PENT) ∫ ∗ ∗ ∗ Nk+1|k+1 (X, Z) = Dk+1|k+1 (x|Z (k) , X k , Z, X)dx. (26.109) • Hedging: Use multisample hedging or single-sample PIMS hedging to eliminate Z: ideal

∗

∗

P IM S N k+1|k+1 (X) = Nk+1|k+1 (X, Zk+1 ).

(26.110)

• Dynamicization: Modify PENT to account for sensor dynamics: nonideal

N

ideal

∗

k+1|k+1 (X) ∗

∗1

∗

∗

∗

∗

= N k+1|k+1 (X) · f k+1|k (X|Z (k) )

∗s

(26.111)

∗

where, if X = { x, ..., x} with |X| = s, ∗

∗

∗

f k+1|k (X|Z (k) ) = s! · δ

∗

s,|X|

s ∗j ∏ ∗j ∗j f k+1|k ( x|Z k ).

(26.112)

j=1

• Optimization: Determine the best set of future sensor states: nonideal

∗

X k+1 = arg sup N

∗

(26.113)

k+1|k+1 (X).

∗

X ∗j

∗

∗j

∗j

∗

∗j

• Sensor measurement-updates: Update f k+1|k (x|Z k ) to f k+1|k+1 (x|Z k+1 ) using Bayes’ rule. ∗

• Target measurement-update: Employing X k+1 as the set of sensor states at ∗ ∗ time tk+1 , update Dk+1|k (x|Z (k) , X k ) to Dk+1|k+1 (x|Z (k+1) , X k+1 ) using the measurement-update equations for one of the multisensor PHD filter approaches described in Chapter 10.

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• Repeat. The section is organized as follows: 1. Section 26.3.2.1: The filtering equations for the approach. 2. Section 26.3.2.2: The two-sensor case of the pseudosensor approximation. 3. Section 26.3.2.3: The pseudosensor approximation for an arbitrary number of sensors. 4. Section 26.3.2.4: Optimization-hedging using the PIMS-hedged PENT objective function. 5. Section 26.3.2.5: A simple example of PENT-based control with PHD filters. 6. Section 26.3.2.6: Optimization using the corresponding PIMS-hedged PENTI objective function. 26.3.2.1

Multisensor, Single-Step PHD Filter Control: Filtering Equations

The filtering equations for the approach are as follows: • Sensor time-updates: ∗j

∗j ∗j

f k+1|k ( x|Z k ) =

∫

∗j

∗j ∗j

∗j

∗j

∗j

∗

f k+1 ( x| x ′ ) · f k|k ( x ′ |Z k )dx′ .

(26.114)

• Target time-update (neglecting target spawning): Dk+1|k (x|Z

(k)

∗

,X

(k)

)

=

bk+1|k (x) +

∫

pS (x′ )

(26.115) ∗

·fk+1|k (x|x′ ) · Dk|k (x′ |Z (k) , X (k) )dx′ . • Sensor measurement-updates: ∗j

∗j ∗j

∗j

∗j

∗j

∗j

∗j ∗j

f k+1|k+1 ( x|Z k+1 ) ∝ f k+1 ( z k+1 | x) · f k+1|k ( x|Z k ).

(26.116)

• Target measurement-update: measurement-update equations for one of the multisensor PHD filter approaches of Chapter 10, or the pseudosensor approximation.

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Advances in Statistical Multisource-Multitarget Information Fusion

26.3.2.2

Multisensor, Single-Step PHD Filter Control: Pseudosensor

It was remarked earlier, in Sections 10.4 and 10.3, that a strictly rigorous development of multisensor PHD filters leads to combinatorily complex formulas. The “parallel combination” approach described in Section 10.6 is tractable, but perhaps too complicated for the purpose of sensor management. The iterated-corrector approach in Section 10.5 is simple and tractable, but has limitations when the probabilities of detection of the sensors are too different. Consequently, a different, approximate, approach is needed for sensor management (although not for the multitarget detection and tracking part of sensor management). Such an approach, the basic elements of which were first proposed in [169], pp. 271-276, was introduced in Section 23.3.5 and is described in more detail in this section. The fundamental concept is this: • Model multiple sensors as a single fictitious “pseudosensor,” using of a suitably-defined probability of detection and likelihood function. • Directly apply the single-sensor sensor management approach of Section 26.3.1 to the multisensor case. For conceptual clarity, begin with the two-sensor case. Suppose that we have • A single target. 1

∗1 ∗2

2

• Two sensors with measurement spaces Z,Z and sensor state spaces X,X. ∗1

∗2

• Respective sensor probabilities of detection pD (x, x) and pD (x, x). • Respective sensor likelihood functions 1 1

∗1

f k+1 (z|x, x) 2 2

∗2

f k+1 (z|x, x)

1

∗1

=

Lz1 (x, x)

=

Lz2 (x, x)

2

(26.117)

∗2

(26.118)

• Both sensors are clutter-free. 12

At time tk+1 , the two sensors collect a joint measurement set Z from the target. There are four possibilities: 12

• Z = ∅ (neither sensor detected the target).

Approximate Sensor Management

12

979

1

• Z = {z} (the first sensor detected the target but the second did not). 12

2

• Z = {z} (the second sensor detected the target but the first did not). 12

1

2

• Z = {(z, z)} (both sensors detected the target). Given that there is a single target and no clutter, the two sensors behave like a single sensor—the “pseudosensor.” This sensor is described by the following models: • Pseudosensor measurement space: 12

1

2

1

2

(26.119)

Z = Z ⊎ Z ⊎ (Z × Z) with integral defined as ∫ ∫ ∫ ∫ 12 1 2 12 12 12 1 12 2 g ( z )d z = 1 g (z)dz + 2 g (z)dz + 1 Z

Z

12

1

2

1

2

(26.120)

g (z, z)dzdz. 2

Z×Z

• Pseudosensor probability of detection (the probability that at least one of the sensors will detect the target): 12

∗1 ∗2

p D (x, x, x)

∗1

∗2

=

1 − (1 − pD (x, x))(1 − pD (x, x)) (26.121)

=

pD (x, x) + pD (x, x)

∗1

∗2

∗1

(26.122)

∗2

−pD (x, x) · pD (x, x). • Pseudosensor likelihood function: ∗1

12

∗1 ∗2

L z1 (x, x, x)

=

∗2

pD (x, x) · (1 − pD (x, x)) ∗1 ∗2

12

p D (x, x, x) ∗1

12

∗1 ∗2

L z2 (x, x, x)

=

1

∗2

(1 − pD (x, x)) · pD (x, x)) ∗1 ∗2

12

∗1

· Lz1 (x, x)(26.123) (26.124)

p D (x, x, x) 2

∗2

·Lz2 (x, x) ∗1

12

∗1 ∗2

L (z, 1 2 (x, x, x) z)

=

∗2

pD (x, x) · pD (x, x)) ∗1 ∗2

12

p D (x, x, x) 1

∗1

2

∗2

·Lz1 (x, x) · Lz2 (x, x).

(26.125)

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Advances in Statistical Multisource-Multitarget Information Fusion

The likelihood function is well defined since, as is easily verified, ∫

12

∗1 ∗2

12

L 12 (x, x, x)d z = 1 z

∗1 ∗2

for every x, x, x. Now suppose that: • The resolution of the sensors is good. • There are multiple targets that are well separated. • The sensors are corrupted by sparse clutter. In this case, the target-generated measurements still behave as though generated by the pseudosensor. For example, if the measurements are positions and if both sensors collect measurements from a target, then these measurements will typically appear as pairs located near the targets and will be clearly associated with them. Consequently, the pseudosensor model will serve as a reasonable approximation of the multisensor-multitarget system, as long as targets are not too close together and clutter is not too dense. This is what is meant by the terminology, “pseudosensor approximation.” While this approximation would clearly be inadequate for the purpose of multitarget tracking, it will suffice for the purpose of reducing computational complexity for sensor management. However, since this approximation also becomes combinatorial as the number of sensors increases, additional approximations will be necessary. 26.3.2.3

Multisensor, Single-Step PHD Filter Control: General Pseudosensor Approximation

These definitions can be generalized to more than two sensors in the obvious fashion. Let the likelihood function for the jth sensor be j

∗j

j j

∗j

Lj (x, x) = f k+1 (z|x, x) z

∗j

and let its FoV be pD (x, x). Then

(26.126)

Approximate Sensor Management

• General pseudosensor measurement space:    ⊎ j ⊎ 1..s Z =  Z ⊎ 

j1

1≤j1