Control, Computation and Information Systems: First International Conference on Logic, Information, Control and Computation, ICLICC 2011, Gandhigram, ... in Computer and Information Science, 140) 3642192629, 9783642192623

Table of contents :
Title Page
Preface
Organization
Table of Contents
1. Control Theory and Its Applications
Existence and Uniqueness Results for Impulsive Functional Integro-Differential Inclusions with Infinite Delay
Introduction
Preliminaries
Existence Result
References
An Improved Delay-Dependent Robust Stability Criterion for Uncertain Neutral Systems with Time-Varying Delays
System Description and Problem Statement
MainResult
Numerical Example
Conclusion
References
Design of PID Controller for Unstable System
Introduction
Need for Controller Tuning
Ziegler Nichols (Z-N) [1, 2]
Chien Hrones Reswick (CHR) [3]
Genetic Algorithm (GA) [6, 7]
Proposed PID Controller Design for Unstable Electronic Circuit
Conclusion
References
New Delay-Dependent Stability Criteria for Stochastic TS Fuzzy Systems with Time-Varying Delays
Introduction
Problem Formulation and Preliminaries
MainResults
Numerical Simulation
Conclusion
References
H∞ Fuzzy Control of Markovian Jump Nonlinear Systems with Time Varying Delay
Introduction
Model Formulation
MainResults
Numerical Simulations
Conclusion
References
Assessing Morningness of a Group of People byUsing Fuzzy Expert System and Adaptive Neuro Fuzzy Inference Model
Introduction
Mode of Data Collection
Designing Fuzzy Expert System
The ANFIS Model Formulation
Conclusions
References
Optimal Control for Navier-Stokes Takagi-Sugeno Fuzzy Equations Using Simulink
Introduction
Statement of the Problem
Simulink Solution of MRDE
Procedure for Simulink Solution
Numerical Example
Solution Obtained Using Simulink
Conclusion
References
Multi-objective Optimization in VLSI Floorplanning
Introduction
VLSI Floorplanning Problem
Objective Function and Constraint
B*tree Encoding in AMOSA
B*tree Coded Archived Multiobjective Simulated Annealing Algorithm(AMOSA)
AMOSA Algorithm
Cooling Schedule and Local Search
Creation of Neighboring Solution
Obtaining the True Pareto-Front
Test Results
Parameters, Results and Discussion
Conclusion
References
Approximate Controllability of Fractional Order Semilinear Delay Systems
Introduction
Preliminaries
Controllability Results
Example
References
2. Computational Mathematics
Using Genetic Algorithm for Solving Linear Multilevel Programming Problems via Fuzzy Goal Programming
Introduction
Formulation of MLPP
GA Scheme for MLPP
FGP Problem Formulation
Construction of Membership Functions
FGP Model Formulation
An Illustrative Example
Conclusion
References
Intuitionistic Fuzzy Fractals on Complete and Compact Spaces
Introduction
Metric Fractals
Intuitionistic Fuzzy Metric Space
Hausdorff Intuitionistic Fuzzy Metric Space
Fractals in the Intuitionistic Fuzzy Metric Space
Conclusion
References
Research on Multi-evidence Combination Based on Mahalanobis Distance Weight Coefficients
Introduction
An Overview of Dempster Theory and Some Improvement
Improved Combination Method Based on Mahalanobis Distance Weight Coefficients
Simulation Analysis
If Data Provided Regular
If Interference Occurs in the Second Sensor
If Interferences Occur in Both the Second and Fifth Sensor
Conclusions
References
Mode Based K-Means Algorithm with Residual Vector Quantization for Compressing Images
Introduction
Mode Based K-Means with RVQ
Algorithm
Results and Discussion
Conclusion
References
Approximation Studies Using Fuzzy Logic in Image Denoising Process
Introduction
Interpolation Using Fuzzy Systems
Proposed Algorithm
Definition (Distance Measure in Colour Space)
Definition (Weight in Fuzzy Noise Reduction) [4, 9]
Weights in RGB Colour Space
Fuzzy Optimization
Optimization Problem
Peak Signal–to-Noise Ratio (PSNR) [9, 12]
Experimental Studies
Conclusion
References
An Accelerated Approach of Template Matching for Rotation, Scale and Illumination Invariance
Introduction
Proposed Methodology
Multi-resolution Based Wavelet Decomposition
Zernike Moments
Improving Speed of the Zernike Moments Calculation
Matching Based on Zernike Moments
Experimental Results
Test Images
Conclusion
References
Image Edge and Contrast Enhancement Using Unsharp Masking and Constrained Histogram Equalization
Introduction
Basic Histogram Equalization
The Proposed Method for Edge and Contrast Enhancement
Modified HE Transformation Function
Weighing and Thresholding
Unsharp Masking
Results and Discussion
Conclusion
References
Application of MOPSO and MOCLPSO for the Optimal Placement of TCSCs
Introduction
Problem Formulation
Objectives for the Problem
Application of MOPSO and MOCLPSO for the Multi-Objective Optimization Problem
TOPSIS Method of Decision Making
Numerical Results and Discussion
IEEE 14 Bus System
IEEE 118 Bus System
Conclusion
References
Torsional Body Forces in a Kelvin-Voigt-Type Visco-Elastic Half Space
Introduction
Formulation of Problem
Method of Solution
Decaying Torsional Body Force
Special Case
Numerical Results and Discussions
Conclusion
References
Medical Image Binarization Using Square Wave Representation
Introduction
Methods and Materials
Thresholding Performance Evaluation
Materials
Results and Discussion
Conclusion
References
Solving Two Stage Transportation Problems
Introduction
Preliminaries
A Procedure for Finding an Optimal Solution to Two Stage Transportation Problem
Conclusion
References
Tumor Growth in the Fractal Space-Time with Temporal Density
Introduction
Methods
Results and Discussions
Conclusion
References
The Use of Chance Constrained Fuzzy Goal Programming for Long-Range Land Allocation Planning in Agricultural System
Introduction
Problem Formulation
Construction of Membership Goals of Fuzzy Goals
Deterministic Equivalent of Chance Constraints
FGP Model Formulation
FGP Model Formulation of the Problem
Description of Fuzzy Goals and Chance Constraints
Description of Chance Constraints
An Illustrative Example: A Case Study
Conclusion
References
A Fuzzy Goal Programming Method for Solving Chance Constrained Programming with Fuzzy Parameters
Introduction
Fuzzy CCP (FCCP) Model Formulation
Formulation of FP Model
Lower Decomposed Sub Problem (LDSP)
Upper Decomposed Sub Problem (UDSP)
Construction of Membership Functions
FGP Model Formulation
An Illustrative Example
Conclusions
References
Fractals via Ishikawa Iteration
Introduction
Preliminaries
Experiments and Results
Conclusion
References
Numerical Solution of Linear and Non-linear Singular Systems Using Single Term Haar Wavelet Series Method
Introduction
STHWS Technique
Single Term Haar Wavelet
Single Term Haar Wavelet Series
Operational Matrix of the Haar Wavelet
Analysis of STHWS Technique
Solution of Linear Singular Systems via STHWS
Solution of Non-linear Singular Systems via STHWS
Numerical Examples
Conclusions
References
3. Information Sciences
Cryptographic Image Fusion for Personal ID Image Authentication
Introduction
Previous Works on Watermarking
Proposed Algorithm
Results and Discussion
Finger Print Watermarking
Signature Watermarking
Personal Identification Marks Watermarking
Conclusion
References
Artificial Neural Network for Assessment of Grain Losses for Paddy Combine Harvester a Novel Approach
Introduction
Materials and Methods
Materials
Field Test Parameters
Machine Operational Parameters
Artificial Neural Network
Results and Discussion
Statistical Characterization of Data
Optimal Artificial Neural Network Selection
Conclusion
References
Implementation of Elman Backprop for Dynamic Power Management
Introduction
Elman Model for DPM
Rescaled Range Analysis
Elman Backprop Neural Networks
Experimental Setup
Experimentation
Results and Discussions
Conclusion
References
Comparison of Fuzzy and Neural Network Models to Diagnose Breast Cancer
Introduction
Material
Neural Network Model
Fuzzy Model
Fuzzy C-Means Algorithm
Methods
Experiment and Results
Conclusion
References
A Secure Key Distribution Protocol for Multicast Communication
Introduction
Literature Survey
Key Distribution Protocol Based on Greatest Common Divisor and Euler’s Totient Function (Proposed Method)
GC Initialization
Member Initial Join
Rekeying
Tree Based Approach
Simulation Results
Conclusion
References
Security-Enhanced Visual Cryptography Schemes Based on Recursion
Introduction
Visual Cryptography
The Existing Model
The 2-out-of-2 Threshold Construction
The Consistent Image Size Visual Cryptography Scheme
Security Analysis of the Existing VCS
Proposed Method
Experimental Results
Analysis of Experimental Results
Conclusions
References
Context-Aware Based Intelligent Surveillance System for Adaptive Alarm Services
Introduction
Related Works
Detection of a Moving Object
Face Recognition
Context-Aware
Intelligent Surveillance System Suggested
System Structure
Context Information Agent for Adaptive Alarm Services
Conclusion
References
Modified Run-Length Encoding Method and Distance Algorithm to Classify Run-Length Encoded Binary Data
Introduction
Proposed Run-Length Encoding Method
Modified RLE Algorithm
Description of the Distance Measure
Results and Discussion
Conclusion
References
A Fast Fingerprint Image Alignment Algorithms Using K-Means and Fuzzy C-Means Clustering Based Image Rotation Technique
Introduction
The Fingerprint Image and Minutia
The Importance of Fingerprint Image Alignment Algorithms
Previous Works
Registration Using Mutual Information
The Proposed Fingerprint Image Alignment Algorithms
Pre-processing the Input Fingerprint Image
K-Means Algorithm
Fuzzy C-Means Algorithm
K-Means and Fuzzy C-Means Algorithm for Fingerprint Image Rotation
Implementation Results and Analysis
Fingerprint Database Used for Evaluation
Sample Set of Results
Conclusion and Future Work
References
A Neuro Approach to Solve Lorenz System
Introduction
Statement of the Problem
Runge - Kutta Butcher Solution
Neural Network Solution
Structure of the FFNN
Results and Discussion
Non-chaotic Solution
Chaotic Solutions
References
Research on LBS-Based Context-Aware Platform for Weather Information Solution
Introduction
Background Research
LBS Application Technology
Context-Aware Technology
LBS-Based Context-Aware Platform for Weather Information Solution
Conclusion
References
A New Feature Reduction Method for Mammogram Mass Classification
Introduction
Shape and Margin Features of Mass
Experimental Results
Conclusion
References
Chaos Based Image Encryption Scheme
Introduction
Chaos and Its Unique Characteristics
Proposed Image Encryption Algorithm Based on Chaotic Map
Experimental Results
Performance Analysis
Conclusion
References
Weighted Matrix for Associating High-Level Features with Images in Web Documents for Image Retrieval
Introduction
Weighted Matrix for Measuring the Relevancy of Keywords with Images in HTML Documents
Experimental Results
Conclusion
References
Author Index

Citation preview

Communications in Computer and Information Science

140

P. Balasubramaniam (Ed.)

Control, Computation and Information Systems First International Conference on Logic, Information, Control and Computation, ICLICC 2011 Gandhigram, India, February 25-27, 2011 Proceedings

13

Volume Editor P. Balasubramaniam Department of Mathematics Gandhigram Rural Institute Deemed University Gandhigram, Tamil Nadu, India E-mail: [email protected]

ISSN 1865-0929 e-ISSN 1865-0937 ISBN 978-3-642-19262-3 e-ISBN 978-3-642-19263-0 DOI 10.1007/978-3-642-19263-0 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: Applied for CR Subject Classification (1998): F.1-2, I.2, I.4-5, H.3, J.3, C.2

© Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Computational science is a main pillar of most of the recent research, industrial and commercial activities. We should keep track of the advancements in control, computation and information sciences as these domains touch every aspect of human life. Information and communication technologies are the innovative thrust areas of research to study stability analysis and control problems related to human needs. With this in mind the International Conference on Logic, Information, Computation & Control 2011 (ICLICC 2011) was organized by the Department of Mathematics, Gandhigram Rural Institute - Deemed University, Gandhigram, Tamil Nadu, India, during February 25–27, 2011. This volume contains papers that were peer reviewed and presented at ICLICC 2011. The conference provided a platform to academics from different parts of the world, industrial experts, researchers and students to share their knowledge, discuss various issues and find solutions to problems. Based on the scientific committee’s reviews that composed of 80 researchers from all over the world we accepted only 52 papers for presentation at ICLICC 2011 out of 278 research papers submitted. Out of 19% of accepted papers only 14% of the research contributions were selected for publication. This volume comprises three parts consisting of 39 accepted papers. Control theory and its real-time applications are presented in Chap. 1. Chapter 2 consists of papers related to information sciences, especially image processing-cumcryptography, and neural networks. Finally, papers that involve mathematical computations and its applications to various fields are presented in Chap. 3.

Acknowledgements As a Conference Chair of ICLICC 2011, I thank all the national funding agencies for their grant support that contributed toward the successful completion of the international conference. ICLICC 2011 was supported in part by the following funding agencies. 1. University Grant Commission (UGC) - Special Assisstanceship Programme (SAP) - Departmental Research Support II (DRS II) & UGC - Unassigned grant, Government of India, New Delhi. 2. Council for Scientific and Industrial Research (CSIR), New Delhi, India. 3. Department of Science & Technology (DST), Government of India, New Delhi, India. I would like to express my sincere thanks to S.M. Ramasamy, Vice-Chancellor, Gandhigram Rural Institute (GRI) - Deemed University (DU), Gandhigram, Tamil Nadu, India, for his motivation and support. I also extend my profound

VI

Preface

thanks to all faculty members of the Department of Mathematics, the Department of Computer Applications, and the administrative staff members of GRI DU, Gandhigram. I especially thank the Honorary Chairs, Co-chairs, Technical Program Chairs, Organizing Chair, and all the committee members who worked as a team by investing their time to make ICLICC 2011 a great success. I am grateful to the keynote speakers who kindly accepted our invitation. Especially, I would like to thank Jong Yeoul Park (Pusan National University, Pusan, Republic of Korea), Francesco Zirilli (Univerist´ a de Roma la Sapienza, Italy), Lim Chee Peng (University Sains Malaysia, Malaysia), Ong Seng Huat, Wan Ainun M.O., Kumaresan Nallasamy, and Kurunathan Ratnavelu (Univesity of Malaya, Malaysia), for presenting plenary talks and making ICLICC 2011 a grand event. A total of 80 experts on various topics from around the world reviewed the submissions. I express my greatest gratitude for spending their valuable time to review and sort out the papers for presentation at ICLICC 2011. I thank Microsoft Research CMT for providing a wonderful tool Conference Management Toolkit for managing the papers. I also thank Alfred Hofmann for accepting our proposal to publish the papers in Springer series. Last but not the least, I express my sincere gratitude to S. Jeeva Sathya Theesar and the team of research scholars of the Department of Mathematics, GRI-DU for designing the website http://iclicc2011.co.in and managing the entire publication related work. February 2011

P. Balasubramaniam

ICLICC 2011 Organization

ICLICC 2011 was organized by the Department of Mathematics, Gandhigram Rural Institute - Deemed University, with the support of UGC - SAP (DRS II).

Organization Structure Honorary Chairs: Patron: Program Chair: Co-chairs: Technical Committee Chairs: Organizing Chair: Organizing Team:

V. Lakshmikantham, Xuerong Mao, S.K. Ntouyas, R.P. Agarwal, J.H. Kim S.M. Ramasamy (Vice Chancellor, GRI - DU) P. Balasubramaniam M.O. Wan Ainun, M. Sambandham, N.U. Ahmed S. Parthasarathy, R. Uthayakumar, R. Rajkumar P. Muthukumar J. Sahabtheen, R. Rakkiappan, R. Chandran, R. Sathy, T. Senthil Kumar, C. Vidhya, S. Lakshmanan, S. Jeeva Sathya Theesar, V. Vembarasan, G. Nagamani, M. Kalpana, D. Eswaramoorthy

Referees J.Y. Park C. Alaca M.S. Mahmoud V.N. Phat W. Feng X. Li Z. Wang C.K. Ahn H. Akca W. Zhang A. Dvorak I.G. Tsoulos S. Chen Z. Cen L.H. Lan N.F. Law

W.C. Siu J.P. Fang K. Balachandran K. Somasundaram A. Boulkroune N. Kumaresan K. Sakthivel S. Prasath S. Ponnusamy E. Karthikeyan A.V.A. Kumar S. Muralisankar S. Baskar C. Chandrasekar T. Kathirvalavakumar D. AshokKumar

A.K.B. Chand G. Mahadevan P. Pandian A. Thangam G. Ganesan P. Shanmugavadivu I. Kasparraj P.B. VinodKumar V.L.G. Nayagam A. Vadivel S. Domnic A.S.V. Murthy D. Senthilkumar K. Ramakrishnan M.C. Monica R. Raja

Table of Contents

1. Control Theory and Its Applications Existence and Uniqueness Results for Impulsive Functional Integro-Differential Inclusions with Infinite Delay . . . . . . . . . . . . . . . . . . . . Park Jong Yeoul and Jeong Jae Ug

1

An Improved Delay-Dependent Robust Stability Criterion for Uncertain Neutral Systems with Time-Varying Delays . . . . . . . . . . . . . . . . . . . . . . . . . K. Ramakrishnan and G. Ray

11

Design of PID Controller for Unstable System . . . . . . . . . . . . . . . . . . . . . . . Ashish Arora, Yogesh V. Hote, and Mahima Rastogi

19

New Delay-Dependent Stability Criteria for Stochastic TS Fuzzy Systems with Time-Varying Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathiyalagan Kalidass

27

H∞ Fuzzy Control of Markovian Jump Nonlinear Systems with Time Varying Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Jeeva Sathya Theesar and P. Balasubramaniam

35

Assessing Morningness of a Group of People by Using Fuzzy Expert System and Adaptive Neuro Fuzzy Inference Model . . . . . . . . . . . . . . . . . . Animesh Biswas, Debasish Majumder, and Subhasis Sahu

47

Optimal Control for Navier-Stokes Takagi-Sugeno Fuzzy Equations Using Simulink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kumaresan Nallasamy, Kuru Ratnavelu, and Bernardine R. Wong

57

Multi-objective Optimization in VLSI Floorplanning . . . . . . . . . . . . . . . . . P. Subbaraj, S. Saravanasankar, and S. Anand Approximate Controllability of Fractional Order Semilinear Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Sukavanam and Surendra Kumar

65

73

2. Computational Mathematics Using Genetic Algorithm for Solving Linear Multilevel Programming Problems via Fuzzy Goal Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bijay Baran Pal, Debjani Chakraborti, and Papun Biswas Intuitionistic Fuzzy Fractals on Complete and Compact Spaces . . . . . . . . D. Easwaramoorthy and R. Uthayakumar

79 89

X

Table of Contents

Research on Multi-evidence Combination Based on Mahalanobis Distance Weight Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Si Su and Runping Xu

97

Mode Based K-Means Algorithm with Residual Vector Quantization for Compressing Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Somasundaram and M. Mary Shanthi Rani

105

Approximation Studies Using Fuzzy Logic in Image Denoising Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sasi Gopalan, Madhu S. Nair, Souriar Sebastian, and C. Sheela

113

An Accelerated Approach of Template Matching for Rotation, Scale and Illumination Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tanmoy Mondal and Gajendra Kumar Mourya

121

Image Edge and Contrast Enhancement Using Unsharp Masking and Constrained Histogram Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Shanmugavadivu and K. Balasubramanian

129

Application of MOPSO and MOCLPSO for the Optimal Placement of TCSCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S.T. Jaya Christa and P. Venkatesh

137

Torsional Body Forces in a Kelvin-Voigt-Type Visco-Elastic Half Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P.C. Pal and Dinbandhu Mandal

145

Medical Image Binarization Using Square Wave Representation . . . . . . . . K. Somasundaram and P. Kalavathi

152

Solving Two Stage Transportation Problems . . . . . . . . . . . . . . . . . . . . . . . . . P. Pandian and G. Natarajan

159

Tumor Growth in the Fractal Space-Time with Temporal Density . . . . . . P. Paramanathan and R. Uthayakumar

166

The Use of Chance Constrained Fuzzy Goal Programming for Long-Range Land Allocation Planning in Agricultural System . . . . . . . . . Bijay Baran Pal, Durga Banerjee, and Shyamal Sen

174

A Fuzzy Goal Programming Method for Solving Chance Constrained Programming with Fuzzy Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Animesh Biswas and Nilkanta Modak

187

Fractals via Ishikawa Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bhagwati Prasad and Kuldip Katiyar

197

Table of Contents

Numerical Solution of Linear and Non-linear Singular Systems Using Single Term Haar Wavelet Series Method . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Prabakaran and S. Sekar

XI

204

3. Information Sciences Cryptographic Image Fusion for Personal ID Image Authentication . . . . . K. Somasundaram and N. Palaniappan Artificial Neural Network for Assessment of Grain Losses for Paddy Combine Harvester a Novel Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sharanakumar Hiregoudar, R. Udhaykumar, K.T. Ramappa, Bijay Shreshta, Venkatesh Meda, and M. Anantachar Implementation of Elman Backprop for Dynamic Power Management . . . P. Rajasekaran, R. Prabakaran, and R. Thanigaiselvan Comparison of Fuzzy and Neural Network Models to Diagnose Breast Cancer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Abdul Hameed and M. Bagavandas A Secure Key Distribution Protocol for Multicast Communication . . . . . . P. Vijayakumar, S. Bose, A. Kannan, and S. Siva Subramanian

213

221

232

241 249

Security-Enhanced Visual Cryptography Schemes Based on Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thomas Monoth and P. Babu Anto

258

Context-Aware Based Intelligent Surveillance System for Adaptive Alarm Services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ji-hoon Lim and Seoksoo Kim

266

Modified Run-Length Encoding Method and Distance Algorithm to Classify Run-Length Encoded Binary Data . . . . . . . . . . . . . . . . . . . . . . . . . . T. Kathirvalavakumar and R. Palaniappan

271

A Fast Fingerprint Image Alignment Algorithms Using K-Means and Fuzzy C-Means Clustering Based Image Rotation Technique . . . . . . . . . . . P. Jaganathan and M. Rajinikannan

281

A Neuro Approach to Solve Lorenz System . . . . . . . . . . . . . . . . . . . . . . . . . . J. Abdul Samath, P. Ambika Gayathri, and A. Ayisha Begum

289

Research on LBS-Based Context-Aware Platform for Weather Information Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jae-gu Song and Seoksoo Kim

297

A New Feature Reduction Method for Mammogram Mass Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Surendiran and A. Vadivel

303

XII

Table of Contents

Chaos Based Image Encryption Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Shyamala

312

Weighted Matrix for Associating High-Level Features with Images in Web Documents for Image Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Sumathy and P. Shanmugavadivu

318

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

327

Existence and Uniqueness Results for Impulsive Functional Integro-Differential Inclusions with Infinite Delay Park Jong Yeoul1 and Jeong Jae Ug2 1

Department of Mathematics, Pusan National University, Busan 609-735, South Korea [email protected] 2 Department of Mathematics, Dongeui University, Busan 614-714, South Korea

Abstract. In this paper, we discuss the existence of mild solutions of first-order impulsive functional integro-differential inclusions with scalar multiple delay and infinite delay. The result is obtained by using a fixed point theorem for condensing maps due to Martelli. Keywords: Impulsive functional differential equation; Uniqueness; Infinite delay.

1

Introduction

This paper is concerned with the existence of solutions for impulsive functional integro-differential inclusions with scalar multiple delay and infinite delay y  (t) ∈ F (t, yt ,



t

k(t, s, ys )ds) + 0

− − y(t+ k ) − y(tk ) = Ik (y(tk )),

y(t) = φ(t),

n∗ 

y(t − Ti ) a.e. t ∈ J\{t1 , t2 · · · , tn },(1)

i=1

k = 1, 2, · · · , m,

t ∈ (−∞, 0],

(2) (3)

where J = [0, b], n∗ ∈ {1, 2, · · ·}, k : J × J × B → Rn , F : J × B × Rn → P(Rn ) is a multivalued map, P(Rn ) is the family of all subsets of Rn , φ ∈ B, B is called a phase space that will be defined later and Ik ∈ C(Rn , Rn ), k = 1, 2, · · · , m are given functions satisfying some assumptions that will be specified later. For every t ∈ [0, b], the history function yt : (−∞, 0] → Rn is defined by yt (θ) = y(t+ θ) for θ ∈ (−∞, 0]. We assume that the histories yt : (−∞, 0] → Rn , yt (θ) = y(t + θ), belong to abstract phase space B. Impulsive differential and partial differential equations have become important in recent years in mathematical models of real processes and they arise in phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics. See the monographs of Bainov and Simeonov [1], Lakshmikantham et al. [4], Samoilenko and Perestyuk [11] and the references P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 1–10, 2011. c Springer-Verlag Berlin Heidelberg 2011 

2

J.Y. Park and J.U. Jeong

therein. The boundary value problems for impulsive functional differential equations with unbounded delay were studied by Benchohra et al. [2]. The problems of functional integro-differential inclusions with infinite delay have been studied by many researcher’s. Recently, the existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay was studied by Ouahab [10]. Nieto and Rodr´ıguez-L´opez [9] considered the first-order impulsive integro-differential equations with periodic boundary value conditions and proved some maximum principles. Luo and Nieto [7] improved and extended the results of Nieto and Rodr´ıguez-L´opez [9]. Chang and Nieto [3] proved the existence of the solutions of the impulsive neutral integral-differential inclusions with nonlocal initial conditions in α-norm by a fixed point theorem for condensing multi-valued maps. Lin and Hu [6] proved the existence of mild solutions for a class of impulsive neutral stochastic functional integro-differential inclusions with nonlocal initial conditions and resulvent operators. They generalized the results of Chang and Nieto [3] to the stochastic setting. Motivated by Ouahab [10], we consider the existence of solutions for impulsive functional integro-differential inclusions with scalar multiple delay and infinite delay. 1.1

Preliminaries

Let C([0, b], Rn ) be the Banach space of all continuous functions from [0, b] into Rn with norm y∞ = sup{y(t) : 0 ≤ t ≤ b}. A measurable function y : [0, b] → Rn is Bochner integrable if and only if |y| is Lebesgue integrable. Let L1 ([0, b], Rn ) be the Banach space of continuous functions y : [0, b] → Rn which are Lebesgue integrable and normed by  b ||y||L1 = |y(t)|dt 0

for all y ∈ L ([0, b], R ). A multivalued mapping G : Rn → P(Rn ) is convex(closed) valued if G(x) is convex(closed) for all x ∈ Rn . G is bounded on bounded sets if G(B) = ∪x∈B G(x) is bounded in Rn for any bounded set B of Rn , i.e., sup {sup{|y| : y ∈ G(x)}} < ∞. 1

n

x∈B

G is called upper semicontinuous(u.s.c.) on Rn if for each x0 ∈ Rn the set G(x0 ) is a nonempty closed subset of Rn and if for each open set B of Rn containing G(x0 ) there exists an open neighborhood V of x0 such that G(V ) ⊆ B. G is said to be completely continuous if G(B) is relatively compact for every bounded subset B of Rn . If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c if and only if G has a closed graph, i.e., xn → x∗ , yn → y∗ , yn ∈ Gxn imply y∗ ∈ Gx∗ . In the following BCC(Rn ) denotes the set of all nonempty bounded closed and convex subset of Rn . A multivalued map G : [0, b] → BCC(Rn ) is said to be measurable if for each x ∈ Rn the function h : [0, b] → R defined by h(t) = d(x, G(t)) = inf{|x − z| : z ∈ G(t)},

Impulsive Functional Integro-Differential Inclusions

3

belongs to L1 ([0, b], R). An upper semicontinuous map G : Rn → P(Rn ) is said to be condensing if for any subset B ⊆ Rn with α(B) = 0, we have α(G(B)) < α(B), where α denotes the Kuratowski measure of noncompactness. G has a fixed point if there is x ∈ Rn such that x ∈ G(x). Lemma 2.1([8]). Let X be a Banach space and N : X → BCC(X) be a condensing map. If the set Ω = {y ∈ X : λy ∈ N y for some λ > 1} is bounded, then N has a fixed point.

2

Existence Result

In order to define the phase space and the solution of (1)-(3) we shall consider the space + − n P C = {y : (−∞, b] → Rn , y(t− k ), y(tk ) exist with y(tk ) = y(tk ), yk ∈ C(Jk , R )},

where yk is the restriction of y to Jk = (tk , tk+1 ), k = 1, 2, · · · , m. Let  · P C be the norm in P C defined by ||y||P C = sup{|y(s)| : 0 ≤ s ≤ b}. We will assume that B satisfies the following axioms: + (A) If y : (−∞, b] → Rn , b > 0 and y0 ∈ B and y(t− k ), y(tk ) exist with y(tk ) = − y(tK ), k = 1, 2, · · · , m, then for every t ∈ [0, b]\{t1, t2 , · · · , tm } the following conditions hold: (i) yt is in B and yt is continuous on [0, b]\{t1 , t2 , · · · , tm }, (ii) ||yt ||B ≤ k(t) sup{|y(s)| : 0 ≤ s ≤ t} + M (t)||y0 ||B , (iii) |y(t)| ≤ H||yt ||B , where H ≥ 0 is a constant, k : [0, ∞) → [0, ∞) is continuous, M : [0, ∞) → [0, ∞) is locally bounded and k, H, M are independent of y(·). (A1) For y(·) in A, yt is B-valued continuous function on [0, b]\{t1 , t2 , · · · , tm }, (A2) The space B is complete. Set Bb = {y : (−∞, b] → Rn |y ∈ P C ∩ B}. Let us state by defining what we mean by a solution of problem (1)-(3). Definition 3.1. We say that the function y ∈ Bb is a mild solution of problem (1)-(3) if y(t) = φ(t) for all t ∈ (−∞, 0], the restriction of y(·) to the interval [0, ∞) is continuous and there exists v(·) ∈ L1 ([0, ∞), Rn ) such that v(t) ∈ t F (t, yt , 0 k(t, s, ys )ds) and such that y satisfies the integral equation y(t) = φ(0)+

n∗   i=1



0

−Ti

φ(s)ds +

t

v(s)ds + 0

n∗   i=1

0

t−Ti

y(s)ds+

 0 0. By Dynkin’s formulat, we have for each where a = mink∈S (λmin (−Φ ku rt = k, k ∈ S  E [V (xt , k, t) − V (ϕ(t), r0 )]  −a

T

  E xT (t)x(t) dt

0

hence

 E

T

 T

x (t)x(t) dt 0



1 V (ϕ(t), r0 ). a

However, from (7) it is easy to see that for any t  0,   E {V (xt , k, t)}  b E xT (t)x(t) , where b = mink∈S (λmin (Pk )) > 0. From the above two inequalites, we get   T   T xT (t)x(t) dt + b−1 V (ϕ(t), r0 ). E x (t)x(t)  ab−1 E 0

Thus applying Gronwall-Bellman lemma to this inequality yields,   E xT (t)x(t)  a−1 (1 − exp(−ab−1 T ))V (ϕ(t), r0 ). As T → ∞, there exists c such that   T

lim E

t→∞

0

xT (t)x(t)|(ϕ(t), r0 )

 c V (ϕ(t), r0 )  c

sup

−τ  t0

|ϕ(t)|2 .

H∞ Fuzzy Control of Delayed MJFS

43

From (2), the closed loop MJFS defined in (6) is stochastically asymptotically stable. In order to cater the H∞ control problem, let us consider the H∞ performance index for given γ > 0 as  J =E

∞

z T (t)z(t) − γ 2 wT (t)w(t) dt

(15)

0

On the other hand, z T (t)z(t) − γ 2 wT (t)w(t) =

r 

T μj (xT (t) GTjk + xT (t − τ (t))GTdjk + wT (t)Hwjk )×

j=1

(Gjk x(t) + Gdjk x(t − τ (t)) + Hwjk w(t)) − γ 2 wT (t)w(t) +JV (xt , k, t) − JV (xt , k, t)

under zero initial conditions we have (jj  )

J  ζ T (t)Ξku ζ(t),

 where ζ(t) = x(t − τ (t)) x(t) x(t − δ1 ) . . . x(t − N δ) . . . w(t) z(t) , ⎤ ⎡˜ ˜ ψ11 ψ12 0 ψ˜14 ψ˜15 ⎢  ψ˜22 ψ˜23 ψ˜24 ψ˜25 ⎥ ⎥ ⎢ (jj  ) ˜ ˜ ˜ ⎥ Ξku = ⎢ ⎢   ψ33 ψ34 ψ35 ⎥ ⎣    ψ˜44 0 ⎦     ψ˜55 T  with ψ˜33 = −γ 2 ; ψ˜15 = Gdjk ; ψ˜25 = Gjk 0 · · · 0 0 · · · 0 0 ; ψ˜35 = Hwjk ; ψ˜55 = −I. (jj  ) In fact J  0 if and only if Ξku < 0. This leads to z(t) 2 ≤ γ w(t) 2 . That is  ∞  ∞ z T (t)z(t) dt  γ 2 wT (t)w(t) dt . E 0

0 (jj  ) Ξku

< 0 is not an LMI, as it involves various nonlinear   (jj  ) matrix terms. Pre and post multiplying Ξku with diag Pk−1 Pk−1 I R−1 I ˜ α = P˜k Qα P˜k , R ˜ α = P˜k Rα P˜k , and the H∞ control gain and letting P˜k = Pk−1 , Q  (jj ) −1 matrices be Kj  k = Yj  k P˜k , we get Φ˜ku which is given in (11) with exception that It is clearly seen that

(1) w1

T = Ajk P˜k + P˜k ATjk + YjT k Bjk + Bjk Yj  k +

s 

˜1 πkk P˜k Pk P˜k + Q˜1 − R

k =1 −1 and ψ˜ . From the property of Markovian jump transition matrix, the 44 = −R s ˜ −1 P˜k  term k =1 πkk P˜k Pk P˜k can be written as P¯ −πkk P˜k and −R−1 = −P˜k R ˜ − 2P˜k . This leads to LMIs (9) and (10). Hence the proof is complete. R

44

4

S. Jeeva Sathya Theesar and P. Balasubramaniam

Numerical Simulations

In this section, we apply the above derived criterion to H∞ control of a modified computer simulated single link robot arm [6]. The time varying delayed model can be given as x˙ 1 (t) = μx2 (t) + (1 − μ)x2 (t − τ (t)) Mk gl μD(t) (1 − μ)D(t) 1 sin(x1 (t)) − x2 (t) − x2 (t − τ (t)) + u(t) + aw(t) x˙ 2 (t) = − Jk Jk Jk Jk z(t) = x1 (t) + 0.2w(t), k = 1, 2, 3, (16)

where x1 (t), x2 (t), u(t), and z(t) are the angle of the arm, the angular velocity, the control input, and the control output, respectively. w(t) is the exogenous disturbance attenuation input. Let the dynamics jumps between the three states of mass Mk and the inertia Jk such that M1 = J1 = 1, M1 = J1 = 5, and M1 = J1 = 10. The transition ⎡ rate matrix of⎤ the markovian jump op−0.3 0.25 0.05 eration modes is given as Π = ⎣ 0.1 −0.2 0.1 ⎦. The length of the arm 0.03 0.07 −0.1 l = 0.5, acceleration of gravity g = 9.81, and Damping D(t) ∈ [1.8, 2.2]. Also here μ ∈ [0, 1] is the constant retarded coefficient.For TS fuzzy modelsin(x1 )−ρx1 x1 (1−ρ) , x1 = 0 ; ing of (16), we use the membership functions μ1 (x1 ) = 1 x1 = 0  x1 −sin(x1 ) x1 (1−ρ) , x1 = 0 μ2 (x) = , where ρ = 10−2 /π. The MJFS defined in (6) can 0 x1 = 0 be easily obtained matices    with following x1 (t) 0 μ 0 μ 0 μ x(t) = , A12 = , A13 = , , A11 = −gl −2μ −gl −0.4μ −gl −0.2μ x (t)  2   0 μ 0 μ 0 μ , A22 = , A23 = , A21 = −ρgl −2μ −ρgl −0.4μ −ρgl −0.2μ   0 1−μ 0 1−μ Ad11 = Ad21 = , Ad12 = Ad22 = , 0 −2(1 − μ) 0 −0.4(1 − μ)    0 1−μ 0 0 , B11 = B21 = , B12 = B22 = , Ad13 = Ad23 = 1 0.2  0 −0.2(1 − μ)   0 0 B13 = B23 = , Bw1 = Bw2 = , G1 = G2 = 1 0 , and 0.1 0.1 Gw1 = Gw2 = 0.2. The main purpose of giving this model is to find the desirable H∞ state feedback controller for time varying delayed single link robot arm model with performance level γ. Now the LMIs (9) and (10) are solved numerically for j, j  = 2, k = 3, and N the number of decomposition by Matab LMI toolbox. The maximum allowable delay limit τ is given with H∞ performance level γ. For N = 3, and γ = 1, μ = 0.9, ϕ(t) = [0.5 ∗ π, −1], ∀t ∈ [0.4635, 0], and r0 = 1 the solution states of (16) is simulated and plotted in Fig. 1with Markovian switching signal that changes the mode randomly according to . For reference Fig. 2 is provided

H∞ Fuzzy Control of Delayed MJFS

45

Table 1. Performance of the H∞ control criterions derived in Threorem 1, for time varying delayed single link robot arm model (16) H∞ Performance level / τ N=2 N=3 N=5 γ = 0.3 0.2274 0.3276 0.4430 γ=1 0.2384 0.3486 0.4635

to show how system states behave in three different modes separately. It is worth noting that the results presented in [6] is only for constant delay and cannot solve the H∞ control problem of model (16).

8 x1 6

2

x

2

4

r

t

2

state

0

1

−2 −4 −6

0

−8 −10

0

50

40

30

20

10

0

10

20

time

30

40

50

time

Fig. 1. State trajectory of model (15) with Markovian switching signal rt

8

6

4

x1

x1

x

6

3

x

4

2

2

x1 2

4

x

2

2 1 0

0

0 state

state

state

2

−2

−2

−4

−4

−6

−6

−8

−1 −2 −3 −4

−8

0

10

20

30

40

time

50

−10

−5 0

10

20

30 time

40

50

−6

0

10

20

30

40

50

time

Fig. 2. State trajectories of (15) at three different modes

Also to cater the need of models that consists of constant delay considering 0  τ˙ (t)  μ (having additional information on delay), we can extend the Theorem 1, based on [7].

5

Conclusion

In this paper, we have studied the H∞ control problem of MJFS with time varying delay. The less conservative delay dependent sufficient criterions are derived. Using the single link robot arm model, the effectiveness of the conditions

46

S. Jeeva Sathya Theesar and P. Balasubramaniam

are tested numerically. In fact, there are cases when noise and network dropout can reduce the stability and performance of the systems, that will be our future research.

References 1. Xiong, J., Lam, J.: Stabilization of discrete-time Markovian jump linear systems via time-delayed controllers. Automatica 42, 747–753 (2006) 2. Xu, S., Lam, J., Mao, X.: Delay-dependent H∞control and filtering for uncertain Markovian jump systems with time varying delays. IEEE Trans. Circuits Syst. I, Reg. Papers 45, 2070–2077 (2007) 3. Wu, H., Cai, K.: Mode-independent robust stabilization for uncertain Markovian jump nonlinear systems via fuzzy control. IEEE Trans. Syst. Man, Cybern, Part B 36, 509–519 (2006) 4. Nguang, S.K., Assawinchaichote, W., Shi, P., Shi, Y.: Robust H∞ control design for uncertain fuzzy systems with Markovian jumps: an LMI approach. In: Proc. of Amer. Contr. Conf., pp. 1805–1810 (2005) 5. Dong, J., Yang, G.: Fuzzy controller design for Markovian jump nonlinear systems. Int. J. Control, Automation, and Systems 5, 712–717 (2007) 6. Zhang, Y., Xu, S., Zhang, J.: Delay-dependent robust H∞ control for uncertain fuzzy Markovian jump systems. Int. J. Control, Automation, and Systems 7, 520– 529 (2009) 7. Zhang, X.M., Han, Q.: A delay decomposition approach to delay dependent stability for linear systems with time-varying delays. Int. J. of Robust Non. Cont. 19, 1922– 1930 (2009)

Assessing Morningness of a Group of People by Using Fuzzy Expert System and Adaptive Neuro Fuzzy Inference Model Animesh Biswas1,*, Debasish Majumder2, and Subhasis Sahu3 1

Department of Mathematics, University of Kalyani, Kalyani – 741235, India 2 Department of Mathematics, JIS College of Engineering, Kalyani – 741235, India 3 Department of Physiology, University of Kalyani, Kalyani – 741235, India {abiswaskln,debamath}@rediffmail.com, [email protected]

Abstract. In this paper two computational systems, one is based on fuzzy logic system and other is based on adaptive neuro fuzzy inference system (ANFIS), are developed for assessing morningness of a group of people and the result is compared for finding the best system in the assessment process. In fuzzy rule based system, the linguistic terms for assessing morningness are quantified by using fuzzy logic and a fuzzy expert system is developed. On the other side, an ANFIS model is generated to assess data for training and testing the model in accordance with a preference scale adopted to quantify the responses of subjects for a reduced version of Morningness-Eveningness Questionnaire (rMEQ). The result reflects that ANFIS is able to assess morningness better than fuzzy expert inference system. Keywords: Chronotypology, rMEQ, Fuzzy Expert System, Mamdani Fuzzy Inference System, ANFIS, Morningness.

1 Introduction Assessing morningness is now getting importance in various fields since human performance is the key to the success. Morningness, or the preference for morning or evening activities, is an individual difference in circadian rhythms with potential applications in optimizing work schedules, sports performance and academic activities. Circadian rhythms, cyclic fluctuation in many physiological functions, are thought to influence adjustment to shift-work. The researchers are interested in studying morningnes and eveningness orientation by developing several self reporting questionnaires. To measure these parameters often referred to collectively as morningness. Several self reporting questionnaires have been developed to measure diurnal type or distinction between ‘Larks’ and ‘Owls’ [1, 2, 3, 4, 5, 6]. Among the questionnaires, Morningness-Eveningness Questionnaires (MEQ) of Horne and Ostberg [1], Circadian *

Corresponding author.

P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 47–56, 2011. © Springer-Verlag Berlin Heidelberg 2011

48

A. Biswas, D. Majumder, and S. Sahu

Typology Questionnaires (CTQ) of Folkard et.al. [3] and Composite Morningness Questionnaires (CMQ) of Smith et.al. [6] were widely used and well accepted. Again, among the three cronotypological questionnaires the reduced version of MEQ (rMEQ) was short, easily understandable and may be considered as the best morningnesseveningness predictor for Indian people [7]. In the present study rMEQ type questionnaire is selected to assess the morningnes of a group of people. In rMEQ, the different types of questions or linguistic terms are asked to fill up the questionnaire. The linguistic terms are very often fuzzily defined. In this context fuzzy logic [8, 9] is seemed to be very much useful to handle the collected data in a proper justified way. The study of fuzzy logic began as early as the 1960s. In the 1970s, fuzzy logic was combined with expert systems to become a fuzzy logic system that was then implemented in many industrial [10, 11], medical [12] and environmental [13] applications. But the application of fuzzy logic in the field of ergonomics, especially, assessing morningness of the group of people is yet to appear in the literature. Again, no standard method is found for transforming human knowledge or experience into the rule base or data base of a fuzzy inference system. Also, there is a need of effective methods for tuning the membership functions so as to minimize the output error. In this perspective, ANFIS [14] can serve as a basis for constructing a set of fuzzy if-then rules with appropriate membership functions to generate the stipulated input-output pairs. In the present study, two types of analysis viz., Fuzzy Expert System based on Mamdani Fuzzy Inference Process and ANFIS based on Takagi-Sugeno Fuzzy Control System are developed in systematic ways and the achieved result is compared in the context of Indian Environment and Western Climatic Environment.

2 Mode of Data Collection The sample consists of 530 post-graduate students randomly taken from University of Kalyani, West Bengal, INDIA who answered the survey item on voluntary basis. Almost fifty percent of the students are resident students and fifty percent are day boarders. They all maintain diurnal schedule. All of them are acquainted with English. All students reported to maintain diurnal schedule. After describing details about the study, the English version of rMEQ was handed over to them. They filled up the questionnaire and return it. The study was conducted between 11:00 a.m. to 4:00 p.m. In the present study the parameters which are considered as fuzzy inputs are Average Rising Time for Feeling Best (X1), Feelings within First Half an Hour after waken up in the morning (X2), Sleeping Time (X3), Feeling Best Time in a Day (X4) and Self Assessment about Morningness (X5) and are measured on a scale ranging from 0 to 5, from 0 to 4, from 0 to 5, from 0 to 5 and from 0 to 6, respectively. Output parameter Morningness (Z) is a linguistic variable and is measured in points on a scale from 0 to 6.

3 Designing Fuzzy Expert System Fuzzy expert system is viewed as an inference system in which expert’s knowledge is used along with fuzzy control theory especially in time of defining fuzzy sets of each

Assessing Morningness of a Group of People

49

linguistic variable and generating rule. The process of fuzzification of the inputs, evaluation of the rules and aggregation of all required rules and making an inference is known as fuzzy control. The type of control theory which is used in the present study to measure the morningness is based on Mamdani type fuzzy controller. The inputs to the fuzzy logic models are linguistic variables which are presented by fuzzy sets. In Mamdani model, the outputs are expressed in terms of linguistic variables described by some fuzzy sets. The fuzzy sets for the inputs are related to the output fuzzy sets through IF-THEN rules, which describe the knowledge about the behavior of the system. The outline of the used model is shown in Fig. 1.

Fig. 1. Outline model of morningness fuzzy expert system

In the proposed approach, rules are generated by the combination of collected data and expert experiences for the fuzzy rule based system. In the present study, arithmetic means of evaluated points corresponding to the responses given by 530 students for each question are calculated first. The evaluated values corresponding to five different linguistic variables are found as X1 = 4.00, X2 = 2.93, X3 = 3.45, X4 = 3.24 and X5 = 3.28. These numerical values are fuzzified by using fuzzification process. For each input linguistic variable, four triangular fuzzy sets are defined as shown in Fig. 2 – Fig. 6 and for the output parameter, three triangular types of fuzzy sets are defined as shown in Fig. 7. After fuzzification of the inputs, the degree to which each part of the antecedent is satisfied for each rule is determined. The fuzzified values and linguistic variables show which rules will be selected to be fired in the fuzzy implication process. In the present study, 16 rules are generated with the help of expert’s opinion as shown in Fig. 8. If the antecedent of a given rule has more than one part, the fuzzy operator AND is applied to obtain only one membership value that represents the result of the antecedent for that rule. Then, this membership value is used for obtaining the output value. In the present study, MIN (minimum) function is used for AND operator. In the next step the aggregation process is performed to combine the derived outputs of each rule into a single fuzzy set. Aggregation only works once for the output variable Morningness, just prior to the final step, defuzzification. Here, the MAX (maximum) method is being used in the aggregation process.

50

A. Biswas, D. Majumder, and S. Sahu

Fig. 2. Membership Functions for Input Variable Fig. 3. Membership Functions for Input Variable Average Rising Time for Feeling Best Feelings within First Half an Hour after waken up in the morning

Fig. 4. Membership Functions for Input Variable Fig. 5. Membership Functions for Input Variable Sleeping Time Feeling Best Time in a Day

Fig. 6. Membership Functions for Input Variable Fig. 7. Membership Functions for Fuzzy Output Self Assessment about Morningness Parameter Morningness

Fig. 8. Snapshot of Fuzzy rules (partial)

The implication, aggregation and defuzzification calculations have been performed by using Fuzzy Rule Viewer of software MATLAB 7.1 which is also used to define the input and output parameters. The output screen which shows implication of the antecedent, aggregation of the consequent and defuzzification process for Morningness is shown in the following Fig. 9.

Assessing Morningness of a Group of People

51

Fig. 9. The Output Screen

Now, three fuzzy sets are developed to assess the morningness in the following manner and the defuzzified value is put into the sets to find the degree of morningness of the group of students in the University.

Fig. 10. Measurement of the Degree of Morningness

The membership value of morningness corresponding to 4.54, is found as 0.51, 0.23, 0 (Fig. 10). The result shows that the group of students of University of Kalyani is morning oriented and the fuzzy membership degree corresponding to the fuzzy set ‘Morning (M)’ is 0.51 and corresponding to the fuzzy set ‘Intermediate (I)’ is 0.23.

4 The ANFIS Model Formulation The ANFIS architecture is designed for Takagi-Sugeno fuzzy inference system. A typical ANFIS consists of five layers which perform different actions in the ANFIS

52

A. Biswas, D. Majumder, and S. Sahu

are described below. Every layer consists of some nodes (circular or square type). The circular type nodes represent nodes that are fixed whereas the square type nodes designate nodes that have parameters which are to be learnt (Fig. 11). For simplicity, a system that has two inputs x and y and one output z has been illustrated below and the rule base of two IF-THEN rules of first order Takagi-Sugeno model has been used for this purpose. Rule 1: if

and

then

Rule 2: if

and

then

where x and y are inputs; and ( 1,2 are appropriate fuzzy sets; , and 1,2 contribute to the output of the system. ( 1,2) are output parameters and (

Fig. 11. Architecture of ANFIS network

The outputs of each of the five layers of ANFIS architecture are described as follows: Layer 1. Every node i in this layer is an adaptive node with a node function , for

1,2;

, for

3,4.

where 1, 2 and 3, 4 are the linguistic terms associated with the 1, 2 and 3,4 can adopt inputs x and y, respectively and any membership function, e.g., triangular, trapezoidal, generalized bell-shaped, Gaussian functions or other types. In fact, 1,2,3,4) specify the degrees to which the given inputs satisfy the corresponding linguistic terms. Layer 2. Every node i in this layer is a fixed node which multiplies the signals coming from the nodes of the previous layer and sends the product out. The outputs 1, 2 of this layer are represented as: , where

1,2) represent the firing strength of the

1,2. rule.

Assessing Morningness of a Group of People

53

Layer 3. All nodes in this layer are also fixed nodes. The node of this layer calcurule’s firing strength to the sum of the firing strengths of all lates the ratio of the rules. The outputs 1, 2 of this layer are represented as: ,

∑

where

1,2.

1, 2 are known as normalized firing strengths.

Layer 4. Every node i in this layer is an adaptive node with a node function , where , 1,2 are the outputs of the layer 3 and nomials (for a first order Sugeno model). Parameters layer are referred to as consequent parameters.

1,2 1, 2 are first order polyand 1, 2 in this

,

Layer 5. The single node in this layer is a fixed layer which computes the overall output as the summation of all incoming signals, i.e., ∑

∑

∑

.

In this paper ANFIS toolbox of software MATLAB 7.1 is used for creating an ANFIS model for which the following stages are performed to assess the morningness of a group of people: 1. Loading 530 pairs sample data (450 pairs for training purpose and 80 pairs for testing purpose) as shown in Fig.12. 2. Generating a Fuzzy Inference System (FIS) model (Fig. 13) using subtractive clustering method with the following parameters as shown in Table 1. Table 1. Parameters of subtractive clustering method Parameters

Range of Influence

Squash Factor

Accept Ratio

Reject Ratio

Value

0.5

1.51

0.5

0.15

3. Training the FIS for 100 steps to tune all the modifiable parameters using hybrid learning algorithm which is a combination of back propagation algorithm and least means square method and setting error of tolerance at zero as there is a no information about how the error behave (Fig. 14). 4. Testing the FIS for validity (Fig. 15).

54

A. Biswas, D. Majumder, and S. Sahu

Fig. 12. Plotted data (training and testing)

Fig. 13. The Structure of FIS

Fig. 14. Diagram of error rate in different training stages

Fig. 15. Diagram of the prediction result

Thus an ANFIS model is developed for assessing morningness of a group of people and it is found that the average testing error is negligibly small. Now, considering the values corresponding to five different linguistic variables evaluated from the preference scale values of rMEQ type questionnaire are taken as X1 = 4.00, X2 = 2.93, X3 = 3.45, X4 = 3.24 and X5 = 3.28. The assessment through the newly generated ANFIS model is found as Z = 3.38 (Fig. 16).

Fig. 16. Diagram of ANFIS rules and output

Fig. 17. Measurement of the Degree of Morningness

The membership value of morningness corresponding to 0.13,

0.81,

3.38, is found as

0 (Fig. 17).

Assessing Morningness of a Group of People

55

The result obtained by proceeding through this process is found as ‘Intermediate’ type with membership value 0.81 and morning oriented with membership value 0.13.

5 Conclusions In this study two methodologies are developed for assessing morningness in a justified way. The first approach is based on fuzzy expert system. In this method the rules which are considered in the modeling process are based on expert’s expertise on that domain. The result shows that the group of students is morning oriented. Again, in the second methodology, an ANFIS method is developed. In this process it is found that the group of people is less morning oriented as that of first approach. In view of physiological perspective, the later method is more justifiable than the former one. Also ANFIS model is used as an alternative way to measure morningness in which the use of hybrid learning procedure is advantageous as the proposed architecture can refine fuzzy if-then rules obtained from human experts to describe the input-output behavior of a complex system. Moreover, if human expertise is not available, we can still set up intuitively reasonable initial membership functions and start the learning process to generate a set of fuzzy if-then rules to approximate a desired data set. Also, the structure of ANFIS is identified by using subtractive clustering method which concerns with the selection of an appropriate input-space partition style and the number of membership functions on each input and is equally important to the successful applications of ANFIS. Effective partition of input-space can decrease the number of the rules and thus increase the speed in both learning and application phases. Further the ANFIS model can be improved by using genetic algorithm in the learning process of modifiable parameters. However, it is hoped that the proposed methodologies may open up new horizon in the way of assessing morningness through fuzzy control. Acknowledgements. The authors remain grateful to the anonymous referees for their constructive comments and valuable suggestions in improving the quality of the paper. Also the authors are thankful to all the subjects who volunteered for the study and gladly given consent.

References 1. Horne, J.A., Ostberg, O.: A Self Assessment Questionnaire to Determine Morningness, Eveningness. Int. J. Chronobio. 4, 97–110 (1976) 2. Horne, J.A., Ostberg, O.: Individual Differences in Human Circadian Rhythm. Biological Psycho. 5, 179–190 (1977) 3. Folkard, S., Monk, T.H., Lobban, M.C.: Toward a Predictive Test of Adjustment to Shift Work. Ergonomics 22, 79–91 (1979) 4. Torsvall, L., Akerstedt, T.: A Diurnal Type Scale Construction, Consistency and Validation in Shift Work. Environment and Health 6, 283–290 (1980) 5. Moog, R.: Morningness- evening Types and Shift Work a Questionnaire Study. In: Reinberg, A., Vieux, N., Andlaner, P.C. (eds.) Night and Shift Work: Biological and Social Aspects, pp. 481–488. Pergamon Press, Oxford (1981)

56

A. Biswas, D. Majumder, and S. Sahu

6. Smith, C.S., Reilly, C., Midkiff, K.: Evaluation of Three Circadian Rhythm Questionnaires with Suggestion for Improved Measures of ‘Morningness’. J. Appl. Psycho. 74, 728–738 (1989) 7. Sahu, S.: An Ergonomic Study on Suitability of Choronotypology Questionnaires on Bengalee (Indian) Population. Indian J. Biological Sci. 15, 1–11 (2009) 8. Zadeh, L.A.: Fuzzy Sets. Inf. and Control. 8, 338–353 (1965) 9. Pal, B.B., Biswas, A.: A Fuzzy Multilevel Programming Method for Hierarchical Decision Making. In: Pal, N.R., Kasabov, N., Mudi, R.K., Pal, S., Parui, S.K. (eds.) ICONIP 2004. LNCS, vol. 3316, pp. 904–911. Springer, Heidelberg (2004) 10. Buckley, D.I.: The Nature of Structural Design and Safety. Ellis Horwood, Chichester (1980) 11. Gurcanli, G.E., Mungen, U.: An Occupational Safety Risk Analysis Method at Construction Sites using Fuzzy Sets. Int. J. Industrial Ergonomics 39, 371–387 (2009) 12. Vila, M.A., Delgado, M.: On Medical Diagnosis using Possibility Measures. Fuzzy Sets and Syst. 10, 211–222 (1983) 13. Cao, H., Chen, G.: Some Applications of Fuzzy Sets of Meteorological Forecasting. Fuzzy Sets and Syst. 9, 1–12 (1983) 14. Venugopal, C., Devi, S.P., Rao, K.S.: Predicting ERP User Satisfaction – an Adaptive Neuro Fuzzy Inference System (ANFIS) Approach. Intelligent Inform. Mgmt. 2, 422–430 (2010)

Optimal Control for Navier-Stokes Takagi-Sugeno Fuzzy Equations Using Simulink Kumaresan Nallasamy, Kuru Ratnavelu, and Bernardine R. Wong Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia [email protected]

Abstract. In this paper, the unsteady Navier-Stokes Takagi-Sugeno (T-S) fuzzy equations (UNSTSFEs) are represented as a differential algebraic system of strangeness index one by applying any spatial discretization. Since such differential algebraic systems have a difficulty to solve in their original form, most approaches use some kind of index reduction. While processing this index reduction, it is important to take care of the manifolds contained in the differential algebraic equation (DAE) /singular system (SS) for each fuzzy rule. The Navier-Stokes equations are investigated along the lines of the theoretically best index reduction by using several discretization schemes. Applying this technique, the UNSTSFEs can be reduced into DAE. Optimal control for NavierStokes T-S fuzzy system with quadratic performance is obtained by finding the optimal control of singular T-S fuzzy system using Simulink. To obtain the optimal control, the solution of matrix Riccati differential equation (MRDE) is found by solving differential algebraic equation (DAE) using Simulink approach. The solution of Simulink approach is equivalent or very close to the exact solution of the problem. An illustrative numerical example is presented for the proposed method. Keywords: Differential algebraic equation, Matrix Riccati differential Equation, Navier-Stokes equation, Optimal control and Simulink.

1 Introduction A fuzzy system consists of linguistic IF-THEN rules that have fuzzy antecedent and consequent parts. It is a static nonlinear mapping from the input space to the output space. The inputs and outputs are crisp real numbers and not fuzzy sets. The fuzzification block converts the crisp inputs to fuzzy sets and then the inference mechanism uses the fuzzy rules in the rule-base to produce fuzzy conclusions or fuzzy aggregations and finally the defuzzification block converts these fuzzy conclusions into the crisp outputs. The fuzzy system with singleton fuzzifier, product inference engine, center average defuzzifier and Gaussian membership functions is called as standard fuzzy system [18]. Two main advantages of fuzzy systems for the control and modelling applications are (i) uncertain or approximate reasoning, especially difficult to express a mathematical P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 57–64, 2011. © Springer-Verlag Berlin Heidelberg 2011

58

N. Kumaresan, K. Ratnavelu, and B.R. Wong

model (ii) decision making problems with the estimated values under incomplete or uncertain information [20]. Stability and optimality are the most important requirements in any control system. For optimality, it seems that the field of optimal fuzzy control is totally open. Singular systems contain a mixture of algebraic and differential equations. In that sense, the algebraic equations represent the constraints to the solution of the differential part. These systems are also known as degenerate, descriptor or semi-state and generalized state-space systems. The system arises naturally as a linear approximation of system models or linear system models in many applications such as electrical networks, aircraft dynamics, neutral delay systems, chemical, thermal and diffusion processes, large-scale systems, robotics, biology, etc., see [3,4,5,11]. As the theory of optimal control of linear systems with quadratic performance criteria is well developed, the results are most complete and close to use in many practical designing problems. The theory of the quadratic cost control problem has been treated as a more interesting problem and the optimal feedback with minimum cost control has been characterized by the solution of a Riccati equation. Da Prato and Ichikawa [6] showed that the optimal feedback control and the minimum cost are characterized by the solution of a Riccati equation. Solving the Matrix Riccati Differential Equation (MRDE) is a central issue in optimal control theory. The needs for solving such equations often arise in analysis and synthesis such as linear quadratic optimal control systems, robust control systems with H2 and H∞-control [22] performance criteria, stochastic filtering and control systems, model reduction, differential games etc. One of the most intensely studied nonlinear matrix equations arising in Mathematics and Engineering is the Riccati equation. This equation, in one form or another, has an important role in optimal control problems, multivariable and large scale systems, scattering theory, estimation, detection, transportation and radiative transfer [7]. The solution of this equation is difficult to obtain from two points of view. One is nonlinear and the other is in matrix form. Most general methods to solve MRDE with a terminal boundary condition are obtained on transforming MRDE into an equivalent linear differential Hamiltonian system [8]. By using this approach, the solution of MRDE is obtained by partitioning the transition matrix of the associated Hamiltonian system [17]. Another class of methods is based on transforming MRDE into a linear matrix differential equation and then solving MRDE analytically or computationally [12,15,16]. However, the method in [14] is restricted for cases when certain coefficients of MRDE are non-singular. In [8], an analytic procedure of solving the MRDE of the linear quadratic control problem for homing missile systems is presented. The solution K(t) of MRDE is obtained by using K(t)=p(t)/f(t), where f(t) and p(t) are solutions of certain first order ordinary linear differential equations. However, the given technique is restricted to single input. Simulink is a MATLAB add-on package that many professional engineers use to model dynamical processes in control systems. Simulink allows creating a block diagram representation of a system and running simulations very easily. Simulink is really translating block diagram into a system of ordinary differential equations. Simulink is the tool of choice for control system design, digital signal processing (DSP) design, communication system design and other simulation applications [1]. This paper focuses upon the implementation of Simulink approach to compute optimal control for linear singular fuzzy system.

Optimal Control for Navier-Stokes Takagi-Sugeno Fuzzy Equations Using Simulink

59

Although parallel algorithms can compute the solutions faster than sequential algorithms, there has been no report on Simulink solutions for MRDE. This paper focuses upon the implementation of Simulink approach for solving MRDE in order to get the optimal solution. This paper is organized as follows. In section 2, the statement of the problem is given. In section 3, solution of the MRDE is presented. In section 4, numerical example is discussed. The final conclusion section demonstrates the efficiency of the method.

2 Statement of the Problem In computational fluid dynamics, a typical example of the equations of gas dynamics under the assumptions of incompressibility is the Navier Stokes equation [19]. It consists of as many differential equations as the dimension of the model indicates and the condition of incompressibility, see e.g.[13]: ∂u = –u.∇+ν Δu –∇p + f ∂t 0 = ∇.u

(1)

These equations, together with the appropriate initial and boundary conditions, are yet d to be solved in Ωx [0,T], where Ω is a bounded open domain in R [d = (2 or 3) dimension of the model] and T the endpoint of the time interval. Two dimensional cases are considered here for easy computation and simplification. The results are valid for a three dimensional model as well. The domain of reference is assumed to be rectangular. This is indeed a restriction. But, at the exact places, it will be pointed out, whether some techniques may be generalized to other domains or not. After applying the method of lines (MOL), i.e. carrying out a spatial discretization by finite difference or finite element techniques, the equations in (1) can be written as a differential algebraic system. Mu′(t) = K(u)u(t) –Bp(t) +f(t) T 0 = B u(t)

(2)

See [2]. Here u(t), p(t) and f(t) are approximations to the time and space dependent quantities u, p and f of (4). The matrix M is symmetric and positive definite. Quantity B stands for discrete gradient operator, while K(u) represents linear and nonlinear velocity terms. The DAE (2) is of a higher index (i.e. non-decoupled), since pressure p does not appear in the algebraic condition. If we assume that B is a full column rank, then the differentiation index is two. Since p is only determined up to an additive constant, B has, in general, a rank deficiency which causes the undeterminedness of at least one solution component. The concept of the differentiation index can not be applied to such systems. Kunkel and Mehrmann [9] have generalized the index concept to the case of over and underdetermined DAEs. Their so called strangeness index (or s-index) μ is the number of additional block columns needed in the derivative array [10] that can filter out a strangeness free system by transformations from the left. This system then represents a DAE of differentiation index one with possibly undetermined components

60

N. Kumaresan, K. Ratnavelu, and B.R. Wong

or a system of ordinary differential equations. Therefore, μ is one lower than the differentiation index if the system is a DAE of at least differentiation index one without undeterminedness. For ordinary differential equations (differentiation index zero), μ is defined as zero. The system (2) is of a higher index, namely s-index 1. Such systems have a difficulty to solve in their original form because of the differential and algebraic components and strangeness [9]. It would be appropriate to remove this strangeness before solving the DAE. In most of the Navier-Stokes solution techniques, this is done without explicitly mentioning that an index reduction is carried out. If the index reduction is omitted, the results may become unsatisfactory, especially in the unsteady case. In [21], examples are computed when a steady state is reached and it is stated that satisfactory smoothness is achieved. But, this is completely impractical, if long time computations are carried out. A characterization is also guaranteed by the concept of s-index when the DAE has an unique solution. However, this is not the main advantage of this approach over the usual concept of the differentiation index. The biggest progress seems to be that [9] provides a way to reformulate the higher index DAE as a strangeness free system with the same dimension and same solution structure as the original system. In other words, it is possible to rewrite a DAE of higher index in a so called normal form of s index zero. This form not only reflects the manifold included in the original system but also all the hidden manifolds. Thus, using strangeness free normal form, a consideration of all manifolds is ensured which makes this approach superior over other index reduction variants. Moreover, the derivative term is not transformed so that no errors in time are caused by the index reduction strategy for Navier Stokes equations. By this index reduction process, the linear differential algebraic system of arbitrary sindex μ, E x′(t)=A(t)x(t)+f(t), under suitable assumptions [10] can be transformed into strangeness free normal form by means of utmost μ.3+2 rank decisions. This procedure is described in [9]. Consider the two dimensional Navier-Stokes (T-S) fuzzy equations ∂u = –ui.∇+ν Δui –∇p + fi(t) ∂t

(3)

0 = ∇.ui The above equation can be transformed into singular T-S fuzzy system using the method described in [9]. The singular time-invariant system by taking fi(t) = 0 in (3) Ei x′(t)=Aix(t)+Biu(t), x(0)=x0, n

(4)

where the matrix Ei is singular, x(t) ∈ R is a generalized state space vector and u(t) m nxn nxm ∈ R is a control variable. Ai∈ R and Bi∈R are known coefficient matrices associated with x(t) and u(t) respectively, x0 is given initial state vector and m ≤ n. In order to minimize both state and control signals of the feedback control system, a quadratic performance index is usually minimized:

Optimal Control for Navier-Stokes Takagi-Sugeno Fuzzy Equations Using Simulink

61

tf J=

½

∫[xT(t)Qx(t)+uT(t)Ru(t)]dt,

T

T

x (tf)Ei SEix(tf)+ ½

0 nxn

nxn

where the superscript T denotes the transpose operator, S∈R and Q∈R are symmetric and positive definite (or semidefinite) weighting matrices for x(t), R∈Rm xm is a symmetric and positive definite weighting matrix for u(t). It will be assumed that |sEi-Ai| ≠ 0 for some s. This assumption guarantees that any input u(t) will generate one and only one state trajectory x(t). If all state variables are measurable, then a linear state feedback control law –1

u(t)= – R

T

Bi Ki(t)Eix(t),

nxn

where Ki(t) ∈R is a symmetric matrix and the solution of MRDE. The relative MRDE for the linear singular fuzzy system (4) is T

T

T

T

–1

Ei Ki′(t)Ei + Ei Ki(t)Ai + Ai Ki(t)Ei +Q – Ei Ki(t) BiR

T

Bi Ki(t)Ei = 0

(5)

T

with the terminal condition Ki(tf)= Ei SEi. In the following section, the MRDE (5) is going to be solved for Ki(t) in order to get the optimal solution.

3 Simulink Solution of MRDE Simulink is an interactive tool for modelling, simulating and analyzing dynamic systems. It enables engineers to build graphical block diagrams, evaluate system performance and refine their designs. Simulink integrates seamlessly with MATLAB and is tightly integrated with state flow for modelling event driven behavior. Simulink is built on top of MATLAB. A Simulink model for the given problem can be constructed using building blocks from the Simulink library. The solution curves can be obtained from the model without writing any codes. As soon as the model is constructed, the Simulink parameters can be changed according to the problem. The solution of the system of differential equation can be obtained in the display block by running the model. 3.1 Procedure for Simulink Solution Step 1. Select the required number of blocks from the Simulink Library. Step 2. Connect the appropriate blocks. Step 3. Make the required changes in the simulation parameters. Step 4. Run the Simulink model to obtain the solution.

4 Numerical Example Consider the optimal control problem: Minimize tf J= ½ xT(tf)EiTSEix(tf)+ ½ ∫[xT(t)Qx(t)+uT(t)Ru(t)]dt, 0

62

N. Kumaresan, K. Ratnavelu, and B.R. Wong

subject to the linear singular fuzzy system Ri : If xj is Tji(mji,σji), i = 1, 2 and j = 1, 2,3, then Ei x′(t)=Aix(t)+Biu(t), x(0)=x0 , where S= 1.1517 0.1517 0.1517 1

A2=

–2 2 0 –4

Ei = ,

Bi = ,

1 0

0 0

A1= ,

0 R=1, Q = 1 ,

–1 1 0 –2

,

1 1 1 1

The numerical implementation could be adapted by taking tf =2 for solving the related MRDE of the above linear singular fuzzy system with the matrix A1 . The appropriate matrices are substituted in MRDE. The MRDE is transformed into differentia algebraic equation (DAE) in k11 and k12. The DAE can be changed into a system of differential equations by differentiating the algebraic equation. In this problem, the value of k22 of the symmetric matrix K(t) is free and let k22=0. Then the optimal control of the system can be found out by the solution of MRDE. 4.1 Solution Obtained Using Simulink The Simulink model is constructed for MRDE. The Simulink model is shown in Figure 1. The numerical solution of MRDE is calculated by Simulink and displayed in Table 1. The numerical solution curve of MRDE by Simulink is illustrated in Figure 2.

Fig. 1. Simulink Model for MRDE

Optimal Control for Navier-Stokes Takagi-Sugeno Fuzzy Equations Using Simulink

63

Fig. 2. Simulink Curve for MRDE Table 1. Simulink Solution of MRDE t 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

k11 0.0003 0.0007 0.0015 0.0033 0.0074 0.0164 0.0368 0.0828 0.1891 0.4467 1.1517

k12 -0.9997 -0.9993 -0.9985 -0.9967 -0.9926 -0.9836 -0.9632 -0.9172 -0.8109 -0.5533 0.1517

Similarly the solution of the above system with the matrix A2 can be found out using Simulink.

5 Conclusion The optimal control for Navier-Stokes fuzzy equations can be obtained by finding the optimal control for DAE. The optimal control is found out by solving MRDE using Simulink. To obtain the optimal control, the solution of MRDE is computed by solving Differential algebraic equation (DAE) using Simulink. The Simulink solution is equivalent to the exact solution of the problem. Accuracy of the solution computed by Simulink approach to the problem is qualitatively better. A numerical example is given to illustrate the proposed method. Acknowledgements. The funding of this work by the UMRG grant (Account No: RG099/10AFR) is gratefully acknowledged.

64

N. Kumaresan, K. Ratnavelu, and B.R. Wong

References 1. Albagul, A., Khalifa, O.O., Wahyudi, M.: Matlab and Simulink in Mechatronics Education. Int. J. Eng., Edu. (IJEE) 21, 896–905 (2005) 2. Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical solution of Initial Value Problems in Differential Algebraic Equations. SIAM, North Holland (1989) 3. Campbell, S.L.: Singular systems of differential equations. Pitman, Marshfield (1980) 4. Campbell, S.L.: Singular systems of differential equations II. Pitman, Marshfield (1982) 5. Dai, L.: Singular control systems. Lecture Notes in Control and Information Sciences. Springer, New York (1989) 6. Da Prato, G., Ichikawa, A.: Quadratic control for linear periodic systems. Appl. Math. Optim. 18, 39–66 (1988) 7. Jamshidi, M.: An overview on the solutions of the algebraic matrix Riccati equation and related problems. Large Scale Systems 1, 167–192 (1980) 8. Jodar, L., Navarro, E.: Closed analytical solution of Riccati type matrix differential equations. Indian J. Pure and Appl. Math. 23, 185–187 (1992) 9. Kunkel, P., Mehrmann, V.: A new class of discretization methods for the solution of linear differential algebraic equations. SIAM J. Numer. Anal. 33, 1941–1961 (1996) 10. Kunkel, P., Mehrmann, V.: Local and global invariants of linear differential algebraic equations and their relation. Electronics Trans. Numer. Anal. 4, 138–157 (1996) 11. Lewis, F.L.: A survey of linear singular systems. Circ. Syst. Sig. Proc. 5(1), 3–36 (1986) 12. Lovren, N., Tomic, M.: Analytic solution of the Riccati equation for the homing missile linear quadratic control problem. J. Guidance. Cont. Dynamics 17, 619–621 (1994) 13. Quartapelle, I.: Numerical Solution of the incompressible Navier-Stokes Equations. Birkhäuser, Basel (1993) 14. Razzaghi, M.: A Schur method for the solution of the matrix Riccati equation. Int. J. Math. and Math. Sci. 20, 335–338 (1997) 15. Razzaghi, M.: Solution of the matrix Riccati equation in optimal control. Information Sci. 16, 61–73 (1978) 16. Razzaghi, M.: A computational solution for a matrix Riccati differential equation. Numerical Math. 32, 271–279 (1979) 17. Vaughu, D.R.: A negative exponential solution for the matrix Riccati equation. IEEE Trans. Automat. Control 14, 72–75 (1969) 18. Wang, L.X.: Stable and optimal fuzzy control of linear systems. IEEE Trans. Fuzzy Systems 6(1), 137–143 (1998) 19. Weickert, J.: Navier-Stokes equations as a differenctial algebraic system. Preprint SFB 393/96-08, Technische Universität Chemnitz-Zwickau (1996) 20. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning. Inform. Sciences, Part I: (8), 199–249, Part II:(8), 301–357, Part III:(9), 43–80 (1975) 21. Zeng, S., Wesseling, P.: Multigrid solution of the incompressible Navier-Stokes equations in general coordinates. SIAM J. Numer. Anal. 6, 1764–1784 (1994) 22. Zhou, K., Khargonekar, P.: An algebraic Riccati equation approach to H ∞  optimization. Systems Control Lett. 11, 85–91 (1988)

Multi-objective Optimization in VLSI Floorplanning P. Subbaraj, S. Saravanasankar, and S. Anand National Centre for Advance Research in Discrete Mathematics (n-CARDMATH), Kalasalingam University, India [email protected]

Abstract. This paper describes use of a multiobjective optimization method, AMOSA to the VLSI floorplanning problem. The proposed model provides for decision maker choice from among the different tradeoff solutions. The objective is to minimize the area and wirelength with the fixed out line constraint. AMOSA is improved with B*tree and B*tree coded AMOSA is introduced to improve the performance of AMOSA for VLSI floorplanning problem. The MCNC benchmarks are considered as test system. The results are compared and validated. Keywords: VLSI, Floorplanning, Simulated Annealing Algorithm and AMOSA.

1

Introduction

Floorplanning is an important stage in the Very Large Scale Integrated circuit (VLSI) design process. After partitioning the relative location of the different modules in the layout, needs to be decided before the placement (placing the cells in the exact location) stage, with the objective of minimizing the total area occupied by the circuit. Floorplanning proved to be NP hard [7]. For years many algorithms have been developed applying stochastic search methods, such as simulated annealing and genetic algorithms, PSO, however only a few of them work explicitly on multi objective optimization [6,4]. Most of the literatures used the weighted sum approach to minimize the area and wire length simultaneously. This approach always has a difficulty to assign the weights and may results in undesirable bias towards a particular objective. In this paper we proposes B*tree coded AMOSA for VLSI floorplanning problem. The solution is encoded as a B*tree. Proposed algorithm has the advantages of AMOSA and usage of B*tree produce less space complexity [23].

2

VLSI Floorplanning Problem

The VLSI floorplanning problem has two objectives, minimizing the area and minimizing the wirelength. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 65–72, 2011. c Springer-Verlag Berlin Heidelberg 2011 

66

2.1

P. Subbaraj, S. Saravanasankar, and S. Anand

Objective Function and Constraint

The objective function of the problem is formulated as follows. The Cost function Φ for a floor plan solution F is given by,   A minΦ(F1 ) = (1) + K3 (arcost)2 normA   W.L minΦ(F2 ) = (2) + K3 (arcost)2 normW.L Where, Φ(F1 )= Area Objective. Φ(F2 )= Wirelength Objective. A= area of the current solution F . normA = average area of 1000 random iteration. W.L = wire length of the current solution F . normW.L = average wirelength of 1000 random iteration. K3 = weight factor for aspect ratio is assigned as one. arcost = ( Current floorplan aspect ratio - Desired floorplan aspect ratio ). Desired aspect ratio = 1.0. Wire length of a net is calculated by the half perimeter of the minimal rectangle enclosing the centers of the modules. In this work Aspect Ratio (AR) is considered to be the given constraint for the modern fixed outline floorplanning problem. For a collection of blocks with total area (A) and a given maximum permissible percentage of dead space (Γ ), a fixed outline with an Aspect Ratio R∗ (i.e. height/width) is constructed .The height H ∗ and width W ∗ of the outline are defined as in [1]; Only solutions satisfying the fixed outline constraint are considered as feasible. ( ( (3) H ∗ = (1 + Γ )AR∗ , W ∗ = (1 + Γ )A/R∗ 2.2

B*tree Encoding in AMOSA

Floorplan of a VLSI circuit can be divided into two categories. 1. The slicing structure [16,22]. 2. The non-slicing structure [8,14,21]. A non slicing structure can be represented by, sequence pair [12], boundary slicing grid (BSG) [14,9,11,13,15], Otree [8] and B*tree [23]. Yun-chih chang et al., proposed a B*tree representaion based on ordered binary trees and the admissible placement [8]. Yun-chich chang proved [23] that B* tree representation runs 4.5 times faster, consumes about 60 percentage lesser memory and results in smaller silicon area than the Otree [8]. Normal AMOSA having sequence representaion. Due to these advantages of B* tree, it has been used in AMOSA to represent the solution.

3

B*tree Coded Archived Multiobjective Simulated Annealing Algorithm(AMOSA)

Classical optimization methods can at best find one solution in one simulation run, thereby making those methods inconvenient to solve multiobjective optimization problems. Evolutionary algorithms, on the other hand, can find multiple optimal solutions in one single run due to their population approach.

Multi-objective Optimization in VLSI Floorplanning

67

The AMOSA algorithm is described in [2]. In this work B*tree coded multiobjective simulated annealing algorithm (AMOSA) is selected to solve VLSI floorplanning problem, because it has been demonstrated to be among the most efficient algorithm for multiobjecitve optimization on a number of benchmark problems [2]. In addition, it has been shown that AMOSA outperforms two other contemporary multiobjective evolutionary algorithms: Pareto-archived evolution strategy (PAES) and Nondominated sorting genetic algorithm-II (NSGA-II) in terms of finding a diverse set of solutions and in converging near the true Paretooptimal set. 3.1

AMOSA Algorithm

AMOSA is a simulated annealing based multiobjective optimization algorithm that incorporates the concept of archive in order to provide a set of tradeoff solutions for the VLSI floorplanning problem. Simulated Annealing algorithm has been proved [5,10] to be very simple compared to Genetic Algorithm [20] and particle swarm optimization [3] methodologies. Simulated annealing [10] is widely used for floorplanning [17,19,18,5]. It is an optimization scheme with non zero probability for accepting inferior (uphill) solutions. The probability depends on the difference of the solution quality and the temperature. The probability is typically defined by ' & (4) prob = min 1, e(−Δc/T ) Where ΔC is the difference between cost of the neighboring state and current state, and T is the current temperature. In the classical annealing schedule, the temperature is reduced by a fixed ratio λ, for each iteration of the annealing process. The AMOSA algorithm incorporates the concept of an archive where the non-dominated solutions seen so far are stored. Two limits are kept on the size of the archive: a hard or strict limit denoted by HL, and a soft limit denoted by SL. The algorithm begins with the initialization of a number (γ × SL, 0 < γ < 1 ) of solutions each of which represents a state in the search space. The multiple objective functions are computed. Each solution is refined by using simple hillclimbing and domination relation for a number of iterations. Thereafter the non-dominated solutions are stored in the archive until the size of the archive increases to SL. If the size of the archive exceeds HL, a single-linkage clustering scheme is used to reduce the size to HL. Then, one of the points is randomly selected from the archive. This is taken as the current-pt, or the initial solution, at temperature T=Tmax. The current-pt is perturbed to generate a new solution named new-pt, and its objective functions are computed. The domination status of the new-pt is checked with respect to the current-pt and the solutions in the archive. A quantity called amount of domination Δdoma,b between two solutions a and b is defined as follows: Δdoma,b =

M ) i=1,fi (a)=fi (b)

fi (a) − fi (b) Ri

(5)

68

P. Subbaraj, S. Saravanasankar, and S. Anand

where fi (a) and fi (b) are the ith objective values of the two solutions and Ri is the corresponding range of the objective function. Based on domination status different cases may arise viz., accept the (i) new-pt, (ii) current-pt or (iii) a solution from the archive. Again, in case of overflow of the archive, clustering is used to reduce its size to HL. The process is repeated N times for each temperature that is annealed with a cooling rate of α(< 1) till the minimum temperature Tmin (Tmin = 1 ∗ e−4 ) is attained. The process thereafter stops, and the archive contains the final non-dominated solutions. 3.2

Cooling Schedule and Local Search

To implement the cooling schedule of AMOSA, temperature updating function Tn is formulated as shown in equation 6. The temperature is reduced as per the cooling schedule. Tn = Tn−1 ∗ α (6) Local search is performed for every Annealing temperature. Number of local searches is decided by the equation 7. N = {k1 ∗ SCKT }

(7)

Where, SCKT =Size of the circuit. k1 =A constant assigned with a value of 400. k1 gives more exploration capability to AMOSA. The parameter SCKT gives flexibility to the algorithm. When the size of the circuit increases, the local search also will increase. 3.3

Creation of Neighboring Solution

To get the neighboring solution, Yun-Chih Chang proposed a Fast-SA method, where the perturbation of B ∗ tree with the perturbation probability is done using only one of the following three methods [23]; 1) Rotate a block, 2) Move a node/block to another place and 3) Swap two nodes/blocks. The same procedure has been adopted in B*tree coded AMOSA.

4

Obtaining the True Pareto-Front

An alternative approach, weighted sum method with (single objective) FastSA, is experimented to solve the VLSI Floorplanning problem. It is used to validate the results obtained using AMOSA. A pareto optimal set is defined as a set of solutions that are not dominated by any feasible member of the search space; they are optimal solutions of the multi objective optimization problem. To evaluate the performance of AMOSA, a reference Pareto-optimal front is needed. One of the classical methods is weighted sum method. Using this method only one solution may be found along the pareto-front in a single simulation run. the FastSA is used to obtain the true pareto-front. the sum of weights is always 1. The weight value for the first objective function is w1 . The weight w1 is linearly varied from 0 to 1 produce the pareto optimal front.

Multi-objective Optimization in VLSI Floorplanning

5

69

Test Results

The proposed B*coded AMOSA is implemented in C++ programming language on Intel Pentium-D 3.20GHz IBM machine. B*tree coded AMOSA and FastSA are tested with MCNC (Microelectronics center of North Carolina) benchmarks to find the floorplan solutions with the dual objectives of minimizing the wire length and area. Details of the benchmarks is shown in Table 1. Table 1. Details of MCNC bench marks MCNC Circuit #Modules # Nets # I/O Pad # Pins hp 11 83 45 309 ami33 33 123 42 522 ami 49 49 408 22 953

5.1

Parameters, Results and Discussion

The results are sensitive to algorithm parameters, typical of heuristic techniques. Hence, it is required to perform repeated simulations to find suitable values for the parameters. The best parameters for the AMOSA, selected through twenty test simulation runs. The parameters are, Alpha (α)=0.84. Initial Temp (Tini )=100. Soft Limit (SL)=101. Hard Limit (HL)=51. The benchmark has different sizes of layouts, as discussed in section 5. The extreme solution of all benchmarks is shown in table 2 and shown on the pareto-front of Fig. 1(a), 1(b) and 1(c) Identification of pareto-front. The pareto-optimal front obtained by different using AMOSA and FastSA for benchmark ami49, ami33, and hp is shown in Fig. 1(a), 1(b) and 1(c) respectively. It shows the trade of between the area and wirelength. In single objective optimization FastSA produces the solution in different runs for different weights as described in section 4. In contrast, B*tree coded AMOSA produces the pareto optimal front in a single simulation run and the execution time is less ( 200 sec for ami49 ) compare to the single objective optimization process. Table 2. Extreme solutions of FastSA and AMOSA ami49 Bench mark

Area (mm2 ) FastSA 36.65 48.22 B*tree 36.65 coded AMOSA 47.16

WL (mm) 1833 837.4 1833

ami33 Area (mm2 ) 1.19 1.408 1.19

WL (mm) 114.5 57.64 114.5

hp Area (mm2 ) 9.173 15.08 9.173

WL (mm) 275.4 178.6 275.4

725.6 1.366 51.09 10.73 157.2

70

P. Subbaraj, S. Saravanasankar, and S. Anand

(a) ami49

(b) ami33

(c) hp

Fig. 1. Pareto-optimal front of : (a), (b) and (c)

Trade-off analysis. From the Fig. 1(a), 1(b) and 1(c) it is clear that the increase in Area gives decrease in wirelength, and this directly causes variation in white space. This observation will help the decision maker to select a single solution from the pareto-optimal front. Form the Fig. 1(c), it is very clear that Fast SA can not able to find the optimal solutions when comparing to AMOSA.

6

Conclusion

In this paper an attempt is made to solve multiobjective VLSI floorplanning problem using B*tree coded AMOSA. Multiobjective optimization enables evolution of tradeoff solutions between different problem objectives through identification of the Pareto-front. The B*tree coded AMOSA algorithm provides that this can be done with very high computational efficiency and provides more exploration capability. This approach is quite flexible so that other formulations using different objectives and/or a larger number of objectives are possible.Proposed approach has been compared with the FastSA method, from the results it is clear that, “B*tree coded AMOSA” enabling the decision maker to understand their options, ultimately resulting in lower White space in lower simulation time. Acknowledgement. The authors thank the Department of Science and Technology, New Delhi, Government of India. For the support under the project no: SR/S4/MS: 326/06.

Multi-objective Optimization in VLSI Floorplanning

71

References 1. Adya, S.N., Markov, I.L.: Fixed-outline floorplanning through better local search. In: ICCD 2001: Proceedings of the International Conference on Computer Design: VLSI in Computers & Processors, pp. 328–334. IEEE Computer Society, Austin (2001) 2. Bandyopadhyay, S., Saha, S., Maulik, U., Deb, K.: A simulated annealing based multi-objective optimization algorithm: Amosa. IEEE Transactions on Evolutionary Computation 12, 269–283 (2008) 3. Chen, G., Guo, W., Chen, Y.: A PSO-based intelligent decision algorithm for vlsi floorplanning. In: Soft Computing - A Fusion of Foundations, Methodologies and Applications, vol. 14, pp. 1329–1337 (2010) 4. Chen, J., Chen, G., Guo, W.: A discrete PSO for multi-objective optimization in vlsi floorplanning. In: ISICA 2009: Proceedings of the 4th International Symposium on Advances in Computation and Intelligence, pp. 400–410. Springer, Heidelberg (2009) 5. Chen, T.-C., Chang, Y.-W.: Modern floorplanning based on b*-tree and fast simulated annealing. IEEE Trans on CAD 25, 510–522 (2006) 6. Fernando, P., Katkoori, S.: An elitist non-dominated sorting based genetic algorithm for simultaneous area and wirelength minimization in vlsi floorplanning. In: Proceedings of 21st International Conference on VLSI Design, pp. 337–342 (2008) 7. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979) 8. Guo, P.-N., Cheng, C.-K., Yoshimura, T.: An O-tree representation of non-slicing floorplan and its applications. In: DAC 1999: Proceedings of the 36th Annual ACM/IEEE Design Automation Conference, pp. 268–273. ACM, New York (1999) 9. Kang, M., Dai, W.: General floorplanning with L-shaped, T-shaped and soft blocks based on bounded slicing grid structure. In: Proceedings of Asia and South Pacific -Design Automation Conference, pp. 265–270 (1997) 10. Kirpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220, 671–680 (1983) 11. Murata, H., Fujiyoshi, K., Kaneko, M.: VLSI/PCB placement with obstacles based on sequence-pair. In: ISPD 1997: Proceedings of the 1997 International Symposium on Physical Design, pp. 26–31. ACM, New York (1997) 12. Murata, H., Fujiyoshi, K., Nakatake, S., Kajitani, Y.: Rectangle-packing-based module placement. In: ICCAD 1995: Proceedings of the 1995 IEEE/ACM International Conference on Computer-aided Design, pp. 472–479. IEEE Computer Society, Washington, DC (1995) 13. Murata, H., Kuh, E.S.: Sequence-pair based placement method for hard/soft/preplaced modules. In: ISPD 1998: Proceedings of the 1998 International Symposium on Physical Design, pp. 167–172. ACM, New York (1998) 14. Nakatake, S., Fujiyoshi, K., Murata, H., Kajitani, Y.: Module placement on BSGstructure and IC layout applications. In: ICCAD 1996: Proceedings of the 1996 IEEE/ACM International Conference on Computer-aided Design, pp. 484–491. IEEE Computer Society, Washington, DC (1996) 15. Nakatake, S., Fujiyoshi, K., Murata, H., Kajitani, Y.: Module placement on BSGstructure with Pre-Places modules and rectilinear modules. In: Proceedings of the Asia and South Pacific Design Automation Conference (ASP-DAC 1998), pp. 571– 576 (1998)

72

P. Subbaraj, S. Saravanasankar, and S. Anand

16. Otten, R.H.: Automatic floorplan design. In: DAC 1982: Proceedings of the 19th Design Automation Conference, pp. 261–267. IEEE Press, Piscataway (1982) 17. Sechen, C., Sangiovanni-Vincentelli, A.: The timberwolf placement and routing package. IEEE J.Solid State Circuits 20, 510–522 (1985) 18. Sechen, C.: Chip-planning, placement, and global routing of macro/custom cell integrated circuits using simulated annealing. In: DAC 1988: Proceedings of the 25th ACM/IEEE Design Automation Conference, pp. 73–80. IEEE Computer Society Press, Los Alamitos (1988) 19. Sechen, C., Sangiovanni-Vincentelli, A.: Timberwolf 3.2: a new standard cell placement and global routing package. In: DAC 1986: Proceedings of the 23rd ACM/IEEE Design Automation Conference, pp. 432–439. IEEE Press, Piscataway (1986) 20. Tang, M., Sebastian, A.: A genetic algorithm for vlsi floorplanning using o-tree representation. In: Rothlauf, F., Branke, J., Cagnoni, S., Corne, D.W., Drechsler, R., Jin, Y., Machado, P., Marchiori, E., Romero, J., Smith, G.D., Squillero, G. (eds.) EvoWorkshops 2005. LNCS, vol. 3449, pp. 215–224. Springer, Heidelberg (2005) 21. Wang, T.-C., Wong, D.F.: An optimal algorithm for floorplan area optimization. In: DAC 1990: Proceedings of the 27th ACM/IEEE Design Automation Conference, pp. 180–186. ACM, New York (1990) 22. Wong, D.F., Liu, C.L.: A new algorithm for floorplan design. In: DAC 1986: Proceedings of the 23rd ACM/IEEE Design Automation Conference, pp. 101–107. IEEE Press, Piscataway (1986) 23. Wu, G.-M., Wu, S.-W., Chang, Y.-W., Chang, Y.-C.: B*-trees: A new representation for non-slicing floorplans. In: Proceedings of the 37th Design Automation Conference, pp. 458–463 (2000)

Approximate Controllability of Fractional Order Semilinear Delay Systems N. Sukavanam and Surendra Kumar Indian Institute of Technology Roorkee, Rorkee 247667, Uttarakhand, India {nsukvfma,sk001dma}@iitr.ernet.in

Abstract. In this paper, we prove the approximate controllability for a class of semilinear delay control systems of fractional order of the form dα y(t) = Ay(t) dtα y0 (θ) = φ(θ),

+ v(t) + f (t, yt , v(t)), t ∈ [0, τ ]; θ ∈ [−h, 0],

where A is linear operators and f is a nonlinear operator defined on appropriate Banach space. Keywords: Controllability, Fractional order delay system, Infinite dimensional system.

1

Introduction

Let V be a Banach space and C([−h, 0], V ) be the Banach space of all continuous functions from [−h, 0] to V . The norm on C([−h, 0], V ) is defined as xt C =

sup xt (θ) V , for t ∈ (0, τ ].

−h≤θ≤0

Consider the following fractional order semilinear delay equation  dα y(t) dtα = Ay(t) + v(t) + f (t, yt , v(t)), t ∈ (0, τ ]; θ ∈ [−h, 0], y0 (θ) = φ(θ),

(1)

where 0 < α < 1; A : D(A) ⊆ V → V is a closed linear operator with dense domain D(A) generating a C0 -Semigroup T (t); the state y(·) and the control function u(·) take values in V ; the operator f : [0, τ ] × C([−h, 0], V ) × V → V is nonlinear. If y : [−h, τ ] → V is a continuous function then yt : [−h, 0] → V is defined as yt (θ) = y(t + θ) for θ ∈ [−h, 0] and φ ∈ C([−h, 0], V ). For the above system (1) controllability results are not available in the literature. But several authors have proved controllability of fractional order system of the form (1), when the nonlinear term is independent of the control v [1,2,3,4]. In [5,6] the approximate controllability of first order delay control systems of the form (1) (when α = 1) has been proved when f is a function of both yt and v. In this paper, we prove the approximate controllability of the system (1), when 0 < α < 1. The paper is organized as follows: In Section 2, we present some necessary preliminaries. The approximate controllability of the system (1) is proved in Section 3. In Section 4 an example is given to illustrate our result. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 73–78, 2011. c Springer-Verlag Berlin Heidelberg 2011 

74

2

N. Sukavanam and S. Kumar

Preliminaries

In this section some basic definitions which are useful for further development are given. Definition 1. A real function f (t) is said to be in the space Cα , α ∈ R if there exists a real number p > α, such that f (t) = tp g(t), where g ∈ C[0, ∞) and it is said to be in the space Cαm if and only if f (m) ∈ Cα , m ∈ N . Definition 2. The Riemann-Liouville fractional integral operator of order β > 0 of function f ∈ Cα , α ≥ −1 is defined as  t 1 (t − s)β−1 f (s)ds. I β f (t) = Γ (β) 0 where Γ is the Euler gamma function. Note that if f ∈ Cα the above integral exists only under the constants β > 0 and p > α ≥ −1. m Definition 3. If the function f ∈ C−1 and m is a positive integer then we can define the fractional derivative of f (t) in the Caputo sense as

1 dα f (t) = α dt Γ (m − α)



t

(t − s)m−α−1 f m (s)ds, m − 1 ≤ α < m.

0

For more details about the theory of fractional differential equations we refer the books [7,8,9]. The state function is defined as  yt (0), t ∈ (0, τ ]; yt (θ) = y0 (θ), θ ∈ [−h, 0]. Definition 4. A function y(·, φ, v) ∈ C([−h, τ ], V ) is said to be a mild solution of (1) if it satisfies 

t 1 T (t)φ(0) + Γ (α) (t − s)α−1 T (t − s)[v(s) + f (s, ys , v(s))]ds, t ∈ (0, τ ]; 0 yt (θ) = φ(θ),

θ ∈ [−h, 0],

on [−h, τ ]. Let y(τ, φ(0), v) be the state value of system (1) at time τ corresponding to the control v and the initial value φ(0). The system (1) is said to be approximate controllable in time interval [0, τ ], if for every desired final state x1 and  > 0 there exists a control function v ∈ V such that the solution x(t, φ(0), v) of (1) satisfies x(τ, φ(0), v) − x1 < 

Approximate Controllability of Fractional Order Semilinear Delay Systems

75

Definition 5. The set Kτ (f ) = {x(τ, φ(0), v) ∈ V : x(·, φ, v) ∈ C([−h, τ ], V ) is a mild solution of (1)} is called the Reachable set of the system (1). Definition 6. The system (1) is said to be approximate controllable on [0, τ ] if Kτ (f ) = V where Kτ (f ) denote the closure of Kτ (f ). When f = 0, then system (1) is called corresponding linear system denoted by (1)∗ . In this case, Kτ (0) denotes the reachable set of the linear system (1)∗ . It is clear that (1)∗ is approximately controllable if Kτ (0) = V.

3

Controllability Results

In this section, we prove the approximate controllability of the semilinear system (1) under suitable conditions. Consider the linear system with a control u as  dα x(t) dtα = Ax(t) + u(t), t ∈ (0, τ ]; (2) x0 (θ) = φ(θ), θ ∈ [−h, 0], Assumptions[H]. Consider the following assumptions: (H1) The linear system (2) is approximate controllable. (H2) The nonlinear operator f (t, xt , u(t)) satisfies the Lipschitz condition i.e. there exists a positive constant l > 0 such that f (t, xt , u(t)) − f (t, yt , v(t)) V ≤ l[ xt − yt C + u(t) − v(t) V ] for all xt , yt ∈ C([−h, 0], V ); u, v ∈ V and t ∈ (0, τ ]. Theorem 1. Under assumptions (H1)-(H2) the semilinear control system (1) is approximate controllable if l < 1. Proof. Let xt (θ) be a mild solution of (2) corresponding to a control u and consider the following semilinear system  dα y(t) dtα = Ay(t) + f (t, yt , v(t)) + u(t) − f (t, xt , v(t)), t ∈ (0, τ ]; (3) θ ∈ [−h, 0], y0 (θ) = φ(θ), On comparing (1) and (3), it can be seen that the control function v satisfies the following condition v(t) = u(t) − f (t, xt , v(t)). Later in the proof we show that there exists a unique v(t) satisfying the above equation. The mild solution of (2) is 

t 1 T (t)φ(0) + Γ (α) (t − s)α−1 T (t − s)u(s)ds, t ∈ (0, τ ]; 0 xt (θ) = (4) φ(θ), θ ∈ [−h, 0],

76

N. Sukavanam and S. Kumar

and the mild solution of (3) is ⎧

t 1 (t − s)α−1 T (t − s) ⎨ T (t)φ(0) + Γ (α) 0 yt (θ) = ×[f (s, ys , v(s)) + u(s) − f (s, xs , v(s))]ds, t ∈ (0, τ ]; ⎩ φ(θ), θ ∈ [−h, 0].

(5)

If θ ∈ [−h, 0], then from (4) and (5), we get x0 (θ) − y0 (θ) = 0

(6)

and if t ∈ (0, τ ], then we have  t 1 xt (0) − yt (0) = (t − s)α−1 T (t − s)[f (s, xs , v(s)) − f (s, ys , v(s))]ds Γ (α) 0 Taking norm on both sides, we get xt (0) − yt (0)V ≤

1 Γ (α)

≤

M Γ (α)

 

t 0 t 0

(t − s)α−1 T (t − s)f (s, xs , v(s)) − f (s, ys , v(s))V ds (t − s)α−1 f (s, xs , v(s)) − f (s, ys , v(s))V ds

where M is a constant such that T (t) ≤ M , for all t ∈ [0, τ ]. Using assumption (H2), we have  t Ml xt (0) − yt (0) V ≤ (t − s)α−1 xs − ys C ds (7) Γ (α) 0 From (6) and (7), we get xt (θ) − yt (θ) V ≤

Ml Γ (α)



t

(t − s)α−1 xs − ys C ds, ∀ − h ≤ θ ≤ 0

0

This implies that xt − yt C ≤

Ml Γ (α)



t

(t − s)α−1 xs − ys C ds

0

Hence Gronwall’s inequality implies that x(t) = y(t) for all t ∈ [−h, τ ]. From this we infer that every solution of the linear system with control u is also a solution of the semilinear system with control v. Hence, Kτ (f ) ⊃ Kτ (0). Now since Kτ (0) is dense in V (by assumption (H1)) it follows that Kτ (f ) is dense in V . Hence the system (1) is approximate controllable. The proof will be complete if we show that there exists a v(t) ∈ V such that v(t) = u(t) − f (t, xt , v(t)) Let v0 ∈ V and vn+1 = u − f (t, xt , vn ), for n = 1, 2, · · ·. Then, we have vn+1 − vn V ≤ f (t, xt , vn−1 ) − f (t, xt , vn ) V ≤ l vn − vn−1 V ≤ ln v1 − v0 V

Approximate Controllability of Fractional Order Semilinear Delay Systems

77

Since l < 1, R.H.S. of above inequality tends to zero as n → ∞. Therefore, the sequence vn is Cauchy in V . Since V is a complete normed linear space, vn converges to an element v ∈ V . Now (u − vn+1 ) − f (t, xt , v) V = f (t, xt , vn ) − f (t, xt , v) V ≤ l vn − v V Since R.H.S. of above inequality tends to zero as n → ∞, we get f (t, xt , v) = lim (u − vn+1 ) = u − v n→∞

Hence v = u − f (t, xt , v). Uniqueness of v can be proved easily. This completes the proof.

4

Example

In this section we present an example to illustrate our results. Let V = L2 (0, π) d2 y d2 and A = dx 2 with D(A) consisting of all y ∈ V with dx2 and y(0) = 0 = y(π). Put φn (x) = ( π2 )1/2 sin(nx); 0 ≤ x ≤ π, n = 1, 2, · · ·, then {φn , n = 1.2, · · ·} is an orthonormal base for V and φn is the eigenfunction corresponding to the eigenvalue λn = −n2 of the operator A, n = 1, 2, ···. Then the C0 -semigroup T (t) generated by A has exp(λn t) as the eigenvalues and φn as their corresponding eigenfunctions [10]. Let us consider the following semilinear control system of the form ∂α ∂2 y(t, x) = y(t, x) + u(t, x) + f (t, y(t − h, x), u(t, x)); t ∈ (0, τ ], 0 < x < π ∂tα ∂t2 y(t, 0) = y(t, π) = 0; t > 0 y(t, x) = φ(t, x); t ∈ [−h, 0],

(8)

where φ(t, x) is continuous. If the conditions (H1) and (H2) are satisfied, then the approximate controllability follows from Theorem 3.1. For example, if we consider the function f as f (t, z, v) = l[ z φ3 + v φ4 ], where l ∈ (0, 1), then f satisfies (H2) with Lipschitz constant l < 1. From Theorem 3.1 the approximate controllably of (8) follows.

References 1. Abd El-Ghaffar, A., Moubarak, M.R.A., Shamardan, A.B.: Controllability of Factional Nonlinear Control System. Journal of Fractional Calculus 17, 59–69 (2000) 2. Balachandran, K., Park, J.Y.: Controllability of Fractional Integrodifferential Systems in Banach Spaces. Nonlinear Analysis: Hybrid Systems 3, 363–367 (2009) 3. Bragdi, M., Hazi, M.: Existence and Controllability Result for an Evolution Fractional Integrodifferential Systems. Int. J. Contemp. Math. Sciences 5(19), 901–910 (2010)

78

N. Sukavanam and S. Kumar

4. Tai, Z., Wang, X.: Controllability of Fractional-Order Impulsive Neutral Functional Infinite Delay Integrodifferential Systems in Banach Spaces. Applied Mathematics Letters 22, 1760–1765 (2009) 5. Dauer, J.P., Mahmudov, N.I.: Approximate Controllability of Semilinear Functional Equations in Hilbert Spaces. J. Math. Anal. Appl. 273, 310–327 (2002) 6. Sukavanam, N., Tomar, N.K.: Approximate Controllability of Semilinear Delay Control Systems. Nonlinear Functional Analysis and Applications 12(1), 53–59 (2007) 7. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) 8. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) 9. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999) 10. Naito, K.: Controllability of Semilinear Control Systems Dominated by the Linear Part. SIAM J. Control and Optimization 25(3), 715–722 (1987)

Using Genetic Algorithm for Solving Linear Multilevel Programming Problems via Fuzzy Goal Programming Bijay Baran Pal1,*, Debjani Chakraborti2, and Papun Biswas3 1

Department of Mathematics, University of Kalyani, Kalyani-741235, W.B., India Tel.: +91-33-25825439 [email protected] 2 Department of Mathematics, Narula Institute of Technology, Kolkata-700109, W.B., India [email protected] 3 Department of Electrical Engineering, JIS College of Engineering, Kalyani-741235, W.B., India [email protected]

Abstract. This article presents a fuzzy goal programming (FGP) procedure for modeling and solving multilevel programming (MLP) problems by using genetic algorithm (GA) in a large hierarchical decision making system. In the proposed approach, an GA scheme is introduced first for searching of solutions at different stages and thereby solving the problem and making decision in the order of hierarchy of execution of decision powers of the decision makers (DMs) located at different hierarchical levels. In the proposed GA scheme, Roulette-wheel selection scheme, single point crossover and random mutation are adopted to search a satisfactory solution in the hierarchical decision system. To illustrate the potential use of the approach, a numerical example is solved. Keywords: Multilevel Programming, Fuzzy Programming, Fuzzy Goal Programming, Goal Programming, Genetic Algorithm, Membership Function.

1 Introduction The MLP is a special field of study in the area of mathematical programming for solving decentralized planning problems with multiple DMs involving different objectives in a large hierarchical decision organization. In MLP problems (MLPPs), one DM is located at each of the different hierarchical levels and each control a decision vector and an objective function separately in the decision making context. The execution of the decision power is sequential from an upper-level to a lower-level, and each DM tries to optimize his/her own benefit, where the objectives often conflict each other in the decision making situation. In a practical decision situation, although the execution of the decision powers is sequential, the decisions of the upper-level DMs are often affected by the reactions of the lower-level DMs due to their dissatisfactions with the decisions of the upper-level DMs. *

Corresponding author.

P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 79–88, 2011. © Springer-Verlag Berlin Heidelberg 2011

80

B.B. Pal, D. Chakraborti, and P. Biswas

As a consequence, most of the hierarchical decision organizations are frequently faced with the problems of distribution of decision powers to the DMs for overall benefit of the organization. During the last four decades, MLPPs as well as bilevel programming problems (BLPPs) as a special case of MLPPs have been studied deeply in [1], [2], [3], [4], [5], [6] from the view point of their potential use to different decision problems. But the use of the classical approaches often lead to the paradox that the decision power of a lower-level DM dominates that of a higher-level DM. To overcome this situation, an ideal point dependent solution approach based on the preferences of the DMs for achievement of their objectives has been introduced by Wen and Hsu [7]. But their method does not always provide a satisfactory decision in a highly conflicting hierarchical decision situation. In order to overcome the shortcomings of the classical approaches, fuzzy programming (FP) [8] approach to hierarchical decision problems has been introduced by Lai [9] in 1996 to solve BLPPs as well as MLPPs. The solution concept of Lai has been further extended by Shih et al. [10] and Shih and Lee [11] with a view to make a balance of decision powers to the DMs. In these approaches, the elicited membership functions for the fuzzy goals are redefined again and again to reach a satisfactory decision in the solution search process. To avoid such a computational difficulty, a goal satisficing method in goal programming GP [12] for minimizing the regrets of the DMs in case of a BLPP has been studied by Pal and Moitra [13]. However, the extensive study on fuzzy MLPPs is at an early stage. Now, in a decision making environment, GAs initially introduced by Holland [14] which based on natural selection and population genetics have appeared as a flexible and robust tools for solving multiobjective decision problems [15]. The GA based solution approaches to decision problems in the framework of GP in [12], [16] have been studied [17] in the past. The use of GAs to FP formulations of BLPPs as well as MLPPs has been studied in [18], [19] by the pioneer researchers in the field. The potential use of GA based on GP method to Interval-valued MLPPs has been studied [20] in the recent past. However, the GA based solution method to the FGP formulations of BLPPs as well as MLPPs is yet to circulate in the literature. In this paper, the efficient use of GA to MLPPs for distribution of proper decision powers to the DMs within a hierarchical decision structure is presented. In the model formulation, the individual decision of each of the DMs is determined first by employing the proposed GA scheme to define the fuzzy goals of the problem. Then, the optimal decision of the problem under the framework of minsum FGP is determined through the proposed GA solution search process. A numerical illustration is provided to expound the approach. Now, the formulation of the MLPP is presented in the Section 2.

2 Formulation of MLPP In the hierarchical decision situation, let X = ( x1 , x 2 ,⋅ ⋅ ⋅, x n ) be the vector of decision variables involved with different levels of the decision system. Then, let Z i be the objective function and X i be the control vector of the decision variables of the i -th level DM, i = 1,2,..., I; I ≤ n where ∪ {X i = 1,2,..., I } = X i

i

Using Genetic Algorithm for Solving Linear MLPPs via FGP

81

Then, the general MLPP within a hierarchical nested decision structure can be presented as: Find X ( X1 , X 2 ,... . , X I ) so as to: max Z 1 ( X) = X1

I

∑

i =1

p1i X i

(top-level)

where, for given X1 ; X 2 , X 3 ,..., X I solve max Z 2 ( X) = X2

I

∑

i =1

p 2i X i

(second-level)

where, for given X1 and X 2 ; X 3 , X 4 ,..., X I solve … … … … … … where, for given X1 , X 2 ,..., X I −1 ; X I solves max Z I ( X) = XI

I

∑

i =1

p Ii Xi

(I -th level) I

subject to ( X1 , X 2 ,..., X I ) ∈ S = { ∑ A i X i ≤ b} , X i ≥ 0, i = 1,2,..., I i =1

(1)

where p ji ( j = 1,2,..., I ) and b are constant vectors, A i (i = 1,2,..., I ) are constant matrices. It is also assumed that S (≠ Φ) is bounded. Now, it is to be observed that the problem (1) is multiobjective in nature and the execution of decision power of the DMs is sequential from the top-level to the bottom level. Now, to develop the fuzzy goals of the problem and then to solve the problem under the framework of the FGP, a GA scheme is introduced in the Section 3.

3 GA Scheme for MLPP In the literature of the GAs, there are a number of schemes [14] for generation of new populations with the use of the different operators: selection, crossover and mutation. Here, the binary coded representation of a candidate solution called chromosome is considered to perform genetic operations in the solution search process. The conventional Roulette wheel selection scheme in [15], single-point crossover [15] and bit-bybit mutation operations are adopted to generate offspring in new population in the search domain defined in the decision making environment. The fitness score of a chromosome v (say) in evaluating a function, say, eval( Ev ) , is based on maximization or minimization of an objective function called the fitness function defined on the basis of DM’s needs and desires in the decision making context. The fitness function is defined as eval( E v ) = ( Z i ) v , i = 1,2,..., I ; v = 1,2,..., pop_size.

(2)

where Z i represents the objective function of the i -th level DM given in (1), and where the subscript v is used to indicate the fitness value of the v -th chromosome, v = 1,2,..., pop_size. The best chromosome with largest fitness value at each generation is determined as:

82

B.B. Pal, D. Chakraborti, and P. Biswas

E * = max{eval( E v ) v = 1,2,..., pop_size}, or, E * = min{eval( E v ) v = 1,2,..., pop_size} , depending on searching of the maximum or minimum value of an objective. Now, formulation of FGP of the problem (1) by defining the fuzzy goals is presented in the Section 4.

4 FGP Problem Formulation To formulate the FGP model of the given problem, the imprecise aspiration levels of the objectives of the DMs at all the levels and the decision vectors of the upper-level DMs as well as the tolerance limits for achieving the respective aspired levels are determined first. Then, they are characterized by their membership functions for measuring the degree of achievement of the aspired goal levels in the decision situation. In the present decision situation, the individual best decisions of the DMs are taken into consideration and they are evaluated by using the proposed GA scheme. Let ( X1i , X i2 ,..., X iI ; Z *i ) be the best decision of the i-th level DM, where

Z i* = max Z i ( X) . X∈S

Then the fuzzy goal objectives can be presented as Z i ) Z i* ,

i = 1,2,..., I .

Similarly, the fuzzy goals for the control vectors of the upper-level DMs appear as X i ) X ii ,

i = 1,2,..., I − 1.

Now, from the view point of considering the relaxation made by an upper-level DM for the benefit of a lower-level DM, the lower-tolerance limit for the i-th fuzzy goal Z i can be determined as: Z ii +1[= Z i ( X1i +1 , X i2+1 ,..., X iI+1 )] < Z i* , i = 1,2,..., I − 1.

Again, since the I-th level DM is at the bottom level, the lower-tolerance limit of the objective Z I can be determined as Z Im [= min( Z i ( X1i , X i2 ,..., X iI ); i = 1,2,...I − 1)] < Z *I

Now, since the DMs are motivated to co-operate each other and a certain relaxation on the decision of each of the upper-level DMs is made for the benefit of a lower level DM, the lower tolerance limit of the decision X i can be determined as X in ( X ii +1 < X in < X ii ), i = 1,2,...I − 1. Then, the construction of the membership functions of the defined fuzzy goals is presented in the Section 4.1.

Using Genetic Algorithm for Solving Linear MLPPs via FGP

83

4.1 Construction of Membership Functions

For the defined lower-tolerance limits of the fuzzy goals, the membership functions can be constructed as follows: ⎧ 1 ⎪ ⎪⎪ Z (X) − Z ii +1 μ Zi [Z i (X)] = ⎨ i * i +1 ⎪ Zi − Zi ⎪ ⎪⎩ 0

if Z i (X) ≥ Z *i , if Z ii +1 ≤ Z i (X) < Z i* , if Z i (X) < Z ii +1, i = 1,2,...,I − 1.

⎧ 1 ⎪ ⎪⎪ Z (X) − Z Im μ Z I [Z I (X)] = ⎨ I * m ⎪ ZI − ZI ⎪ ⎪⎩ 0 ⎧ ⎪ ⎪⎪ X μ Xi [ X i ] = ⎨ i i ⎪ Xi ⎪ ⎪⎩

(3)

if Z I (X) ≥ Z *I , if Z Im ≤ Z I (X) < Z *I ,

(4)

if Z I (X) < Z Im , if X i ≥ X ii ,

1 − X in

if X in ≤ X i < X ii ,

− X in

if X i < X in ,

0

(5)

i = 1, 2, . . . , I − 1.

Now, the FGP model formulation is presented in the following Section 4.2. 4.2 FGP Model Formulation

In the FGP model formulation, the membership functions are transformed into goals by assigning the aspiration level unity and introducing under- and over-deviational variables to each of them. Then, achievement of the aspired goal levels to the extend possible by minimizing the sum of the weighted under-deviational variables in the goal achievement function on the basis of weights of importance of achieving the goals is taken into account. The minsum FGP formulation of the problem can be presented as: Find ( X1 , X 2 ,..., X I ) so as to: I

Minimize Z =

∑ i

and satisfy μ Z i :

=1

wi− d i− +

Z i ( X) − Z ii +1 Z i*

−

Z ii +1

2 I −1 ∑

l = I +1

w l− d l− ,

+ d i− − d i+ = 1, i = 1,2,..., I − 1.

m μ Z I : Z I ( X) − Z I + d − − d + = 1 I I Z *I − Z Im

84

B.B. Pal, D. Chakraborti, and P. Biswas n μ Xi : X i − X i + d − − d + = I , l l l X ii − X in

l = I + i, where i = 1,2,..., I − 1.

d i+ , d i− ≥ 0, d l+ , d l− ≥ 0,

(6)

subject to the given system constraints in (1), where d i+ , d i− represent the over- and under-deviational variables, respectively, and d l+ , d l− ≥ 0, are the vectors of over- and under-deviational variables associated with the respective goals. Z represents the goal achievement function consisting of the weighted under-deviational variables and vectors of weighted under-deviational vari− ables, where the numerical weights wi (> 0), i = 1,2,..., I , and the vector of the nu− merical weights w l (> 0), l = I + 1, I + 2,...,2 I − 1 , represent the relative weights of importance of achieving the goals to their aspired levels, and they are determined as:

wi− = wI− =

1 Z i*

− Z ii +1 1

Z *I

−

Z Im

, i = 1,2,..., I − 1 , and w l− =

1 X ii

− X in

, l = I + i, where i = 1,2,..., I − 1.

Now, since MLPP is a large hierarchical decision structured problem and the objectives at the various decision levels often conflict each other in the decision making situation, the use of conventional FGP solution approach may not always provide the satisfactory decision from the view point of distribution of proper decision power to the DMs in the decision making environment. Here, the GA method as a decision satisficer [14] rather than optimizer can be employed in the solution search process. The fitness function under the GA scheme appears as: eval( E v ) = ( Z i ) v =

I

∑

i =1

wi− d i− +

2 I −1 ∑

l = I +1

w l− d l− , v = 1,2,..., pop_size.

(7)

In, the decision search process, the best chromosome E* with the highest score at a generation is determined as: E * = min{eval( E v ) v = 1,2,..., pop_size} in the genetic search process. The efficient use of the proposed approach is illustrated by a numerical example.

5 An Illustrative Example The following trilevel problem, studied by Anandalingam [1] is considered. Find ( x1 , x 2 , x3 ) so as to

max Z 1 = 7 x1 + 3x 2 − 4 x3 x1

(top-level)

Using Genetic Algorithm for Solving Linear MLPPs via FGP

85

where, for given x1 , x 2 and x3 solve max Z 2 = x 2

(middle-level)

x2

where, for given x1 and x 2 , x3 solves max Z 3 = x3

(bottom-level)

x3

subject to x1 + x2 + x3 ≤ 3, x1 + x2 − x3 ≤ 1, x1 + x2 + x3 ≥ 1, − x1 + x2 + x3 ≤ 1, x3 ≤ 0.5, x1, x2 , x3 ≥ 0

(8)

Now, to solve the problem by employing the proposed GA scheme, the following genetic parameter values are incorporated in the decision search process. • • • •

population size = 100 probability of crossover = 0.8 probability of mutation = 0.08 length of the chromosome = 30.

The GA is implemented using the Programming Language C. The execution is performed in an Intel Pentium IV PC with 2.66 GHz. Clock-pulse and 1 GB RAM. Now, following the procedure, the individual optimal solutions of the three successive levels are obtained as: 1 1 1 (i) ( x1 , x 2 , x 3 ; Z 1 ) = (1.4997, 0, 0.4998 ; 8.4990) (ii) ( x12 , x 22 , x 32 ; Z 2 ) = (0.3567, 1.000, 0.3567 ; 1.000)

(top-level) (middle-level)

3 3 3 (iii) ( x1 , x 2 , x 3 ; Z 3 ) = (0.5522, 0.4914, 0.50 ; 0.50)

(bottom-level)

Then, the fuzzy goals are obtained as: Z 1 )8.4990, Z 2 )1, Z 3 )0.5 and

x1 )1.4997, x 2 )1.

Following the procedure, the lower-tolerance limits of the objective goals are determined as:

Z12 = 4.1, Z 23 = 0.4914, Z 3m = 0.3567 . Then, the top-level and middle-level DMs feel that their control variables x1 and x 2 should be relaxed up to 1.35 and 0.8, respectively, for the benefit of the respective lower-level DMs, and not beyond of them. So, x1n = 1.35 ( x12 < 1.35 < x11 ) and

86

B.B. Pal, D. Chakraborti, and P. Biswas

x 2n = 0.8 ( x 23 < 0.8 < x 22 ) act as lower tolerance limits of the decisions x1 and x 2 , respectively. Now, following the procedure and using the above numerical values, the membership functions of the defined fuzzy goals can be constructed by (3), (4) and (5). Then, the resultant FGP model appears as: Find ( x1 , x 2 , x3 ) so as to Minimize Z =

1 1 1 1 1 − d 1− + d 2− + d 3− + d 4− + d5 4.399 0.5086 0.1433 0.1497 0 .2

and satisfy: 7 x1 + 3 x 2 − 4 x 3 − 4.1 + d 1− − d 1+ = 1 4.399 x − 0.4914 μZ2 : 2 + d 2− − d 2+ = 1 0.5086 x − 0.3567 + μ Z3 : 3 + d 3− − d 3 = 1 0.1433 x1 − 1.35 μ x1 : + d 4− − d 4+ = 1 0.1497 x − 0.8 μ x2 : 2 + d 5− − d 5+ = 1 0 .2 μ Z1 :

d q+ , d q− ≥ 0, q = 1,2,.....,5. subject to the given system constraints in (8) Now, employing the GA scheme with the evaluation function in (7), the resultant solution of the problem is obtained as ( x1 , x 2 , x 3 ) = (1.201, 0.398, 0.5) with ( Z 1 , Z 2 , Z 3 ) = (7.319, 0.398, 0.5) The achieved membership values of the objective goals are μ Z1 = 0.7 , μ Z 2 = 0, μ Z 3 = 1 The solution achieved here is the most satisfactory one from the view point of distributing the proper decision powers to the DMs in the decision making environment. Now, the solutions of the problem obtained previously by using the different approaches are presented as follows: (a) The solution of the problem obtained by using the conventional mathematical programming approach [1] is ( x1 , x 2 , x3 ) = (0.5 ,1, 0.5 ) with ( Z1 , Z 2 , Z 3 ) = (4.5 , 1, 0.5) . (b) The solution of the problem obtained by using the conventional FGP approach [21] is ( x1 , x 2 , x3 ) = (1.5, 0, 0.5) with ( Z 1 , Z 2 , Z 3 ) = (8.5, 0, 0.5) .

Using Genetic Algorithm for Solving Linear MLPPs via FGP

87

(c) The solution of the problem obtained by using fuzzy compensatory γ-operator [11] in fuzzy sets is: ( x1 , x 2 , x3 ) = (0.9, 0.6, 0.5) with ( Z 1 , Z 2 , Z 3 ) = (6.1, 0.6, 0.5) A graphical representation of the goal values achieved with the use of different approaches is displayed in the Fig. 1.

Fig. 1. Graphical representation of goal achievement under different approaches

A comparison of the results shows that that the proposed GA based FGP solution is the most satisfactory one from the viewpoint of arriving at a compromise decision by maintaining the hierarchical order of the DMs in the decision making environment.

Note: It is to be noted that the decision of the middle-level DM is improved here under the proposed approach with a least sacrifice of the decision of the top-level DM than the use of conventional FGP approach [21]. As a matter of fact, it may be claimed that, as GA is a volume-oriented (global one) search method and a goal satisficer [15] rather than objective optimizer, the incorporation of the GA solution search process to the FGP formulation of the problem makes the model more effective to make a reasonable balance of execution of decision powers of the DMs in the decision environment.

6 Conclusion The main advantage of the GA solution approach to the FGP model of the problem is that the higher degree of satisfaction with the solution can be searched on the basis of the needs and desires of the DMs in the decision making environment. The proposed approach can easily be extended to solve fuzzy MLPPs having the characteristics of fractional criteria. An extension of the approach to fuzzy multilevel decentralized planning problems is one of the current research problems. However, it is hoped that the proposed approach can contribute to future study in the field of real life large-scale hierarchical decision making problems.

88

B.B. Pal, D. Chakraborti, and P. Biswas

Acknowledgements. The authors are grateful to the anonymous Reviewers and Prof. P. Balasubramaniam, Organizing Chair ICLICC2011, for their helpful suggestions and comments which have led to improve the quality and clarity of presentation of the paper.

References 1. Anandalingam, G.: A mathematical programming model of decentralized multi-level system. Journal of Operational Research Society 39(11), 1021–1033 (1988) 2. Bard, J.F., Falk, J.E.: An explicit solution to multi-level programming problem. Computers and Operations Research 9, 77–100 (1982) 3. Bialas, W.F., Karwan, M.H.: On two level optimization. IEEE Transactions on Automatic Control 27, 211–214 (1982) 4. Bialas, W.F., Karwan, M.H.: Two level linear programming. Management Sciences 30, 1004–1020 (1984) 5. Burton, R.M.: The multilevel approach to organizational issues of the firm-Critical review. Omega 5, 446–457 (1977) 6. Candler, W.A.: A linear bilevel programming algorithm, a comment. Computers and Operations Research 15, 297–298 (1988) 7. Wen, U.P., Hsu, S.T.: Efficient solution for linear bilevel programming problem. European Journal of Operational Research 62, 354–362 (1991) 8. Liu, B.: Theory and Practice of Uncertain Programming, 1st edn. Physica-Verlag, Heidelberg (2002) 9. Lai, Y.J.: Hierarchical optimization. A satisfactory solution. Fuzzy Sets and Systems 77, 321–335 (1996) 10. Shih, H.S., Lai, Y.J., Lee, E.S.: Fuzzy approaches for multi-level programming problems. Computers and Operations Research 23, 73–91 (1996) 11. Shih, H.S., Lee, S.: Compensatory fuzzy multiple level decision making. Fuzzy Sets and Systems 14, 71–87 (2000) 12. Ignizio, J.P.: Goal Programming and Extensions. D.C. Health, Lexington (1976) 13. Pal, B.B., Moitra, B.N.: A fuzzy goal programming approach for solving bilevel programming problems. In: Pal, N.R., Sugeno, M. (eds.) AFSS 2002. LNCS (LNAI), vol. 2275, pp. 91–98. Springer, Heidelberg (2002) 14. Holland, J.H.: Genetic Algorithms and Optimal Allocations of Trials. SIAM Journal of Computing 2(2), 88–105 (1973) 15. Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading (1989) 16. Charnes, A., Cooper, W.W.: Management models and industrial applications of linear programming. John Wiley & Sons, New York (1961) 17. Gen, M., Ida, K., Lee, J., Kim, J.: Fuzzy nonlinear goal programming using genetic algorithm. Computers and Industrial Engineering 33(1-2), 39–42 (1997) 18. Hejazi, S.R., Memariani, A., Jahanshahloo, G., Sepehri, M.M.: Linear bilevel programming solution by genetic algorithm. Computers and Operations Research 29, 1913–1925 (2002) 19. Sakawa, M., Nishizaki, I., Hitaka, M.: Interactive fuzzy programming for multi-level 0-1 programming problems with fuzzy parameters through genetic algorithms. Fuzzy Sets and Systems 117, 95–111 (2001) 20. Pal, B.B., Chakraborti, D., Biswas, P.: A Genetic Algorithm Based Goal Programming Approach to Interval-Valued Multilevel Programming Problems. In: 2nd International Conference on Computing, Communication and Networking Technologies, pp. 1–7 (2010) 21. Pal, B.B., Pal, R.: A linear multilevel programming method based on fuzzy programming. In: Proceedings of APORS 2003, vol. 1, pp. 58–65 (2003)

Intuitionistic Fuzzy Fractals on Complete and Compact Spaces D. Easwaramoorthy and R. Uthayakumar Department of Mathematics, The Gandhigram Rural Institute - Deemed University, Gandhigram - 624 302, Dindigul, Tamil Nadu, India Tel.: +91 451 2452371; Fax: +91 451 2453071 [email protected], [email protected]

Abstract. We develop the Hutchinson-Barnsley theory for generating the intuitionistic fuzzy fractals through iterated function system, which consists of finite number of intuitionistic fuzzy contractive mappings on intuitionistic fuzzy metric space. For that, we prove the existence and uniqueness of fractals in a complete intuitionistic fuzzy metric space and a compact intuitionistic fuzzy metric space by using the intuitionistic fuzzy Banach contraction theorem and the intuitionistic fuzzy Edelstein contraction theorem respectively. Keywords: Fractal Analysis; Iterated Function System; Intuitionistic Fuzzy Metric Space; Contraction; Complete; Compact; Precompact.

1

Introduction

Mandelbrot defined Fractal as a subset of a complete metric space for which the fractal dimension strictly exceeds the topological dimension in 1975 and the theory was popularized by various mathematicians [2,3,7]. The theory of fuzzy sets was introduced by Zadeh in 1965 [15]. Many authors have introduced and discussed several notions of fuzzy metric space [4,8] and fixed point theorems with interesting consequent results in fuzzy metric spaces [4,5,12]. Then the concept of intuitionistic fuzzy metric space was given by Park [10] recently and the subsequent fixed point results in the intuitionistic fuzzy metric spaces were investigated by Alaca and et al. [1]. Hutchinson [7] and Barnsley [2] initiated and developed the theory, called as Hutchinson-Barnsley theory (HB theory), in order to define and construct the fractal as a compact invariant subset of a complete metric space generated by the IFS of contractions, by using the Banach fixed point theorem. Here, we initiate the fractal theory in the intuitionistic fuzzy metric spaces. In this paper, we propose a generalization of Hutchinson-Barnsley theory in order to define and construct the intuitionistic fuzzy IFS fractals through iterated function system. For that, we prove the existence and uniqueness of fractals in 

Corresponding author.

P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 89–96, 2011. c Springer-Verlag Berlin Heidelberg 2011 

90

D. Easwaramoorthy and R. Uthayakumar

a complete intuitionistic fuzzy metric space and a compact intuitionistic fuzzy metric space by using the intuitionistic fuzzy Banach contraction theorem and the intuitionistic fuzzy Edelstein contraction theorem, respectively.

2

Metric Fractals

Now we recall the Hutchinson-Barnsley theory (HB theory) to define and construct the IFS fractals in the complete metric space. Definition 1 ([2]). Let (X, d) be a metric space and Ko (X) be the collection of all non-empty compact subsets of X. Define, d(x, B) := inf y∈B d(x, y) and d(A, B) := supx∈A d(x, B) for all x ∈ X and A, B ∈ Ko (X). The Hausdorff metric or Hausdorff distance (Hd ) is a function Hd : Ko (X) × Ko (X) −→ Ê defined by Hd (A, B) = max {d(A, B), d(B, A)} . Then Hd is a metric on the hyperspace of compact sets Ko (X) and hence (Ko (X), Hd ) is called a Hausdorff metric space. Definition 2 ([2,7]). Let (X, d) be a metric space and fn : X −→ X, n = 1, 2, 3, ..., No (No ∈ Æ) be No - contraction mappings with the corresponding contractivity ratios kn , n = 1, 2, 3, ..., No. The system {X; fn , n = 1, 2, 3, ..., No} is called an Iterated Function System (IFS) or Hyperbolic Iterated Function Syso tem with the ratio k = maxN n=1 kn . Then the Hutchinson-Barnsley operator (HB operator) of the IFS is a function F : Ko (X) −→ Ko (X) defined by F (B) =

No *

fn (B).

n=1

Further, the HB operator (F ) is a contraction mapping on (Ko (X), Hd ). Theorem 1 (HB Theorem [2,7]). Let (X, d) be a complete metric space and {X; fn , n = 1, 2, 3, ..., No; No ∈ Æ} be an IFS. Then, there exists only one compact invariant set A∞ ∈ Ko (X) of the HB operator (F ) or, equivalently, F has a unique fixed point namely A∞ ∈ Ko (X). Definition 3 ([2]). The fixed point A∞ ∈ Ko (X) of the HB operator F described in the Theorem 1 is called the Attractor (Fractal) of the IFS. Sometimes A∞ ∈ Ko (X) is called as Metric Fractal generated by the IFS and so called as IFS Metric Fractal.

3

Intuitionistic Fuzzy Metric Space

Definition 4 ([1,10]). A 5-tuple (X, M, N, ∗, ♦) is said to be an intuitionistic fuzzy metric space if X is an arbitrary (non-empty) set, ∗ is a continuous t-norm [14], ♦ is a continuous t-conorm [14] and M, N are fuzzy sets on X 2 × [0, ∞) satisfying the following conditions:

Intuitionistic Fuzzy Fractals on Complete and Compact Spaces

(a) (b) (c) (d) (e) (f ) (g) (h) (i) (j) (k) (l) (m)

91

M (x, y, t) + N (x, y, t) ≤ 1; M (x, y, 0) = 0; M (x, y, t) = 1 if and only if x = y; M (x, y, t) = M (y, x, t); M (x, y, t) ∗ M (y, z, s) ≤ M (x, z, t + s); M (x, y, ·) : [0, ∞) −→ [0, 1] is left continuous; limt→∞ M (x, y, t) = 1 N (x, y, 0) = 1; N (x, y, t) = 0 if and only if x = y; N (x, y, t) = N (y, x, t); N (x, y, t)♦N (y, z, s) ≥ N (x, z, t + s); N (x, y, ·) : [0, ∞) −→ [0, 1] is right continuous; limt→∞ N (x, y, t) = 0

for all x, y, z ∈ X and t, s > 0. Then (M, N, ∗, ♦) or simply (M, N ) is called an intuitionistic fuzzy metric on X. The functions M (x, y, t) and N (x, y, t) represents the degree of nearness and the degree of non-nearness between x and y in X with respect to t, respectively. Definition 5 ([10]). Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. The open ball B(x, r, t) with center x ∈ X and radius r, 0 < r < 1, with respect to t > 0, is defined as B(x, r, t) = {y ∈ X : M (x, y, t) > 1 − r, N (x, y, t) < r} . Define τ(M,N ) = {A ⊂ X : f or each x ∈ A, there exists t > 0 and r ∈ (0, 1) such that B(x, r, t) ⊂ A}. Then τ(M,N ) is a topology on X induced by an intuitionistic fuzzy metric (M, N ). Definition 6 ([1,11,13]). Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. We say that X is complete if every Cauchy sequence is convergent. It is called compact if every sequence contains a convergent subsequence. We say that X is precompact if for + each 0 < r < 1 and t > 0 there exists a finite subset A of X such that X = a∈A B(a, r, t). Definition 7 ([1]). Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. We say that the mapping f : X −→ X is intuitionistic fuzzy B-contraction (intuitionistic fuzzy Sehgal contraction) if there exists k ∈ (0, 1) such that M (f (x), f (y), kt) ≥ M (x, y, t) and N (f (x), f (y), kt) ≤ N (x, y, t) for all x, y ∈ X and t > 0. Here, k is called the intuitionistic fuzzy B-contractivity ratio of f . We say that the mapping f : X −→ X is intuitionistic fuzzy Edelstein contraction if M (f (x), f (y), ·) ≥ M (x, y, ·) and N (f (x), f (y), ·) ≤ N (x, y, ·) for all x, y ∈ X such that x = y. Here, k is called the intuitionistic fuzzy Edelstein contractivity ratio of f .

92

3.1

D. Easwaramoorthy and R. Uthayakumar

Hausdorff Intuitionistic Fuzzy Metric Space

In [6], Gregori et al. defined the Hausdorff intuitionistic fuzzy metric on intuitionistic fuzzy hyperspace Ko (X) and constructed the Hausdorff intuitionistic fuzzy metric space as follows. Definition 8 ([6]). Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. We shall denote by Ko (X), the set of all non-empty compact subsets of X. Define, M (x, B, t) := supy∈B M (x, y, t), M (A, B, t) := inf x∈A M (x, B, t) and N (x, B, t) := inf y∈B N (x, y, t), N (A, B, t) := supx∈A N (x, B, t) for all x ∈ X and A, B ∈ Ko (X). Then we define the Hausdorff intuitionistic fuzzy metric (HM , HN , ∗, ♦) as follows: HM (A, B, t) = min {M (A, B, t), M (B, A, t)} and HN (A, B, t) = max {N (A, B, t), N (B, A, t)} . Here (HM , HN ) is an intuitionistic fuzzy metric on the hyperspace of compact sets, Ko (X), and hence (Ko (X), HM , HN , ∗, ♦) is called a Hausdorff intuitionistic fuzzy metric space. Now we recall that if (X, U ) is a uniform space, then the Hausdorff-Bourbaki uniformity HU (of U ) on Ko (X), has as a base the family of sets of the form HU = {(A, B) ∈ Ko (X) × Ko (X) : B ⊂ U (A) and A ⊂ U (B)}, where U ∈ U . This definition is valid even upto the set of all non-empty subsets of X instead of Ko (X). Further, if (X, M, N, ∗, ♦) is an intuitionistic fuzzy metric space, then {Un : n ∈ Æ} is a (countable) base for a uniformity U(M,N ) on X compatible with τ(M,N ) , where Un = {(x, y) ∈ X × X : M (x, y, 1/n) > 1 − 1/n, N (x, y, 1/n) < 1/n}, for all n ∈ Æ (See the Lemma 2.6 in [13]). U(M,N ) is called the uniformity induced by (M, N ). Especially, U(HM ,HN ) is exactly the uniformity induced by the Hausdorff fuzzy metric (HM , HN ). In this connection, we frame the following results. Theorem 2. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space . Then, the Hausdorff-Bourbaki uniformity HU(M,N ) , coincides with the uniformity U(HM ,HN ) on Ko (X).

Proof. For each n ∈ Æ, it is enough to prove the following:

{(A, B) ∈ Ko (X) × Ko (X) : B ⊂ Un+1 (A) and A ⊂ Un+1 (B)} ⊂ {(A, B) ∈ Ko (X) × Ko (X) : HM (A, B, 1/(n + 1)) ≥ 1 − 1/(n + 1), HN (A, B, 1/(n + 1)) ≤ 1/(n + 1)} ⊂ {(A, B) ∈ Ko (X) × Ko (X) : HM (A, B, 1/n) ≥ 1 − 1/n, HN (A, B, 1/n) ≤ 1/n} ⊂ {(A, B) ∈ Ko (X) × Ko (X) : B ⊂ Un (A) and A ⊂ Un (B)}. This concludes that, HU(M,N ) = U(HM ,HN ) on Ko (X).

Intuitionistic Fuzzy Fractals on Complete and Compact Spaces

93

Theorem 3. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Then 1. A sequence {x}n∈ is a Cauchy sequence in the intuitionistic fuzzy metric space (X, M, N, ∗, ♦) if and only if it is a Cauchy sequence in the uniform space (X, U(M,N ) ). 2. An intuitionistic fuzzy metric space (X, M, N, ∗, ♦) is complete if only if the uniform space (X, U(M,N ) ) is complete. Proof. It is proved in ([11], Theorem 3.4) that if {x}n∈ is a Cauchy sequence in (X, U(M,N ) ), then {x}n∈ is a Cauchy sequence in the intuitionistic fuzzy metric space (X, M, N, ∗, ♦), so the uniform space (X, U(M,N ) ) is complete whenever (X, M, N, ∗, ♦) is complete intuitionistic fuzzy metric space. It is easy to prove that the converse of these two results. Theorem 4. An intuitionistic fuzzy metric space (X, M, N, ∗, ♦) is precompact if only if the uniform space (X, U(M,N ) ) is precompact. Proof. Since an intuitionistic fuzzy metric space (X, M, N, ∗, ♦) is precompact if and only if every sequence in X has a Cauchy subsequence ([13], Theorem 2.4) and a sequence in (X, M, N, ∗, ♦) is Cauchy if and only if it is Cauchy in the uniform space (X, U(M,N ) ) (by Theorem 3), we realize that (X, M, N, ∗, ♦) is precompact if only if (X, U(M,N ) ) is precompact. Remark 1. It is well known that, if (X, U ) is a uniform space, then (Ko (X), HU ) is complete (respectively precompact) if and only if (X, U ) is complete (respectively precompact) (See [9]). Theorem 5. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Then (Ko (X), HM , HN , ∗, ♦) is complete if and only if (X, M, N, ∗, ♦) is complete. Proof. By Theorem 3, (Ko (X), HM , HN , ∗, ♦) is complete if and only if (Ko (X), U(HM ,HN ) ) is a complete uniform space. Since by Theorem 2, HU(M,N ) = U(HM ,HN ) on (Ko (X), it follows from Remark 1 that (Ko (X), UHM ,HN ) is complete if and only if (X, U(M,N ) ) is complete. Therefore, we conclude that (Ko (X), HM , HN , ∗, ♦) is complete if and only if (X, M, N, ∗, ♦) is complete. Now we can easily prove the following result as in similar arguements in the above Theorem 5, by using Theorem 4 and Remark 1. Theorem 6. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Then (Ko (X), HM , HN , ∗, ♦) is precompact if and only if (X, M, N, ∗, ♦) is precompact. Theorem 7. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Then (Ko (X), HM , HN , ∗, ♦) is compact if and only if (X, M, N, ∗, ♦) is compact. Proof. Since an intuitionistic fuzzy metric space is compact if and only if it is precompact and complete (See Corollary 2.7 in [13]), and by Theorems 5 and 6, we deduce that (Ko (X), HM , HN , ∗, ♦) is compact if and only if (X, M, N, ∗, ♦) is compact.

94

4

D. Easwaramoorthy and R. Uthayakumar

Fractals in the Intuitionistic Fuzzy Metric Space

In this paper, our primer interests are to define intuitionistic fuzzy fractals on complete and compact spaces. Definition 9. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space and fn : X −→ X, n = 1, 2, 3, ..., No (No ∈ Æ) be No - intuitionistic fuzzy Bcontractions (respectively intuitionistic fuzzy Edelstein contractions) with the corresponding contractivity ratios kn , n = 1, 2, 3, ..., No. The system {X; fn , n = 1, 2, 3, ..., No} is called an Intuitionistic Fuzzy Iterated Function System (IF-IFS) in the intuitionistic fuzzy metric space (X, M, N, ∗, ♦). Then the Intuitionistic Fuzzy Hutchinson-Barnsley operator (IF-HB operator) of the IF-IFS is a function F : Ko (X) −→ Ko (X) defined by F (B) =

No *

fn (B).

n=1

Theorem 8. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Let (Ko (X), HM , HN , ∗, ♦) be the corresponding Hausdorff intuitionistic fuzzy metric space. Suppose f : X −→ X is an intuitionistic fuzzy B-contraction with the corresponding contractivity ratio k ∈ (0, 1). Then, HM (f (A), f (B), kt) ≥ HM (A, B, t)

and

HN (f (A), f (B), kt) ≤ HN (A, B, t)

for all A, B ∈ Ko (X) and t > 0. Theorem 9. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Let (Ko (X), HM , HN , ∗, ♦) be the corresponding Hausdorff intuitionistic fuzzy metric space. Suppose f : X −→ X is an intuitionistic fuzzy Edelstein contraction. Then, HM (f (A), f (B), .) ≥ HM (A, B, .)

and

HN (f (A), f (B), .) ≤ HN (A, B, .)

for all A, B ∈ Ko (X) such that A ∩ B = ∅. Lemma 1. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Let (Ko (X), HM , HN , ∗, ♦) be the corresponding Hausdorff intuitionistic fuzzy metric space. If A, B, C, D ∈ Ko (X), then HM (A ∪ B, C ∪ D, t) ≥ min{HM (A, C, t), HM (B, D, t)} and HN (A ∪ B, C ∪ D, t) ≤ max{HN (A, C, t), HN (B, D, t)} for all t > 0. The proofs of the above Theorems and Lemmas are straightforward and are omitted.

Intuitionistic Fuzzy Fractals on Complete and Compact Spaces

95

Theorem 10. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Let (Ko (X), HM , HN , ∗, ♦) be the corresponding Hausdorff intuitionistic fuzzy metric space. Suppose {X; fn , n = 1, 2, 3, ..., No; No ∈ Æ} is an IF-IFS of intuitionistic fuzzy B-contractions with the same contractivity ratio k ∈ (0, 1) for each n ∈ {1, 2, ..., No} . Then the HB operator F is an intuitionistic fuzzy B-contraction with the ratio k. Proof. Lemma 1 and Theorem 8 completes the proof. Similarly we can prove the following result. Theorem 11. Let (X, M, N, ∗, ♦) be an intuitionistic fuzzy metric space. Let (Ko (X), HM , HN , ∗, ♦) be the corresponding Hausdorff intuitionistic fuzzy metric space. Suppose {X; fn , n = 1, 2, 3, ..., No; No ∈ Æ} is an IF-IFS of intuitionistic fuzzy Edelstein contractions. Then the HB operator F is an intuitionistic fuzzy Edelstein contraction. Theorem 12 (HB Theorem for Complete Space). Let (X, M, N, ∗, ♦) be a complete intuitionistic fuzzy metric space. Let {X; fn , n = 1, 2, 3, ..., No; No ∈ Æ} be an IF-IFS of intuitionistic fuzzy B-contractions and F be the IF-HB operator of the IF-IFS. Then, there exists only one compact invariant set A∞ ∈ Ko (X) of the HB operator (F ) or, equivalently, F has a unique fixed point namely A∞ ∈ Ko (X). Proof. Since (X, M, N, ∗, ♦) is a complete intuitionistic fuzzy metric space and by Theorem 5, we have (Ko (X), HM , HN , ∗, ♦) is also complete Hausdorff intuitionistic fuzzy metric space. By Theorem 10 shows that that the HB operator F is an intuitionistic fuzzy B-contraction. Then by the Intuitionistic Fuzzy Banach Contraction Theorem (Theorem 7 in [1]), we conclude that F has a unique fixed point, namely A∞ ∈ Ko (X). Theorem 13 (HB Theorem for Compact Space). Let (X, M, N, ∗, ♦) be a compact intuitionistic fuzzy metric space. Let {X; fn , n = 1, 2, 3, ..., No; No ∈ Æ} be an IF-IFS of intuitionistic fuzzy Edelstein contractions and F be the IFHB operator of the IF-IFS. Then, there exists only one compact invariant set A∞ ∈ Ko (X) of the HB operator (F ) or, equivalently, F has a unique fixed point namely A∞ ∈ Ko (X). Proof. Similar arguements of the Theorem 12, Theorems 7 and 11 completes the proof. Now we define the Intuitionistic Fuzzy Attractor or Fractal on complete space (respectively compact space) as the set A∞ ∈ Ko (X), which is described in the HB Theorem 12 for complete space (respectively HB Theorem 13 for compact space). Such A∞ ∈ Ko (X) is also called as Fractal generated by the IF-IFS of intuitionistic fuzzy B-contractions (respectively intuitionistic fuzzy Edelstein contractions) and so called as Intuitionistic Fuzzy Fractal on complete space (compact space). This study will leads to develop more interesting concepts and properties of fractals in the fuzzy spaces.

96

5

D. Easwaramoorthy and R. Uthayakumar

Conclusion

In this study, we proposed a generalization of Hutchinson-Barnsley theory in order to define and construct the intuitionistic fuzzy IFS fractals with respect to Hausdorff intuitionistic fuzzy metric through iterated function system of intuitionistic fuzzy contractions by using the intuitionistic fuzzy version of two fixed point theorems. Acknowledgments. The research work has been supported by University Grants Commission (UGC - MRP & SAP), New Delhi, India.

References 1. Alaca, C., Turkoglu, D., Yildiz, C.: Fixed points in intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals 29, 1073–1078 (2006) 2. Barnsley, M.: Fractals Everywhere, 2nd edn. Academic Press, USA (1993) 3. Barnsley, M.: SuperFractals. Cambridge University Press, New York (2006) 4. George, A., Veeramani, P.: On some results in fuzzy metric spaces. Fuzzy Sets and Systems 64, 395–399 (1994) 5. Gregori, V., Sapena, A.: On fixed-point theorems in fuzzy metric spaces. Fuzzy Sets and Systems 125, 245–252 (2002) 6. Gregori, V., Romaguera, S., Veeramani, P.: A note on intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals 28, 902–905 (2006) 7. Hutchinson, J.E.: Fractals and self similarity. Indiana University Mathematics Journal 30, 713–747 (1981) 8. Kramosil, I., Michalek, J.: Fuzzy metrics and statistical metric Spaces. Kybernetika 11, 336–344 (1975) 9. Morita, K.: Completion of hyperspaces of compact subsets and topological completion of open-closed maps. General Topology and its Applications 4, 217–233 (1974) 10. Park, J.H.: Intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals 22, 1039–1046 (2004) 11. Park, J.S., Kwun, Y.C., Park, J.H.: Some properties and Baire space on intuitionistic fuzzy metric space. In: Sixth International Conference on Fuzzy Systems and Knowledge Discovery, pp. 418–422. IEEE Press, Los Alamitos (2009) 12. Rodriguez-Lopez, J., Romaguera, S.: The Hausdorff fuzzy metric on compact sets. Fuzzy Sets and Systems 147, 273–283 (2004) 13. Saadati, R., Park, J.H.: On the intuitionistic fuzzy topological spaces. Chaos, Solitons and Fractals 27, 331–344 (2006) 14. Schweizer, B., Sklar, A.: Statistical metric spaces. Pacific Journal of Mathematics 10, 314–334 (1960) 15. Zadeh, L.A.: Fuzzy Sets. Information and Control 8, 338–353 (1965)

Research on Multi-evidence Combination Based on Mahalanobis Distance Weight Coefficients Si Su and Runping Xu No. 3 Wanshou Road, Haidian District, 100036 Beijing, China [email protected]

Abstract. To solve the contradiction in using Dempster’s method for the combination of highly conflicting evidences, an improved evidence combination method is presented based on Mahalanobis distance weight coefficients. First of all, the similarities between evidences are used as an approach to judge whether conflict exists. If there are more than 3 evidences consisting of conflicts, the Mahalanobis Distance algorithm can be used to calculate the distance between each evidence and the others to obtain the evidences’ weight coefficients, which could be transformed into BPA functions by means of the coefficients, and finally the Dempster’s method is used for the combination. Stimulation results show that this method can deal with conflicting evidences effectively, calculate faster than traditional algorithms, reduce more uncertainty in recognizing results, and retain the advantages of Dempster’s method in tackling non-conflicting evidences. Keywords: conflicting evidence; D-S evidence theory; information fusion.

1 Introduction Dempster’s evidence combination theory aims for uncertainty reasoning promoted by Dempster and Shafer at late 60s and early 70s in the 20th century. Oriented towards basic hypothesis set within the frame of discernment, this method is the further expansion of the probability theory applied to sensor measurements fusion in different levels. This theory is an improved method for uncertainty reasoning adapted to multisensor information fusion [1]. Nevertheless, shortcomings also exist in Dempster’s method, that is ineffectiveness in handling highly conflicting evidences. Results of evidence combination are basically right when there is no or little conflicts; whereas results contrary to common sense are arrived at in time of serious conflicts. To solve this problem, retaining the advantages of traditional algorithms and adopting the concept of the Mahalanobis Distance [2] from statistical theory, this paper presents an improved evidence combination method based on evidence weight coefficients. More accurate results of multi-sensor information fusion could be obtained by use of this method.

2 An Overview of Dempster Theory and Some Improvement Let m1 and m2 be functions of Basic Probability Assignment (BPA) of two evidences in the frame of discernment. The Dempster combination formula could be presented as P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 97–104, 2011. © Springer-Verlag Berlin Heidelberg 2011

98

S. Su and R. Xu

m(C ) =

∑ m ( A)m ( B)

A∩ B = C

∈ , A∈Θ, B∈Θ,

In formula 1, C Θ

1

2

(1)

1− k k=

∑ m ( A)m ( B)

A ∩ B =φ

1

2

indicating the

conflicting degree of two evidences. When larger conflict appears, k is close to 1, and the combination result would always be unreasonable [3]. Many scholars have brought improved suggestions to solve this problem which could be generally categorized into two classes: one is improvement of combination rules, and the other is perfection of evidence models [3-9]. Yager first improved D-S evidence theory by classifying the conflicting evidences into set Θ [4]. Though conflicts are removed, some useful information is also lost. When the conflict level is high, recognition results would fall under the set Θ, that is, many fields are unknown, which is useless for decision making. Besides, Yager’s solution could not get rid of “one-vote veto” problem existing in Dempster’s method. When one evidence is totally inconsistent with many, the focal element supported by them in the combination would be fully vetoed [5]. Reference [6] raises weighted additive strategy to process the existing evidence combination. Adding each evidence together in derangement to achieve conformity, conflicts could be removed consequently; then Dempster’s method could be used to synthesize to get the final result. However, this kind of correctness lies on utilizing repeatedly the existing evidences. In Reference [7], Murphy presents when there are many evidences in the system, basic assignment probability of these evidences should be equalized and combined for many times with Dempster’s method. But this is just the simple average of multi-source information without considering the relevance of these evidences. While Reference [8] unfolds an improvement of Murphy’s way by Deng Yong in taking the relevance into account. Using the formula of Euclidean Distance algorithm to calculate the distance between evidences, this method ensures the credibility of evidences greatly. Weighted average of evidences and combination making by Murphy’s method could process conflicting evidences powerfully except for low recognition speed. Combining the advantages of traditional algorithms, this paper overcomes “one vote veto” problem in Dempster and Yager’s method. Judgment should be made first on whether conflicting evidences exist based on degree of similarity between the evidences. Supposing there are over three conflicting evidence combinations, the Mahalanobis Distance algorithm can be used later to calculate the distance between one evidence and other groups of evidences so as to obtain the evidence weight coefficients. Then combination is to be made using the transformed results of BPA functions. This way could ensure the axiom BPA functions should add to 1 and also meet the requirement of exchangeability in combination order. Compared with traditional algorithm, this method possesses faster recognition speed, reduces the uncertainty largely, as well as retains the merit of Dempster’s method in processing non-conflicting evidence combinations.

Research on Multi-evidence Combination

99

3 Improved Combination Method Based on Mahalanobis Distance Weight Coefficients Let the frame of discernment in fusion system include many completed incompatible hypothetical propositions {A1, A2, A3,…, Aw}, the BPA functions of evidence e1, e2, e3,…, en are respectively called as m1, m2, m3,…, mn. Then synthetic procedure could be carried out as follows: 1) Test whether there is conflict between evidences. Evidence ei and ej could find their BPA functions mi and mj, with a maximum of max[ mi ( Ap )] and 1≤ p ≤ w

max[mi ( Aq )] respectively. If Ap and Aq correspond to the same recognition result,

1≤ q ≤ w

there is no evidence conflict; otherwise, mi and mj are treated as two vectors, similarity coefficient between them would be calculated as follows:

δ (mi , m j ) =

miT m j

(2)

mi m j

≤p

Let p be threshold value. If σ(mi,mj) , that means no conflict between ei and ej. Test BPA functions m1, m2, m3,…, mn in pairs to find out conflicting evidences, then go to Formula 2); if no conflicting evidence is discovered after the whole test, then go to Formula 5); 2) Confirm the total number of evidences n. If n=2, go to Formula 3); if n>2, go to Formula 4); 3) Apply the Additive Strategy to process BPA functions m1, m2, and then go to Formula 5); 4) Calculate the distance between one evidence and the other groups of evidences. Except for evidence ei, all remaining evidences could be viewed as a unity G, the average is vector for w dimension, and covariance ∑ is matrix of w w order. Using the Mahalanobis Distance algorithm to calculate the distance between ei and the unity G:

μ

×

d (i ) = (mi − μ ) ∑ −1 (mi − μ )T , i=1,2,…,n

(3)

Here, d(i) indicates the proximity between evidence ei and others. Weight coefficients vector of n dimension is defined as: 2

n ⎧ ⎫ ⎪⎪ ∑ d (i ) − d (i ) ⎪⎪ , i=1,2,…,n β (i ) = ⎨ i =1 n ⎬ ⎪ max[∑ d (i ) − d (i )] ⎪ ⎪⎩ 1≤i ≤n i =1 ⎪⎭

(4)

Here β(i) indicates the importance of evidence ei among the list e1, e2, e3,…, en. By use of weight coefficients, BPA functions m1, m2, m3,…, mn could be converted as following, and then turn to Formula 5).

100

S. Su and R. Xu

∈

mi' ( A) = β (i ) mi ( A) , ∀ A Θ, A≠Θ, i=1,2,…,n

(5)

mi' (Θ) = β (i )mi (Θ) + [1 − β (i )] , i=1,2,…,n

(6)

5) Dempster’s method is used to synthesize the evidences e1, e2, e3,…, en, and the final result could be arrived at. In conclusion, the overall procedure of combination put forward in this paper could be reduced as Figure 1. Evidences

Test whether conflicts exist yes

no

Confirm the number of evidences n

n>2 Calculate the distance d between evidences

Calculate the weight coefficients

β

Calculate the transformed BPA functions

n=2 Process BPA functions in Additive Strategy

Combination in Dempster method

Recognition results

Fig. 1. Total process of combination

4 Simulation Analysis Assuming a maneuvering target is detected in a sea battlefield and identified as the hostile force, five different sensors would provide information for this target. The frame of discernment could be: { A = fighter, B = helicopter, C = UAV }, and the BPA functions of evidences transformed by calculated data from these five sensors could be indicated as below: e1: m1(A) = 0.5, m1(B) = 0.2, m1(C) = 0.3 e2: m2(A) = 0.5, m2(B) = 0.2, m2(C) = 0.3 e3: m3(A) = 0.6, m3(B) = 0.1, m3(C) = 0.3 e4: m4(A) = 0.6, m4(B) = 0.1, m4(C) = 0.3 e5: m5(A) = 0.6, m5(B) = 0.1, m5(C) = 0.3 4.1 If Data Provided Regular Let threshold value for conflict test p=0.5. Six methods, such as Dempster’s, Yager’s, Additive Strategy, Murphy’s in Reference [7], Deng Yong’s in Reference [8] and improved method proposed here are applied respectively to synthesize the combination results. Results are compared in Table 1.

Research on Multi-evidence Combination

101

Table 1. Comparison of different evidence combination methods on condition of no conflict Evidence Combination

e1 + e2 + e3

e1 + e2 + e3+ e4

e1 + e2 + e3+ e4 + e5

Combination Method Dempster Yager Additive Strategy Murphy Deng Yong This Paper Dempster Yager Additive Strategy Murphy Deng Yong This Paper Dempster Yager Additive Strategy Murphy Deng Yong This Paper

m(A)

Recognition Results A

m(B)

m(C)

m(Θ)

0.9417 0.2100

0.0179 0.0040

0.0404 0.0090

0 0.7770

0.9286

0.0195

0.0519

0

A

0.9257 0.9240 0.9417 0.9760 0.1260

0.0198 0.0196 0.0179 0.0031 0.0004

0.0544 0.0564 0.0404 0.0209 0.0027

0 0 0 0 0.8709

A A A A

Θ

0.9695

0.0034

0.0271

0

A

0.9671 0.9654 0.9760 0.9889 0.0756

0.0038 0.0036 0.0031 0.0005 0.0000

0.0291 0.0310 0.0209 0.0106 0.0008

0 0 0 0 0.9235

A A A A

Θ

0.9856

0.0006

0.0138

0

A

0.9843 0.9831 0.9889

0.0007 0.0006 0.0005

0.0150 0.0162 0.0106

0 0 0

A A A

Θ

Many facts are indicated by this table. When sensors are working normally, recognition results using Yager’s method are all unknown term. That is because it takes out conflicting factors in combination, and most BPA functions of conflicting evidences are classified into uncertainty set Θ. This method is applicable in handling conflicting evidences, but as to no or little confliction it shall not apply. By comparison, the other methods could ensure the accuracy of results, and recognition result focalizes as the increasing of evidences supported. Despite this, recognition results applying Dempster’s or the method proposed in this paper possesses less uncertainty compared with others, which indicates the merit of Dempster is maintained in the improved method of this paper. 4.2 If Interference Occurs in the Second Sensor It works out of order because of electromagnetic interference from the rivaling force, and then BPA functions are changed to: e2: m2(A) = 0, m2(B) = 0.2, m2(C) = 0.8 So conflicts between the evidences are obvious. Combination results using these methods are compared in Table 2. From Table 2, the problem “one vote veto” appears in both Dempster’s and Yager’s method when big conflict exists between the evidences. That is when there is only one evidence completely vetos A while most don’t, the final combination results would completely veto A, which is surely unreasonable. Using Yager’s method, most

102

S. Su and R. Xu

Table 2. Comparison of different evidence combination methods on condition that one conflicting evidence exists Evidence Combination

e1 + e2 + e3

e1 + e2 + e3 + e4

e1 + e2 + e3 + e4 + e5

Combination Method Dempster Yager Additive Strategy Murphy Deng Yong This Paper Dempster Yager Additive Strategy Murphy Deng Yong This Paper Dempster Yager Additive Strategy Murphy Deng Yong This Paper

Recognition Results C

m(A)

m(B)

m(C)

m(Θ)

0 0

0.0526 0.0040

0.9474 0.0720

0 0.9240

0.3022

0.0330

0.6648

0

C

0.3169 0.4517 0.6648

0.0298 0.0290 0.0851

0.6533 0.5193 0.2502

0 0 0

C C A

0 0

0.0182 0.0004

0.9818 0.0216

0 0.9780

Θ

0.4721

0.0086

0.5193

0

C

0.4962 0.6959 0.7477 0 0

0.0077 0.0060 0.0490 0.0061 0.0000

0.4962 0.2981 0.2034 0.9939 0.0065

0 0 0 0 0.9935

A, C A A C

0.6439

0.0020

0.3541

0

A

0.6668 0.8434 0.8671

0.0017 0.0011 0.0125

0.3315 0.1556 0.1204

0 0 0

A A A

Θ

C

Θ

BPA functions of conflicting evidences are classified into the uncertainty set Θ. With the increase of evidences supporting A, the uncertainty of recognition results also keeps on increasing. With regard to Additive Strategy, Murphy’s method and Deng Yong’s method, the recognition results are C when there are three evidences, and the correct results could only be arrived at when more evidences appear. It shows low recognition speed of these methods; in practical application, more evidences need to be collected to ensure the accuracy of recognition results. When using Murphy’s method, that is just a simple average of weight coefficients of evidences without considering the degree of practical importance, which makes evidences of little importance affecting overall combination result. Uncertainty factors in recognition results need to be further reduced. Although Deng Yong’s method makes up for the shortcomings of Murphy’s, it can’t compare with the method proposed here in recognition speed and competence in reducing the uncertainty of recognition results. 4.3 If Interferences Occur in Both the Second and Fifth Sensor They work out of order because of strong electromagnetic interference from the rivaling force, and then BPA functions are changed to: e2: m2(A) = 0, m2(B) = 0.2, m2(C) = 0.8 e5: m5(A) = 0.1, m5(B) = 0.8, m5(C) = 0.1

Research on Multi-evidence Combination

103

Table 3. Comparison of different evidence combination methods on condition that two conflicting evidences exist Evidence Combination

e1 + e2 + e3 + e4 + e5

Combination Method Dempster Yager Additive Strategy Murphy DengYong This Paper

Recognition Results C

m(A)

m(B)

m(C)

m(Θ)

0 0

0.1290 0.0003

0.8710 0.0022

0 0.9975

0.5233

0.0748

0.4020

0

A

0.4377 0.6861 0.8334

0.1246 0.0382 0.0218

0.4377 0.2757 0.1448

0 0 0

A, C A A

Θ

In the case of interference coming from multiple sensors simultaneously, combination results using these methods are compared in Table 3. From Table 3, the question “one vote veto” still exists in both Dempster’s and Yager’s method when two sensors are interfered simultaneously. More supporting evidences are needed to arrive at the correct recognition results in Murphy’s method. Although the Additive Strategy and Deng Yong’s method could come at the accurate result, the way raised in this paper could reduce the uncertainty of recognition results better. Besides, summing up the results getting from Tables 1, 2 and 3, this improved method could make sure all BPA functions should add to 1. Moreover, by making use of Dempster’s method to synthesize the transformed BPA functions, it ensures the exchangeability of evidence combination order.

5 Conclusions Aiming at the shortcomings of Dempster’s method in processing conflicting evidences, an improved evidence combination method is presented in this paper based on Mahalanobis Distance evidence weight coefficients. Simulation results indicate that this method could efficiently process conflicting evidence combination, tackle the problem of “one vote veto”, satisfy the need that the sum of BPA functions should be 1, and ensure the exchangeability of evidence combination order. Compared with other traditional methods, this one is possessed of many merits, such as achieving high speed of recognition, efficiently reducing the uncertainty of recognition results and retaining good features of Dempster’s method in coping with non-conflicting evidence combination.

References 1. Chongzhao, H., Hongyan, Z., Zhansheng, D.: Multi-source Information Fusion. Tsinghua University Press, Beijing (2006) 2. Yan, W., Silian, S., Aiqing, W.: Mathematic Statistics And MATLAB Engineering Data Analysis. Tsinghua University Press, Beijing (2007)

104

S. Su and R. Xu

3. Haiyan, L., Zonggui, Z., Xi, L.: Combination of Conflict Evidences in D-S Theory D-S. J. Journal of University of Electronic Science and Technology of China 37, 701–704 (2008) 4. Yager, R.R.: On the Dempster–Shafer framework and new combination rules. J. Information Science 41, 93–137 (1989) 5. Fenghua, L., Lize, G., Yixian, Y.: An Efficient Approach for Conflict Evidence Combination. J. Journal of Beijing University of Posts and Telecommunications 31, 28–32 (2008) 6. Bin, Q., Xiaoming, S.: A Data Fusion Method for Single Sensor under Interfere environment. J. Journal of Projectiles, Rockets, Missiles and Guidance 27, 305–307 (2006) 7. Murphy, C.K.: Combining belief functions when evidence conflicts. J. Decision Support Systems 29, 1–9 (2000) 8. Yong, D., Wenkang, S., Zhenfu, Z.: Efficient Combination Approach of Conflict Evidence. J. Journal Infrared Millimeter and Waves 23, 27–32 (2004) 9. Quan, P., Shanying, Z., Yongmei, C.: Some Research on Robustness of Evidence Theory. J. ACTA Automatic Si 27, 798–805 (2001)

Mode Based K-Means Algorithm with Residual Vector Quantization for Compressing Images K. Somasundaram and M. Mary Shanthi Rani Image Processing lab Department of Computer Science and Applications Gandhigram Rural Institute- Deemed University Gandhigram

Abstract. Image compression plays a vital role in many online applications like Video Conferencing, High Definition Television, Satellite Communication and other applications that demand fast and massive transmission of images. In this paper, we propose a Mode based K-means method that combines K-Means and Residual Vector Quantization(RVQ) for compressing images. Three processes are involved in this approach; Partitioning and Clustering, Pruning and Construction of Master codebook and Residual vector Quantization. Extensive experiments show that this method obtains a fast solution with better compression rate and comparable PSNR than conventional K-Means algorithm. Keywords: K-Means, Mode, Median, Residual Vector Quantization, Break Even Point, Master Codebook, Residual Codebook.

1 Introduction Vector quantization (VQ) is an effective means for lossy compression of speech and digital images. K-Means [1] is a widely used Vector Quantization technique known for its computational efficiency and speed. It has a great number of applications in the fields of image and video compression, Pattern Recognition, and Watermarking, However, it has some pitfalls like apriori fixation of number of clusters and random selection of initial seeds. Wrong choice of initial seeds may yield poor results and results in increase in time for convergence. We begin with an overview of existing methods based on K-Means algorithm. A pioneering work of seeds initialization was proposed by Ball and Hall(BH) [2]. A similar approach named as Simple Cluster Seeking(SCS) was proposed by Tou and Gonzales [3]. The SCS method chooses the first input vector as the first seed and the rest of the seeds are selected if they are ‘d’ distance apart from all selected seeds. The SCS and BH method are sensitive to the parameter ‘d’ and the order of the inputs. Astrahan [4] suggested a method using two distance parameters, d1 and d2 which first computes the density of each point in the dataset. The highest density point is chosen as the first seed and subsequent seed points are chosen in the order of decreasing density subject to the condition that each new seed point be at least at a distance of ‘d2’ from all other previously chosen seed points. The drawback in this approach is that it is very sensitive to the values of d1 and d2 and requires hierarchical clustering. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 105–112, 2011. © Springer-Verlag Berlin Heidelberg 2011

106

K. Somasundaram and M. Mary Shanthi Rani

Kaufman and Rousseau [5] introduced a method that estimates the density through pair wise distance comparisons and initializes the seed clusters using the input samples from the areas with high local density. A significant drawback of the method is its computational complexity which may sometimes be higher than k-means when ‘n’ is large. Katsavounidis et al (KKZ) [6] suggested a parameterless approach, which chooses the first seed near the “edge” of the data, by choosing the vector with the highest norm. Then, it chooses the next seed to be the point that is farthest from the nearest seed in the set chosen so far. This method is very inexpensive (O (kn)) and is easy to implement but is sensitive to outliers. Bradley and Fayyad (BF) [7] proposed an initialization method that is suitable for large datasets. The main idea of their algorithm is to select ‘m’ subsamples from the data set, apply the k-means on each subsample independently, keep the final k centers from each subsample and produce a set that contains mk points. They apply the k-means on this set ‘m’ times: at the first time, the first k points as initial centers and at the second time, the second k points and so on. The algorithm returns the best k centers from this set. They use 10 subsamples from the data set, each of size 1% of the full dataset size. Finally, a last round of k-means is performed on this dataset and the cluster centers of this round are returned as the initial seeds for the entire dataset. This method generally performs better than k-means and converges to the local optimum faster. However, it still depends on the random choice of the subsamples and hence, may obtain poor clusters in an unlucky session. Fahim et al. [8] proposed a method to minimize the number of distance calculations required for convergence. Arthur and Vassilvitskii [9] proposed the k-means++ approach, which is similar to the KKZ method. In k-means++, the next point chosen will be the one with the probability proportional to the minimum distance of this point from already chosen seeds. Note that due to the random selection of first seed and probabilistic selection of remaining seeds, different runs have to be performed to obtain a good clustering. Single Pass Seed Selection (SPSS) [10] algorithm initializes first seed and the minimum distance that separates the centroids is based on the point which is close to more number of other points in the data set. An efficient fast algorithm to generate VQ codebook was proposed by Kekre et al[11] which uses sorting method to generate codebook and the code vectors are obtained using median approach. Recently, Residual VQ has captured the attention of researchers as they reduce memory and computation complexity [12]-[14]. A low bit rate still image compression scheme by compressing the indices of Vector Quantization (VQ) and generating residual codebook was proposed by Somasundaram and Dominic[15]. The proposed method is an innovative approach that combines the merits of mode based seed selection and residual vector quantization The motivation of this approach is multi-fold. Firstly, the use of mode parameter exploits correlation among similar blocks for clustering resulting in low bit rate and preserves image quality as well. Second, the use of divide and conquer strategy results in significant reduction of computation time. Third, residual vector quantization phase encodes the residual image, further enhancing the reconstructed image quality.

Mode Based K-Means Algorithm with RVQ for Compressing Images

107

The rest of the paper is organized as follows: In Section 2, we briefly discuss Mode based K-means with Residual Vector Quantization (RVQ) approach. Experimental results are presented in Section 3. Finally, Section 4 concludes by outlining the merits and demerits of the proposed method.

2 Mode Based K-Means with RVQ Mode based K-means is a simple and fast clustering algorithm. It adopts the divide and conquer strategy by dividing the input data space into classes and apply K-means clustering. Mode based K-means works in three phases. In the first phase, the input vectors are divided into “N” classes based on the mean value of each vector. The distribution of image data determines the number of classes ‘N’. As the range ‘R’ of the value of a pixel in image data is 0-255, the range of the domain of each class is set based on the ratio of ‘R’ to the number of classes(N). The range of the domain of each class Di is calculated as follows; Di = (max_pixel_value – min_pixel_value) / N * I

(1)

where i is the Class_index between 1 and N, max_pixel_value and min_pixel_value are the maximum and minimum pixel values in the input image data respectively. This mean based grouping of input data speeds up convergence subsequently reducing computation time. K-Means algorithm is performed on each class of vectors with initial seeds. The number of initial seeds ‘M[i]’ in each class Ci is proportional to its size and is calculated as M[i] = Class-size[i] / T

(2)

Where ‘T’ is the ratio of total number of training vectors and number of initial seeds ‘K’ and ∑M[i] = K , i = 1…N. The novelty of our approach is using mode for choosing the initial seeds. The initial seeds are the high mode vectors among the class population. High mode vectors are the vectors whose values are mode values of each dimension of the class vectors, in descending order. If mode_vector[i,j] is the ith mode vector, then mode_vector(i,j)= pixel value that occupies ith position in descending order of frequency of occurrence(mode) in jth dimension of all the vectors of a class. (3) In the second phase, the code vectors generated from each class are subjected to pruning based on a minimum (Euclidean) distance threshold “D” with the code vectors of the partially generated code book. A new code vector v1i of a class Ci, will be added to the master codebook only if its Euclidean distance with all of the code vectors of classes C1 ..Ci-1 is greater than a predefined threshold “D”. The value of ‘D’ can be used to tune rate-distortion performance. Increase in the value of ‘D’ results in reduction in bit-rate but at the cost of image quality. Experiments have revealed that the value of ‘D’ should lie in the range 2-4 to achieve better rate-distortion

108

K. Somasundaram and M. Mary Shanthi Rani

performance. Also, pruning of code vectors done at the construction phase significantly reduces the code book size and computation time as well. The third phase constructs the residual codebook. It computes the residual image by computing the difference between the input vector and the code vector. Next, the residual vectors with the same code index are grouped together which results in ‘K’ groups. One median vector (16-element) is constructed from each group by taking the median value of each column in the group. Next, each median vector (16-element) is converted to 4-element by dividing the 16-element vector into four 4-element blocks and representing each block by its mean value. Thus, the values of 4-element vector are the mean values of the four sub blocks. The resulting ‘K’ 4-element vectors form the residual code book. Grouping vectors by their code index provides two advantages. First, input vectors quantized to same code vector are highly correlated which may result in highly correlated residual vector group .This correlation can be utilized to reduce the size of the residual codebook. Second, we don’t require any additional index for residual code book. One master index is used for both the master code book and residual codebook as well. This significantly reduces the index overhead, despite the use of two code books. 2.1 Algorithm 2.1.1 Seed Generation Phase 1. Divide the 512x512 input image data into 4x4 image blocks and convert the blocks into training vectors(16-vecctor) of dimension 16. 2.

Classify the training vectors into “N” classes based on the mean value of its elements of each vector.

3.

Calculate the range of the domain Di of each class using equation (1).

4.

Calculate ‘T’ which is equal to the ratio of total number of training vectors and number of initial seeds ‘K’.

5.

Determine the number of initial seeds ‘M[i]’ for each class using equation (2).

6.

Construct mode vectors whose values are the mode value (value with highest frequency of occurrence) of its respective dimension in descending order until we get the desired number of initial seeds ‘M[i]’ using equation (3).

7.

Perform K-means algorithm on each class with ‘M’ initial seeds.

2.1.2 Pruning and Construction Phase 8. Calculate the Euclidean distance between each of the newly generated code vectors of the current class Ci and the code vectors in the partial codebook constructed from classes C1..Ci-1 so far. 9.

Add only those code vectors to the code book whose distance with all of the partial codebook vectors is greater than a predefined threshold ‘D’.

Mode Based K-Means Algorithm with RVQ for Compressing Images

109

10. Repeat this process for all the classes Ci, i=1 .. N 11. The final code book resulting after step 10 is the initial codebook for the entire data set. 12. Perform a final run of K-means on the the entire data set with this final codebook as initial seeds. 13. The final code book resulting after step 12 is the master codebook for the entire data set. 2.1.3 Residual Vector Quantization (RVQ) Phase 14. Compute the residual vector which is the difference between the input vector and its corresponding code vector. 15. Group all residual vectors having the same master code index. 16. Construct the median vector for each group by calculating the median value of each column of the residual vectors belonging to the same group. 17. Convert each median vector (16-element) to 4-element vector by dividing it into four 2x2 blocks and representing each sub block by its mean value. 18. Construct the residual codebook with the median vectors resulting out of step 17. The master code book, residual code book and the code indices form the compressed stream of data. 2.1.4 Decoding Phase 1 Determine the code vector from the master code book based on the Master code index of each input vector. 2

Retrieve the residual code vector (4-element) from the residual code book using Master code index.

3

Convert the residual 4-element vector to 16-element vector by replacing the pixel values of the 2x2 sub blocks with their respective mean value.

4

Construct the final code vector by adding or subtracting the corresponding residual values of residual vector (16-vector) from the master code vector.

3 Results and Discussion We carried out experiments using the proposed method on several standard images of size 512 x 512 on a machine with core2 duo processor at 2 GHZ using MATLAB 6.5. The time taken for code book generation, bit rate in terms of BPP (Bits Per Pixel) and PSNR values computed for standard images Lena, Barbara, Boat, Peppers and Couple using our method and K-Means algorithm are given in Table 1. We note from

110

K. Somasundaram and M. Mary Shanthi Rani

the Table 1 that the time taken for code book generation with the proposed method reduced by 2 times in the case of Boat image to about 17 times for Peppers when compared to K-Means algorithm. It is further observed that the bit rate has also improved without significant reduction in the image quality measured in terms of PSNR value. Experimental results demonstrate that Mode based k-means shows optimal performance in terms of Computation time, Bit rate and PSNR for a specific value of ‘N’ which can be considered as the Break-Even Point (BEP). Further analysis of results reveals that BEP is image-specific and is equal to c(log2K) for some constant ‘c’ which lies in the range between 0.5 and 1 where K is the number of initial seeds. Moreover, the results show that the value of c depends on the distribution of frequently occurring pixels in input data (hit-no). The value of c is equal to 0.5 for images whose hit-no is less than 200 and 1 for images with hit_no >200. Besides, the size of the final codebook is reduced further in the pruning phase and is found to be less than K thereby enhancing compression rate. It is worth noting that the initial value of K is chosen to be less than 256 to increase bit rate without compromising picture quality by coding the residuals in Residual Vector Quantization phase. In terms of memory requirements, the proposed method requires storage for the master and the residual codebooks, as well as for storing code indices. As the master codebook consists of a total of K 16-element vectors, the size P in bits of the master codebook is given by P = K × 16 × 8 bits. However, the residual codebook requires storage only for K 4-element vectors and as the value of each vector element does not exceed 8, the size Q in bits of the residual codebook is given by Q = K × 4 × 3 bits. Despite using two codebooks, the memory requirements of the proposed method (P + Q) is still less than conventional K-Means. For visual comparison, the original and reconstructed images of Lena by the proposed method are shown in Fig.1 and Fig. 2.

Fig. 1. Original Lena image

Fig. 2. Reconstructed Lena image by Mode based K-means

Mode Based K-Means Algorithm with RVQ for Compressing Images

111

Table 1. Perfomance analysis of Mode based K-Means on test images Image

Method

BEP (N)

K-Means Lena

Barbara

Boat

Peppers

Couple

Bridge

Computation time 1500

Bit rate

K

PSNR

0.125

256

33.01

350

0.113

208

32.812

2075

0.125

256

28.2

Mode Based K-means K_Means

4

Mode based K-means K-Means

8

160

0.115

210

27.972

-

362

0.125

256

31.4

Mode based K-Means K-Means

8

150

0.119

214

31.319

-

3160

0.125

256

31.4

Mode based K-Means Kmeans

8

189

0.119

214

31.176

-

1196

0.125

256

28.898

Mode based K-Means K-Means

4

188

0.118

212

28.699

-

1368

0.125

256

25.723

Mode Based K-Means

8

249

0.116

211

25.454

4 Conclusion Mode Based K-Means proposes a three phase method in which the use of mode parameter for initial seed selection has been investigated. Mode based Kmeans shows superior performance than traditional kmeans with random choice of initial seeds significantly in terms of computation time with comparable PSNR and Bit rate. This would be an ideal choice for applications of image compression where time complexity plays a crucial role like video conferencing, online search engines of image and Multimedia databases.

References [1] Lloyd, S.P.: Least square quantization in PCM. IEEE Trans. Inform. Theory 28, 129–137 (1982) [2] Ball, G.H., Hall, D.J.: PROMENADE-an on-line pattern recognition system. Stanford Research Inst. Menlo Park, Stanford University (1967) [3] Tou, J., Gonzalez, R.: Pattern Recognition Principles, p. 377. Addision-Wesley, Reading (1977) [4] Astrahan, M.M.: Speech analysis by clustering (1970)

112

K. Somasundaram and M. Mary Shanthi Rani

[5] Kaufman, L., Rousseeuw, P.J.: Finding Groups in Data: An Introduction to Cluster Analysis, p. 342. Wiley, New York (1990) [6] Katsavounidis, I., Kuo, C.C.J., Zhen, Z.: A new initialization technique for generalized Lloyd iteration. IEEE. Sig. Process. Lett. 1, 144–146 (1994) [7] Bradley, P.S., Fayyad, U.M.: Refining initial points for K-means clustering. In: Proceeding of the 15th International Conference on Machine Learning (ICML 1998), July 24-27, pp. 91–99. ACM Press, Morgan Kaufmann, San Francisco (1998) [8] Fahim, A.M., Salem, A.M., Torkey, F.A., Ramadan, M.: An efficient enhanced k- means clustering algorithm. J. Zhejiang Univ. Sci. A. 7, 1626–1633 (2006) [9] Arthur, D., Vassilvitskii, S.: k-means++: The advantages of careful seeding. In: Proceeding of the 18th Annual ACM-SIAM Symposium of Discrete Analysis, January 7-9, pp. 1027–1035. ACM Press, New Orleans (2007) [10] Karteeka Pavan, K., Rao, A.A., Dattatreya Rao, A.V., Sridhar, G.R.: Single Pass Seed Selection Algorithm for k-Means. Journal of Computer Science 6(1), 60–66 [11] Kekre, H.B., Sarode, T.K.: An Efficient Fast Algorithm to Generate Codebook for Vector Quantization. In: Proceedings of International Conference on Emerging Trends in Engineering and Technology, Thadomal Shahani, pp. 62–67 [12] Juang, B.H., Gray, A.H.: Multiple stage vector quantization for speech coding. In: Proc. IEEE Int. Conf. Acoust. Speech,Signal Process., vol. 1, pp. 597–600 (1982) [13] Kossentini, F., Smith, M.J.T., Barnes, C.F.: Image Coding Using Entropy- Constrained Residual Vector Quantization. IEEE Trans. Image Processing 4, 1349–1357 (1995) [14] Wu, F., Sun, X.: Image Compression by Visual Pattern Vector Quantization (VPVQ). In: Data Compression Conference, 1068-0314/08 © (2008) [15] Somasundaram, K., Dominic, S.: Lossless VQ Index Coding Scheme for Image Compression. In: IEEE International Conference on Signal and Image Processing, vol. 1, pp. 89–94 (2006)

Approximation Studies Using Fuzzy Logic in Image Denoising Process Sasi Gopalan1, Madhu S. Nair2, Souriar Sebastian3, and C. Sheela4 1 Department of Mathematics, School of Engineering, Cochin University of Science and Technology, Kochi – 682022 2 Department of Computer Science, University of Kerala, Kariavattom, Thiruvananthapuram – 695581 3 Department of Mathematics, St. Albert’s College, Ernakulam, Kerala 4 Department of Mathematics, Federal Institute of Science and Technology, Angamali, Kerala

Abstract. In this work Lossy methods are used for the approximation studies, since it is suitable for natural images such as photographs. When comparing the compression codecs, it used as an approximation to human perception of reconstruction quality. In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation. Fuzzy Logic Systems are blending functions and it has been used to develop a Modified Fuzzy Basis Function (MFBF) and also its approximation capability has been proved. An iterative formula (Lagrange Equation) for the fuzzy optimization has been adopted in this paper. This formula closely relates with the membership function in the RGB colour space. By maximizing the weight in the objective function the noise in the image reduced, so that the filtered image approximates the original image. Keywords: Approximation, Fuzzy, Denoising, color images.

1 Introduction In fuzzy image processing the function approximation is possible only by the modification in the membership functions. The concept of fuzzy basis functions is related with the membership functions [1, 2, 3, 8]. The designed Modified Fuzzy Basis Function (MFBF) facilitates faster approximation when compared to other available filters. The centroid defuzzification method is used with weight as the scaling function and the maximization problem is discussed. That is, by the maximization of weight (for small distances) the function is optimized. An experimental study in connection with our work shows that the designed MFBF can be used more effectively to reduce noise from an image signal. Colours in RGB space are represented by a 3-D vector with first element being red, second being green and third being blue respectively. A colour image C represented by a 2-D array of vectors where (i, j ) = xi , j defines a position in C called pixel and

Ci , j ,1 = xi , j ,1 , Ci , j , 2 = xi , j , 2 and Ci , j ,3 = xi , j ,3 denotes the red,

green and blue components respectively of C. The distance measure of pixels used in MFBF for the approximation studies. In fuzzy image processing the function P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 113–120, 2011. © Springer-Verlag Berlin Heidelberg 2011

114

S. Gopalan et al.

approximation is possible only by the modification in the membership functions. The concept of fuzzy basis functions is related with the membership functions [1, 2, 3, 5, 7, 8]. The designed Modified Fuzzy Basis Function (MFBF) facilitates faster approximation when compared to other available filters. The centroid defuzzification method is used with weight as the scaling function and the maximization problem is discussed. That is, by the maximization of weight (for small distances) the function is optimized. An experimental study in connection with our work shows that the designed MFBF can be used more effectively to reduce noise from an image signal. Colours in RGB space are represented by a 3-D vector with first element being red, second being green and third being blue respectively. A colour image C represented by a 2-D array of vectors where (i, j ) = xi , j defines a position in C called pixel and

Ci , j ,1 = xi , j ,1 , Ci , j , 2 = xi , j , 2 and Ci , j ,3 = xi , j ,3 denotes the red, green and blue components respectively of C. The distance measure of pixels used in MFBF for the approximation studies.

2 Interpolation Using Fuzzy Systems The concept of Fuzzy Basis Function [FBF] is clearly defined in [8]. Hence the entire static mapping for multiple rule antecedent is of the form p

p

p

n

∑ ∑ ⋅ ⋅ ⋅ ⋅∑ a ∏ A

ki i

j

F (a, x ) = g a , x =

k n =1 k n −1 =1 p

k1 =1

p

∑∑

k n =1 k n =1 =1

( xi )

i =1

p

n

⋅ ⋅ ⋅ ⋅∑∏ Aiki ( xi ) k1 =1 i =1

where i = 1, 2,...., n , j = j ( k 1 , k 2 ,...., k n ) The sub index

i of ki indicates the variable xi that is being considered and ki

indicates the fuzzy sets and order of the variable

xi from left to right.

According to [3, 4, 10, 11]

F ( a, x) = g a , x =

m= pn

∑a Φ j

j

( x) = aT Φ ( x)

j =1

That is, F ( a, x ) is presented as a convex linear combination of the basis functions and hence

F ( a, x) = g a , x =

m= pn

∑a Φ j

j

( x) = a j , where Φ j (x) are called fuzzy basis

j =1

functions (FBF). Thus the interpolation capability is satisfied.

3 Proposed Algorithm The basics of noise reduction is clearly defined in [9,10,12].

Approximation Studies Using Fuzzy Logic in Image Denoising Process

115

3.1 Definition (Distance Measure in Colour Space) Let d be the Euclidean distance, RG-red green, NRG- Neighbour Red green. Let

RG = ( xi , j ,1 , xi , j , 2 ) and NRG = ( xi+k , j +l ,1 , xi+ k , j +l , 2 ) . Then

d ( RG, NRG ) =

(x

− xi , j ,1 ) + ( xi + k , j + l ,2 − xi , j ,2 ) 2

i + k , j + l ,1

2

3.2 Definition (Weight in Fuzzy Noise Reduction) [4, 9] The weight function is defined as the product of the membership functions. That is, Weight = Ai , j ( Dist ( RG , NRG )) × Ai , j ( Dist (RB, NRB )) =

min[ Dist ( RG, NRG )), Dist (RB, NRB )]

3.3 Modified Fuzzy Basis Function (MFBF) Here the modified fuzzy filter is defined as n

g (xi , j ) =

m

∑∑x

i − k , j −l i =1 j =1 n m

∑∑ Φ i =1 j =1

where

Φ i−k , j −l ( xi , j )

i − k , j −l

( xi , j )

Φ i −k , j −l is the product of the membership function (weight function) and the

distance function. The maximum weight is equal to unity if the distance between colour pixel and its neighboring pixel is small. This can be brought to triangular form. The Modified Fuzzy Basis Function is

Φ( xi , j ) =

Φ i − k , j − l ( xi , j )

n

m

∑∑ Φ i =1 j =1

i − k , j −l

.

( xi , j )

Therefore by section 2,

g (xi , j ) = ∑∑ xi − k , j − l Φ ( xi , j ) m

n

i =1 j =1

By adjusting the weight means the membership function is modified. This variation in distance between the pixels directly affects the standard deviation of the Gaussian noise. i.e., for small distances its PSNR value will be high. So the best approximation is obtained with the Modified Fuzzy Membership Function (MFBF).

4 Weights in RGB Colour Space According to [10], the non linear function

Φ k ,l can be approximated as a step like

function. The following figure shows that for small distances the weight is maximum.

116

S. Gopalan et al.

Fig. 1. Approximation of weight function

In this case the output of the fuzzy filter is represented as n

gi , j =

n

∑ ∑Φ

k ,l k =−n l =−n n n

∑ ∑Φ

k =−n l =−n

If

( xi , j ) xi − k , j −l , k ,l

( xi , j )

xi. j − xi − k , j − l and xi+k . j +l − xi , j is small then xi − k , j − l and xi + k , j + l are assumed

in the same flat area (here Φ k , l

= 1 ). If not they are assumed in different flat area

( Φ k , l = 0) . The concept of gradient method is discussed in [6]. In the gradient method, a parameter is updated by subtracting a value which is proportional to the gradient of the cost function. 4.1 Fuzzy Optimization The iterative formula is

S k (n + 1) = S k (n) − α

∂E ( n) ∂Φ k

Sk (n) is the value Sk ( Sk is the height of the k th step of the step like function) at time point n, α is a small positive coefficient and E is the mean where

square error. Using this iterative formula we start with an algorithm and by the fine tuning we can choose the suitable rule for the function approximation. Therefore by Banach fixed point theorem we can find fixed points corresponding to different rules. 4.2 Optimization Problem This is a maximization problem. For the maximum weight the desired image approximates with the filtered image. Therefore we define the following Legrange equation as the Objective function

E = Wk ,l − α ( f i , j − g i , j )

Approximation Studies Using Fuzzy Logic in Image Denoising Process

117

E − The error W − The weight factor α − Small positive constant f i. j − Desired output g i , j − Filtered output

(a)

(b) PSNR = 22.10

(c) PSNR = 26.57

(d) PSNR = 27.66

(e) PSNR =28.38

(f) PSNR = 29.33

(a)

(b) PSNR = 18.59

(c) PSNR = 24.63

(d) PSNR = 24.73

(e) PSNR =25.16

(f) PSNR = 25.79

Fig. 2. (a) Original Peppers image (256×256) (b) Noisy image (Gaussian noise, σ = 30) (c) After applying Mean filter (3×3 window) (d) After applying Median filter (5×5 window) (e) After applying FBF with K=3 (7×7 window) and L=2 (5×5 window) (f) MFBF with K=3 (7×7 window) and L=2 (5×5 window)

118

S. Gopalan et al.

4.3 Peak Signal–to-Noise Ratio (PSNR) [9, 12] PSNR is the most commonly used measure of quality of reconstruction of lossy compression (e.g. for image compression). This metric is used to compare results of codec type. Typical values for the PSNR in lossy image and video comparison are between 30 and 50, where higher is better. Therefore,

PSNR[ g (i, j , c) − f (i, j , c)] = 10 log10 [ S 2 / MSE ( g (i, j , c) − f (i, j , c)] where MSE is the mean square error and

∑∑∑ [g 3

MSE[ g i , j ,C − f i , j ,C ] =

N

M

c =1 i =1 j =1

i , j ,C

− f i , j ,C

]

2

3.N .M

f (i, j , c) is the desired colour image and g (i, j , c) is the filtered colour image of size N .M (c = 1,2,3 corresponding to the colours red, green and blue Here

respectively).

(a)

(d) PSNR = 21.52

(b) PSNR = 16.10

(c) PSNR = 22.03

(e) PSNR =22.75

(f) PSNR = 23.61

Fig. 3. (a) Original Gantry crane image (400×264) (b) Noisy image (Gaussian noise, σ = 40) (c) After applying Mean filter (3×3 window) (d) After applying Median filter (3×3 window) (e) After applying FBF with K=3 (7×7 window) and L=2 (5×5 window) (f) After applying Proposed Fuzzy filter with K=3 (7×7 window) and L=2 (5×5 window)

Approximation Studies Using Fuzzy Logic in Image Denoising Process

119

Table 1. Corresponding to different distances and for different filters the PSNR value is calculated and shown PSNR (dB) σ=5

σ = 10

σ = 20

σ = 30

σ = 40

Noisy

34.13

28.12

22.10

18.57

16.08

Mean

28.05

27.70

26.57

25.13

23.72

Median

32.31

30.81

27.66

25.02

22.95

FLF Method

34.12

31.79

28.38

25.85

23.76

MFBF Method

34.22

32.77

29.33

26.51

24.18

5 Experimental Studies The PSNR values for the following figures are verified for small distances. The MFBF is applied in three natural photographs and its approximation is also satisfied. The results are shown in fig. 2 and 3. The PSNR values are shown in Table 1. The PSNR values obtained at different noise levels are shown in fig. 4.

Fig. 4. For restored image approximating noise image for distance (σ = 5 )

6 Conclusion The main claim of this work is the formation of Modified Fuzzy Basis Function (MFBF) in RGB colour space. It is shown that MFBF approximates the noisy image with minimum error and it is better than any other filters.

120

S. Gopalan et al.

References 1. De Luca, A., Termini, S.: A definition of non probabilistic entropy on the setting of fuzzy sets theory. Inform and Control 20, 301–312 (1972) 2. Alimi, A.M., Hassine, R., Selmi, M.: Beta fuzzy logic systems: Approximation Properties in the MISO case. Int. J. Appl. Math. Comput. Sci. 13(2), 225–238 (2003) 3. Kosko, B.: Fuzzy Associative Memories in fuzzy systems. In: Kandel., A. (ed.), AdditionWesley Pub.co., New York (1987) 4. Moser, B.: Sugeno Controllers with a bounded number of rules are nowhere dense. Fuzzy Sets and System 104(2), 269–277 (1999) 5. Kosko, B.: Neural Networks and Fuzzy systems: A Dynamical Approach To Machine Intelligence. Prentice Hall, Inc., Upper Saddle River (1992) 6. Simon, D.: Innovatia Software, Gradient Descent Optimization of fuzzy functions, May 29 (2007) 7. Kerre, E.E.: Mike Nachtegael-2000, Fuzzy Techniques in Image Processing (Studies in Fuzziness and Soft Computing), 1st edn. Physica-Verlag, Heidelberg (2000) 8. Kim, H.M., Mendel, J.M.: Fuzzy Basis Functions: Comparisons with other Basis Functions. IEEE Transactions on Fuzzy systems 3(2), 158–168 (1995) 9. Nair, M.S., Wilscy, M.: Modified Method for Denoising Color Images using Fuzzy Approach, SNPD. In: 2008 Ninth ACIS International Conference on Software Engineering, Artificial Intelligence, Networking, and Parallel/Distributed Computing, pp. 507–512 (2008) 10. Lu, R., Deng, L.: An Image Noise Reduction Technique Based on Fuzzy Rules. In: IEEE International conference on Audio, Language and Image Processing, pp. 1600–1605 (2008) 11. Gopalan, S., Sebastian, S., Sheela, C.: Modified Fuzzy Basis Function and Function Approximation. Journal of Computer & Mathematical Sciences 1(2), 263–273 (2010) 12. Wilscy, M., Nair, M.S.: A New Fuzzy-Based Additive Noise Removal Filter. In: Advances in Electrical Engineering and Computational Science. Lecture Notes in Electrical Engineering, vol. 39, pp. 87–97 (2009), doi:10.1007/978-90-481-2311-7_8

An Accelerated Approach of Template Matching for Rotation, Scale and Illumination Invariance Tanmoy Mondal1 and Gajendra Kumar Mourya2 1 Graphics and Intelligence Based Script Technology Group Centre for Development of Advanced Computing, Pune, India 2 Computational Instrumentation Unit Central Scientific Instruments Organisation, Chandigarh, India [email protected], [email protected]

Abstract. Template matching is the process for determining the presence and location of a certain reference in the scene sub image. The conventional approach like spatial correlation, cross correlation is computationally expensive and error prone when the object in the image is rotated or translated. The conventional approach gives erroneous result in the presence of variation in illumination of the scene sub image. In this paper, an algorithm for a rotation, scale and illumination invariant template matching approach is proposed, which is based on the combination of multi-resolution wavelet technique, projection method and Zernike moments. It is ideally suited for highlighting local feature points in the decomposed sub images, and the result is computationally saving in the matching process. As demonstrated in the result, the wavelet decomposition along with Zernike moments makes the template matching scheme feasible and efficient for detecting objects in arbitrary orientation and rotations. Keywords: Wavelet Decomposition; Zernike Moments; Template matching; Rotation, scale and illumination invariance.

1 Introduction Template matching is the process of determining the maximum match between template of interest and the scene sub image according to the same arbitrary measure, the scene sub image might be taken at different condition. It has a great application in different domains like remote sensing, medical imaging, computer vision etc. In the last few decades template matching approach have been widely applied in image registration and recognition process as illustrated in literatures [1-3]. Various rotation invariant template matching approaches have been proposed. A couple of novel set of template matching algorithm was proposed by Choi et al. [4]. In the first approach, the matching candidates, called as vector sum of circular projections of the sub image are selected using a computationally low cost feature, later frequency domain calculation was adopted to reduce the computational cost of spatial domain. In the second approach, rotation-invariant properties of Zernike moments are applied to perform template matching only on the candidate pixels. Orientation code-based matching is robust for searching objects in cluttered environments even in the cases of illumination fluctuations resulting from shadowing or highlighting etc. Ullah et al. [5] P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 121–128, 2011. © Springer-Verlag Berlin Heidelberg 2011

122

T. Mondal and G.K. Mourya

applied orientation codes for rotation invariant template matching in grey scale images. A new approach of template matching, invariant to image translation, rotation, scaling, proposed by Lin et al. [6]. An amalgamation of “ring-projection transform” to convert the 2D template in a circular region into 1D grey level signal and “parametric template vector” for template matching is achieved in their work. Tsai et al. [7] addressed a wavelet decomposition approach for rotation-invariant template matching. In the matching process, only the pixels with high wavelet coefficients based on norm in the decomposed detail sub image at a lower resolution level to compute the normalized correlation between two compared patterns is used. To perform rotation invariant template matching, they further use the ring-projection transform (RPT), which is invariant to object orientation, to represent an object pattern in the detailed sub image. The proposed method significantly reduces the computational burden of the traditional pixel-by-pixel matching. Cole-Rhodes et al. [8] introduced a registration algorithm that combines a simple yet powerful search strategy based on a stochastic gradient with two similarity measures, correlation and mutual information, together with a wavelet-based multi-resolution pyramid. Mutual information are shown to optimize with one-third the number of iterations required by correlation. Ideally, template matching should be invariant to translation, rotation and scaling in the input scene. However, the matching results of these schemes are sensitive to scale changes. A new template matching approach proposed by Lin et al. [9], in this approach, ring projection transform process is used to convert 2D template in a circular region into a 1D grey level signal as a function of radius. Then, the template matching is performed by constructing a parametric template vector (described by Tanaka et al. [10]) of the 1D grey level signal with differently scaled template of the object. However, the matching scheme becomes computational time expensive, for large searching regions. Template matching based on moments is a popular matching scheme [11, 12]. Moment invariants are properties of connected regions in binary images that are invariant to translation, rotation and scaling. They are useful because they define a simply calculated set of region properties that can be used for shape classification and object recognition. However, they require too much computation because of many integration and multiplication operations. Reddy and Chatterji [13] proposed an FFT-based matching, which is an extension to phase correlation technique. The specialty of this algorithm is its insensitivity to translation, rotation, scaling and noise. To find scale and rotational variation Fourier scaling properties and Fourier rotational properties are used; the translational movement is determined by the phase correlation techniques, but the matching method requires computation of three FFT’s and three IFFT’s, so the computation is apparently too computationally intensive. Therefore the template matching algorithm faces great challenges of computational efficiency, accuracy, and the input scene with the changes of translation, rotation and scale. The rotation invariant moment based method can be helpful but it is computationally expensive and is sensitive to noise. Araujo et al. [14] introduced the Color-Ciratefi that takes into account the color information for color template matching, which is a cumbersome problem due to color constancy. Recently they had proposed [15] a new grey scale template matching algorithm named Ciratefi, invariant to rotation, scale, translation, brightness and contrast. In their recent work for color template matching, a new similarity metric in the CIELAB space is used to obtain invariance to brightness and contrast changes. They demonstrated that Color-Ciratefi is

An Accelerated Approach of Template Matching

123

more accurate than C-color-SIFT.Kim [16] developed another RST-invariant template matching based on Fourier transform of the radial projections named Forapro. Forapro is faster than Ciratefi in a conventional computer but the later seems to be fit to be implemented in FPGA than the former.

2 Proposed Methodology In order to deal with template matching in large size of images a multi-resolution based wavelet decomposition [17] approach is proposed, which drastically reduces the computationally expensiveness of template matching. 2.1 Multi-resolution Based Wavelet Decomposition In case of 2-D image the wavelet transform can be defined as a two dimensional scaling function , , and three two dimensional wavelets , , , and , are required in defining wavelet transform in case of discrete images. Each of this produces results of two dimensional functions. Excluding products that produce one dimensional results like (x), the four remaining products produce the separable scaling function. The pyramid structure wavelet decomposition approach consists of filtering and down sampling horizontally using 1D low pass filter L (with impulse responses ) and high pass filter H (with impulse responses ) to each row in the image , , and produces the coefficient matrix , and , and produces four sub-images x, y , , , , , , for one level decomposition. , is the coarse representation of the image and , , , , Here , , , are detail sub-images, which represents horizontal, vertical and diagonal direction of the image. The detailed 2-D pyramid decomposition algorithm, with periodic boundary conditions applied, can be expressed as follows: Let

h

Then,

= original size of the image , = low pass coefficient of specific wave let basis, 0,1,2,3, … … . 1, where is the support length of the specific filter L. = high pass coefficient of specific wave let basis, j 0,1,2,3, … … . 1, where is the support length of the specific filter H. ∑

,

.

0,1,2, … … .

for ,

1

0,1,2, … … .

,

;

1

,

∑ 0

1

.

2

,

1 and y = 0, 1, 2, 3,……N-1. . 1

for

2

;

, 2 .

,

, 2

1 and y = 0, 1, 2, 3,…… -1.

1

. ,

1

;

, 2 .

, 2

,

124

T. Mondal and G.K. Mourya

A wavelet basis with long supports such as the Harr wavelet with compact support of 2 and a 4-tap Daubechies wavelet are the best choice for the investigated application, as with long support length may results in poor illumination, and can be affected by dominating boundary effect but it’s advantageous factors are, it is less sensitive to boundary effect and computationally efficient than those with long support lengths. The next decomposition level can be obtained by iterating 2-D pyramidal algorithm on the smooth image , which is the coarse representation of the original image. The multi-resolution based wavelet decomposition results in drastic reduction in the size of the image, in effect lessen computational time. Image size reduction level is denoted by 4 , where J denotes decomposition levels. If the decomposed sub-image has an over down-sampling size, it may results in false matching accordingly. As aforementioned three sum images containing, separately horizontal, vertical, and diagonal edge information of object patterns are obtained in decomposition level. These three detailed sub image can be combined in as a single composite detail, sub-image that can simultaneously contain the characteristics of horizontal, vertical, and diagonal edge information in a composite detail sum-image. The composite detail sub-image is given by: ,

,

=

,

,

+

,

+

Where,

;

,

,

,

,

are horizontal, vertical and diagonal detail sub images . Given an image of

, the size of the composite detail sub image

size

,

at resolution level

J is reduced to . Now matching process can be carried out on the composite detail sub image. The pixels with high energy values are considered for matching, these pixels are selected by comparing with threshold value. The threshold for selecting high energy value pixels is determined by Otsu’s method. Depending on the complexity of the image, the high energy valued pixels generally dominate 2030% of the entire composite detail sub image. So matching process become more robust and faster. 2.2 Zernike Moments Zernike moments are based on a set of complex polynomials that form a complete orthogonal set over the interior of the unit circle [18]. Zernike moments are defined as the orthogonal image function on these orthogonal basis functions. The basic , , is given by: , , ; where n is functions , , , | | is even and (+ve) integer, m is the non-zero integer subject to the constraints | | , is the length of the vector from the origin (centre of the image) to x, y and is the angle between the vector and axis in a counter clockwise direction and is the Zernike radial polynomial. The Zernike radial polynomial , is , defined as here

∑

,

,

! | |,

!

,

!

∑

| |,

, ,

. The basis function in equitation 1 is orthogonal thus satisfy

, ,

!

,

,

,

,

; where,

,

1; when a=b and

,

= 0;

An Accelerated Approach of Template Matching

125

otherwise the Zernike moments of order n with repetition m for a diagonal image , ∑∑ , , ; where , is defined function is given by: , , . Zernike moments extracted from a pattern are as the complex conjugate of , perfectly rotation, scale, and translation invariant. For more detailed discussion see [19,20]. 2.3 Improving Speed of the Zernike Moments Calculation By substituting equations 2 in 1 and 5 and reorganizing the term the Zernike moments can be calculated in the following form. ∑∑

,

∑

| |

,

,

=

∑

=

∑

| |

∑∑

, ,

| |

, ,

,

(1)

,

As mentioned earlier, defined in equation (1) , is noted to be the common term in the computation of Zernike moments with the same repetition as shown in figure (1) for the case of repetition m=0 and order n=0. In general to compute Zernike moments up to order N, , need to be computed for each repetition as shown in Table 1. The table demonstrates that , to be computed for each repetition up to order 10. The second row of the table corresponds to the , shown (same colour indicates repetition of , in the calculation of , ; n=0, 1...10) in figure 1. Once all the entries in the table 1 are computed, Zernike moments of any order and repetition can be calculated as a linear combination of , as shown in equation (1). It is worth to be noted that , , does not depend upon the image coordinates; therefore, they are stored on the small lookup table to save further computation. Numerical precision is another issue in the computation of high order Zernike moments. Depending on the image size and the maximum order, double precision arithmetic does not provide enough precision. 2.4 Matching Based on Zernike Moments In our application Zernike moments from order 0 up 10 (say l) is used. Therefore, for this case 36 (say k). Zernike moments are calculated as feature vector of the ,

, ,

,

, ,

,

, ,

,

, ,

, ,

,

, ,

+ + , + , +

, ,

, ,

+

, ,

,

,

, ,

,

,

, ,

,

, ,

,

, ,

,

, ,

,

, ,

, , , ,

, ,

, ,

,

, , , ,

, ,

, , , ,

, ,

, ,

,

Fig. 1. The common terms to compute Zernike moments up to 10 orders and zero repetition

the image. The repetition parameter m is automatically calculated for each order (n), as, n is even, then m will be number of even numbers present between 0 to n and vice

126

T. Mondal and G.K. Mourya

versa. Let be the Zernike moments of the template , , and be the moments of , in the region which coincides with current location of . Then the correlation coefficient at point where

0,1,2, … … . .

1,

, and

Table 1. , is needed to compute Zernike moments to 10 order and m repetition repetition m

0 1 2 3 4 5 6 7 8 9 10

given by:

,

are means of

∑ ∑

∑

and

, respectively.

Table 2. List of Zernike moments and their corresponding mber of features up to order 10 Order

0 1 2 3 4 5 6 7 8 9 10

Zernike Moments

Number of Zernike Moments

1 1 2 2 3 3 4 4 5 5 6

3 Experimental Results Experiments were conducted to measure the performance of the algorithm in terms of speed and matching accuracy. As proposed in this paper, that the combination of multi-resolution based wavelet decomposition approach with accelerated Zernike moment methodology is very useful for computational time inexpensive template matching. The hierarchical Zernike moment approach utilizes the pyramid structure of the image and yielded results comparable in terms of accuracy. 3.1 Test Images The proposed algorithm is tested with a series of experiment conducted under various translations, rotation, scales and also with variation in brightness, contrast change, and added Gaussian noise. As observed from the experiment, the speed of the test merely depends on the size of the template. The following figure (2) reports the results obtained by performing experimentation in all of the images with different settings, and the proposed algorithm succesfully locates the position of each template precisely.

An Accelerated Approach of Template Matching

(a)

(c)

127

(d)

(b)

(e)

(f )

Fig. 2. (a) Sample test image (PCB board), (b) reference template, (c) tested image rotated in an arbitrary angle, (d) image rotated in arbitrary angle with additive Gaussian noise of zero mean variance of 0.025 (e) image with brightness variation, (f) image with contrast variation

4 Conclusion In this paper a robust and computationally inexpensive template matching approach are proposed. The proposed method significantly reduces the number of search pixels by using the composite detail sub image at a low resolution level. The computation of correlation coefficients is only carried out for those high energy-valued pixels in the composite detail sub image. The accelerated Zernike moment representation not only reduces the computational complexity of the normalized correlation but also makes the detection invariant to different transformations. As can be observed from the experiment that the performance is rather good when SNR is more than 20 dB.

References [1] Brown, L.G.: A survey of image registration techniques. ACM Comput. Surv. 24(4), 326–376 (1992) [2] Zitová, B., Flusser, J.: Image registration methods: a survey. Image Vis. Comput. 21(10), 997–1000 (2003) [3] Maintz, J.B.A., Viergever, M.A.: A survey of medical image registration. Med. Image Anal. 2, 1–36 (1998) [4] Choi, M.S., Kim, W.Y.: A novel two stage two template matching method for rotation and illumination invariance. Pattern Recognition 35, 119–129 (2002)

128

T. Mondal and G.K. Mourya

[5] Ullah, F., Kanekoi, S.: Using orientation codes for rotation invariant template matching. Pattern Recognition 37(2), 201–209 (2004) [6] Yi-Hsein, L., Chin-Hsing, C.: Template matching using parametric template vector with translation, rotation and scale invariance. Pattern Recognition 41, 2413–2421 (2008) [7] Tsai, D.M., Chiang, C.H.: Rotation-invariant pattern matching using wavelet decomposition. Pattern Recognition Lett. 23, 191–201 (2002) [8] Cole-Rhodes, A.A., Johnson, K.L., Le Moigne, J., Zavorin, I.: Multi-resolution registration of remote sensing imagery by optimization of mutual information using a stochastic gradient. IEEE Trans. Image Process. 12(11), 1495–1511 (2003) [9] Yi, L., Chen, C.: Template matching using parametric template vector with translation, rotation, scale invariance: Elsevier. Pattern Recognition 41, 2413–2421 (2008) [10] Tanaka, K., Sano, M., Ohara, S., Okudaira, M.: A parametric template method and its application to robust matching. In: IEEE Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 620–627 (June 2000) [11] Liao, S.X., Pawlak, M.: On image analysis by moments. IEEE Trans. Pattern Anal. Mach. Intell. 18(3), 254–266 (1996) [12] Flusser, J., Suk, T.: Affine moment invariants: a new tool for character recognition. Pattern Recognition Lett. 15, 433–436 (1994) [13] Reddy, B.S., Chatterji, B.N.: An FFT-based technique for translation, rotation, and scaleinvariant image registration. IEEE Trans. Image Process 5(8), 1266–1271 (1996) [14] de Araújo, S.A., Yong, K.H.: Color-Ciratefi: A color- based RST-invariant template matching algorithm. In: IWSSIP - 17th International Conference on Systems, Signals and Image Processing (2010) [15] Kim, H.Y., de Araújo, S.A.: Grayscale template-matching invariant to rotation, scale, translation, brightness and contrast. In: Mery, D., Rueda, L. (eds.) PSIVT 2007. LNCS, vol. 4872, pp. 100–113. Springer, Heidelberg (2007) [16] Kim, H.Y.: Rotation-Discriminating Template Matching Based on Fourier Coefficients of Radial Projections with Robustness to Scaling and Partial Occlusion. Pattern Recognition 43(3), 859–872 (2010) [17] Tsai, D.M., Chen, C.H.: Rotation invariant pattern matching using wavelet decomposition, http://citeseerx.ist.psu.edu/viewdoc/ download?doi=10.1.1.108.2705&rep=rep1&type=pdf [18] Amayeh, G.R., Erol, A.: Accurate and Efficient computation of High Order Zernike Moments [19] More, V.N., Rege, P.P.: Devanagari handwritten numeral identification based on Zernike moments. IEEE Xplore, pp. 1–6 (2009) [20] Qader, H.A., Ramli, A.R.: Fingerprint recognition using Zernike moments. The International Arab Journal of Information Tech. 4(4) (October 2007)

Image Edge and Contrast Enhancement Using Unsharp Masking and Constrained Histogram Equalization P. Shanmugavadivu1 and K. Balasubramanian2 1 Department of Computer Science & Applications, Gandhigram Rural Institute – Deemed University, Gandhigram [email protected] 2 Department of Computer Applications, PSNA College of Engineering & Technology, Dindigul [email protected]

Abstract. Histogram Equalization (HE) based methods are the commonly used image enhancement techniques to improve the contrast of an input image by changing the intensity level of the pixels, based on its intensity distribution. This paper presents a novel image contrast and edge enhancement technique using constrained histogram equalization. A new transformation function for histogram equalization is devised and a set of constraints are applied over the histogram of an image prior to the equalization process and then, unsharp masking is applied on the resultant image. The proposed method ensures better contrast and edge enhancement, which is measured in terms of discrete entropy. Keywords: Histogram equalization, image sharpening, edge enhancement, unsharp masking.

1 Introduction Contrast enhancement plays a major role in image processing applications such as digital photography, medical imaging, remote sensing etc. Histogram Equalization (HE) is one of the well-known methods for enhancing the contrast of an image [1]. HE uniformly distributes the intensity of the input image by stretching its dynamic intensity range using cumulative density function. The major problem of HE is the mean shift that attributes to a drastic deviation in the mean brightness of input and output images. HE based techniques are categorized into two as: (i) The Adaptive (or local) HE methods (AHE) (ii) The improved global methods based on histogram equalization or histogram specification (matching) In the AHE [1] methods, equalization is based on the statistics obtained from the histograms of neighborhood pixels. AHE methods usually provide stronger enhancement effects than global methods. However, due to their high computational load, AHE methods are not best suited for real-time applications. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 129–136, 2011. © Springer-Verlag Berlin Heidelberg 2011

130

P. Shanmugavadivu and K. Balasubramanian

The following are the improved global HE methods: • Brightness Preserving Bi-Histogram Equalization (BBHE) BBHE [2] divides the input image into two sub-images based on its mean and the equalization is performed independently for the two sub-images. Unlike HE, BBHE preserves the original brightness of an input image. However, it creates unwanted artifacts. • Dualistic Sub-Image Histogram Equalization (DSIHE) DSIHE [3] is a simple variation of BBHE. It selectively separates the image histogram based on the gray level (threshold level) with cumulative probability density equivalent to 0.5. • Recursive Mean Separate HE (RMSHE) Chen and Ramli proposed a novel method, called as RMSHE [4] which partitions the input image’s histogram recursively based on mean. Here, some portions of histogram among the partitions cannot be expanded much, while the outside regions are expanded significantly that creates the unwanted artifacts. This is a common drawback of most of the existing histogram partitioning techniques since they keep the partitioning point fixed throughout the entire process of equalization. • Minimum Mean Brightness Error Bi-Histogram Equalization (MMBEBHE) In MMBEBHE [5], a threshold level is calculated based on Absolute Mean Brightness Error (AMBE). Using this level, the input image histogram is separated into two and the HE process is then applied to them independently. But it is recorded that the performance of RMSHE is higher than that of MMBEBHE. • Recursive Sub-Image Histogram Equalization (RSIHE) RSIHE [6] is another recursive method which resembles RMSHE. In RSIHE, the histogram is recursively partitioned, based on the median instead of mean. This method produces 2r pieces of sub-histograms. It is very difficult to find the optimal value of ‘r’. • Weighted Thresholded Histogram Equalization (WTHE) WTHE [7] was proposed by Wang and Ward in which the histogram of an input image is modified by weighting and thresholding prior to the histogram equalization. But this technique fails to sharpen and to preserve the original details of the input image, while enhancing the contrast. • Recursively Separated and Weighted Histogram Equalization (RSWHE) In RSWHE [8], the input image histogram is segmented into two or more subhistograms recursively. A weighing process is then applied on those subhistograms independently. HE process is finally applied on the weighted subhistograms. This method is computationally complex since it is recursive and the formulation of weighing module for all sub-histograms is not an easy task to be performed at a faster rate. • Sub-Regions Histogram Equalization (SRHE) Ibrahim and Kong developed a new HE technique called SRHE [9]. This method partitions the image based on the smoothed intensity values, which are obtained by convolving the input image with Gaussian filter.

Image Edge and Contrast Enhancement Using Unsharp Masking and Constrained HE

131

In this paper, a new edge enhancement technique based on the modified HE transformation function of SRHE and the weighing principle used in WTHE is developed. Additionally, the edge and contrast of the images are improved by the application of unsharp masking filter with better scalability.

2 Basic Histogram Equalization Let X={X(i, j)} denote a digital image, where X(i, j) denotes the gray level of a pixel at (i, j). The total number of pixels in the image is N and the image intensity is digitized into L levels that are {X0, X1, ..., XL-1}. So it is obvious that ∀X (i, j ) ∈{ X 0 , X 1 ,...., X L −1 } . Suppose nk denotes the total number of pixels with gray level of Xk in the image, then the probability density of Xk is given as: p( X k ) =

nk , k = 0, 1, ...., L − 1 N

(1)

The plot between p(Xk) and Xk is defined as the histogram of an image. The cumulative density function (CDF) based on the image's PDF is defined as: k

c( X k ) = ∑ p( X i )

(2)

i =0

where k = 0, 1 ,..., L-1 and it is known that c(XL-1)=1. The transformation function of HE is defined as: f ( X k ) = X 0 + ( X L −1 − X 0 ) × c( X k ), k = 0, 1, ...., L − 1

3

(3)

The Proposed Method for Edge and Contrast Enhancement

The proposed technique accomplishes edge and contrast enhancement at three levels through i) Development of modified HE transformation function ii) Classifying input image histograms based on weighing and thresholding and the application of modified HE transformation function and iii) Unsharp masking over the resultant image. 3.1 Modified HE Transformation Function It is evident that there is significant intensity variation between the HEed image and the image obtained after inversing the input image, equalizing and then re-inversing it [9]. It is illustrated in Fig. 1(a) – 1(e). It is prominently visible from Fig. 1(b) and Fig. 1(e) that there exists a qualitative difference due to intensity variations between them. This intensity variation is due to the fact that the low probable intensity values of the input image gain importance after inversion. In the normal HE process, they are almost ignored and hence the contrast and edges of the image are not enhanced significantly.

132

P. Shanmugavadivu and K. Balasubramanian

(a)

(b)

(c)

(d)

(e)

Fig. 1. (a) Original Truck Image (b) HEed Truck Image (c) Negative of Truck Image (d) HEed of Fig.1(c) (e) Negative of Fig.1(d)

In order to exploit the advantages of these two approaches, it is proposed to take the intensity average of Fig. 1(b) and Fig. 1(e). The transformation function f1 for the image with inversion process is given in equation (4). ⎛ X L −1 ⎞ f 1 ( X k ) = X 0 + ( X L −1 − X 0 ) × ⎜⎜1 − ∑ p( X i )⎟⎟ ⎝ i= X k ⎠

The average of the output image fnew is obtained from f and f1 as:

f new ( X k ) =

1 ( f ( X k ) + f 1 ( X k )) 2

Xk ⎞ ⎛ ⎟ ⎜ X 0 + ( X L −1 − X 0 ) × ∑ p ( X i ) + i= X 0 ⎟ 1⎜ = ⎜ ⎟ X L −1 2⎜ ⎞⎟ ⎛ X + ( X L −1 − X 0 ) × ⎜⎜1 − ∑ p ( X i )⎟⎟ ⎟ ⎜ 0 ⎠⎠ ⎝ i= X k ⎝

(4)

Image Edge and Contrast Enhancement Using Unsharp Masking and Constrained HE

133

Xk ⎛ ⎞ ⎜ X 0 + ( X L−1 − X 0 ) × ∑ p ( X i ) + ⎟ ⎟ i= X 0 1⎜ = ⎜ ⎟ X 2⎜ ⎞⎟ ⎛ k −1 X + ( X L−1 − X 0 ) × ⎜⎜ ∑ p ( X i )⎟⎟ ⎜ 0 ⎟ ⎠⎠ ⎝ i= X 0 ⎝ ⎛ ⎛ X k −1 ⎞ ⎞ ⎜ X 0 + ( X L −1 − X 0 ) × ⎜ ∑ p( X i ) + p( X k )⎟ + ⎟ ⎜ ⎟ ⎟ 1⎜ ⎝ i= X 0 ⎠ = ⎜ ⎟ X 2⎜ ⎛ k −1 ⎞ ⎟ ⎜ X 0 + ( X L −1 − X 0 ) × ⎜⎜ ∑ p ( X i )⎟⎟ ⎟ ⎝ i= X 0 ⎠ ⎝ ⎠ X k −1 ⎤ 2 X 0 ( X L−1 − X 0 ) ⎡ = + ⎢ p( X k ) + 2 ∑ p( X i )⎥ 2 2 i= X 0 ⎣ ⎦ X k −1 ⎡ ⎤ f new ( X k ) = X 0 + ( X L −1 − X 0 ) ⎢0.5 p ( X k ) + ∑ p ( X i )⎥ i= X0 ⎣ ⎦

(5)

The HE process of the proposed method uses this transformation function (equation (5)). 3.2 Weighing and Thresholding

Once the probability densities are found using equation (1) for the individual pixels of an input image, the probability density values are modified [7] using the weighing and thresholding method as follows: p c ( X k ) = Τ( p ( X k )) ⎧ pu ⎪ ⎪⎛ p ( X k ) − p l = ⎨⎜⎜ ⎪⎝ p u − p l ⎪0 ⎩

if p ( X k ) > p u

r

⎞ ⎟⎟ * p u ⎠

⎫ ⎪ ⎪ if p l < p ( X k ) ≤ p u ⎬ ⎪ ⎪ if p ( X k ) ≤ p l ⎭

(6)

where pc(Xk) is the weighted and thresholded probability density for the pixels with gray level Xk; Pu is the upper constraint which is computed as Pu = v * max(PDF) where 0.1 < v < 1.0; ‘r’ is the power factor which accepts values as 0.1 < r < 1.0 and Pl is the lower constraint whose value is arbitrarily selected as low as possible (for e.g. Pl=0.0001). The transformation function T(p(Xk)) fixes the original probability density value at an upper threshold Pu and at a lower threshold Pl and transforms all values between the upper and lower thresholds using a normalized power law function with index r>0. This clamping procedure is mainly used for enhancing the contrast of an image. The CDF is obtained using the modified probability density values as in equation (7).

134

P. Shanmugavadivu and K. Balasubramanian

k

cc ( X k ) = ∑ pc ( X i )

(7)

i =0

Then, the HE procedure is applied using the modified HE transformation function given in equation (5). 3.3 Unsharp Masking

The high frequency domain of an image plays a vital role in enhancing its visual appearance, with respect to the edges and contrast. The classic linear unsharp masking (UM) is one of the ideal techniques which is often employed for this purpose [1]. UM produces an edge image g(x,y) from an input image f(x,y) as follows: g ( x, y ) = f ( x, y ) − f smooth ( x, y )

(8)

where fsmooth(x,y) is the smoothed version of f(x,y). This edge image can be used for sharpening by adding it back to the original image as: f sharp ( x, y ) = f ( x, y ) + k * g ( x, y )

(9)

where, ‘k’ is the scaling constant which decides the degree of sharpness of an image. ‘k’ usually accepts the values in the range 0.1 to 1.0. After the input image’s probability density values are weighed, thresholded and equalized, the resultant image is treated with unsharp masking filter which sharpens the image by means of enhancing the edges.

4

Results and Discussion

The proposed image enhancement algorithm is developed in MATLAB 7.0. In this proposed technique, three major parameters which control the enhancement of contrast as well as edges are identified. They are: • the value of ‘v’ for the calculation of Pu • the value of normalized power law function index ‘r’ and • the value of UM scaling constant ‘k’

The optimal values for ‘v’, ‘r’ and ‘k’ are identified through iterative process. The objective function, discrete entropy E(X) is used to measure the richness of details in an image after enhancement [10]. It is defined as: 255

E ( X ) = − ∑ p ( X k ) log 2 ( p ( X k ))

(10)

k =0

In order to evaluate the performance of the proposed method, standard images such as Bottle, Truck, Village, Woman and Putrajaya are chosen. The performance of the proposed method is quantitatively measured in terms of discrete entropy. The entropy values are computed for those test images after the application of our proposed method. The same are calculated for contemporary HE methods such as GHE, BBHE,

Image Edge and Contrast Enhancement Using Unsharp Masking and Constrained HE

135

Table 1. Comparison of Discrete Entropy Values Image

Original

Bottle Truck Village Woman Putrajaya

7.4472 6.5461 7.4505 7.2421 6.8116

(a)

(d)

Proposed Method 7.3717 6.4082 7.3006 7.0981 6.7068

GHE

BBHE

DSIHE

RMSHE

5.9776 5.8041 5.9769 5.9594 5.7758

7.2826 6.3796 7.2455 6.9889 6.6223

7.2477 6.3781 7.2404 6.9824 6.5952

7.2924 6.4163 7.2982 7.0451 6.6436

(b)

(c)

(e)

(f)

Fig. 2. (a) Original Putrajaya image and the resultant images of (b) Proposed Method (c) GHE (d) BBHE (e) DSIHE (f) RMSHE

DSIHE and RMSHE. The comparative entropy values are detailed in the Table 1. It is evident from this table that the proposed method has higher entropy values which are very closer to the original images. This signifies that the proposed method is found to preserve the details of the original image in the enhanced image. The qualitative performance of the proposed and contemporary methods is illustrated using Putrajaya image which is shown in Fig. 2(a) – 2(f). Fig. 2(c) – 2(f) are the enhanced images using the standard methods such as GHE, BBHE, DSIHE and RMSHE respectively. It can be noted from these images that the standard methods failed to enhance the edges of the input image. Fig. 2(b) is the result of the proposed method in which the edges as well as the contrast of the original image are enhanced and the optimal values of ‘v’, ‘r’ and ‘k’ are found to be 0.2, 0.5 and 0.9 respectively. Moreover, Fig. 2(b) clearly shows that the visual perception of the proposed

136

P. Shanmugavadivu and K. Balasubramanian

method is better than those of the other HE techniques and is free from overenhancement. The proposed method is found to produce comparatively better results for other test images too.

5 Conclusion In general, the mean gray level of the input image is different from that of the histogram equalized output image. The proposed method is proved to solve this problem effectively. This method accomplishes two major desired objectives of edge and contrast enhancement of any given input image. It is computationally simple and is easy to implement. It provides subjective scalability in edge as well as contrast enhancement, by appropriately adjusting the control parameters namely ‘v’, ‘r’ and ‘k’. This method suitably finds application in the field of consumer electronics, medical imaging, etc.

References 1. Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 2nd edn. Prentice Hall, Englewood Cliffs (2002) 2. Kim, Y.: Contrast enhancement using brightness preserving bi-histogram equalization. IEEE Transactions on Consumer Electronics 43, 1–8 (1997) 3. Wan, Y., Chen, Q., Zhang, B.: Image enhancement based on equal area dualistic subimage histogram equalization method. IEEE Transactions on Consumer Electronics 45, 68–75 (1999) 4. Chen, S., Ramli, A.R.: Contrast enhancement using recursive mean-separate histogram equalization for scalable brightness preservation. IEEE Transactions on Consumer Electronics 49, 1301–1309 (2003) 5. Chen, S., Ramli, A.R.: Minimum mean brightness error bi-histogram equalization in contrast enhancement. IEEE Transactions on Consumer Electronics 49, 1310–1319 (2003) 6. Sim, K.S., Tso, C.P., Tan, Y.Y.: Recursive sub-image histogram equalization applied to gray-scale images. Pattern Recognition Letters 28, 1209–1221 (2007) 7. Wang, Q., Ward, R.K.: Fast image/video contrast enhancement based on weighted thresholded histogram equalization. IEEE Transactions on Consumer Electronics 53, 757–764 (2007) 8. Kim, M., Chung, M.: Recursively separated and weighted histogram equalization for brightness preservation and contrast enhancement. IEEE Transactions on Consumer Electronics 54, 1389–1397 (2008) 9. Ibrahim, H., Kong, N.S.P.: Image sharpening using sub-regions histogram equalization. IEEE Transactions on Consumer Electronics 55, 891–895 (2009) 10. Wang, C., Ye, Z.: Brightness preserving histogram equalization with maximum entropy: A variational perspective. IEEE Transactions on Consumer Electronics 51, 1326–1334 (2005)

Application of MOPSO and MOCLPSO for the Optimal Placement of TCSCs S.T. Jaya Christa1 and P. Venkatesh2 1

Department of Electrical and Electronics Engineering, Mepco Schlenk Engineering College, Sivakasi – 626 005, Tamil Nadu, India Mobile: 91-9486288111 [email protected] 2 Department of Electrical and Electronics Engineering, Thiagarajar College of Engineering, Madurai – 625 015, Tamil Nadu, India [email protected]

Abstract. The optimal placement of Thyristor Controlled Series Compensators (TCSCs) in transmission systems is formulated as a multi-objective optimization problem with the objective of maximizing transmission system loadability, minimizing transmission loss and minimizing the cost of TCSCs. The thermal limits of transmission lines and voltage limits of load buses are considered as security constraints. Multi-Objective Particle Swarm Optimization (MOPSO) and Multi-Objective Comprehensive Learning Particle Swarm Optimizer (MOCLPSO) are applied to solve this problem. The proposed approach has been successfully tested on IEEE 14 bus system and IEEE 118 bus system. From the results it is inferred that MOCLPSO can obtain an evenly distributed pareto front. A multi-criterion decision-making tool, known as TOPSIS, has been employed to arrive at the best trade-off solution. Keywords: MOCLPSO, System loadability, Thyristor Controlled Series Compensator, Transmission loss, Multi-objective optimization, TOPSIS.

1 Introduction Many real-world problems generally involve simultaneous optimization of multiple, often conflicting objective functions. In recent years, several evolutionary algorithms have been successfully applied to solve multi- objective optimization problems. An algorithm which finds an ideal set of tradeoff solutions which are close to and uniformly distributed across the optimal tradeoff surface can be considered to be well suited for a particular problem. One of the problems faced by Power System Engineers is to increase the power transfer capability of existing transmission systems. This problem can be overcome by installing Flexible AC Transmission Systems (FACTS) devices which has recently gained much attention in the electrical power world [1]. Finding the optimal location of a given number of FACTS devices and their parameters is a combinatorial optimization problem. To solve such a problem, heuristic methods can be used [2]. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 137–144, 2011. © Springer-Verlag Berlin Heidelberg 2011

138

S.T. Jaya Christa and P. Venkatesh

Many researches were carried out on the optimal allocation of FACTS devices considering a single objective of loadability maximization and several techniques have been applied for finding the optimal location of different types of FACTS devices in power systems [3],[4],[5],[6],[7]. Several methods have been proposed to solve multi-objective optimization problems [8],[9]. Recently, multi-objective particle swarm optimization techniques have received added attention for their application in power system problems for FACTS devices placement [10],[11]. Decision making is an important task in multi-objective optimization. Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) was proposed in [12] to solve classical Multi-Criteria Decision Making (MCDM) problems. In this paper, one of the FACTS devices namely Thyristor Controlled Series Compensator (TCSC) is employed to analyse the increase in power transfer capabilities of transmission lines and reduction in transmission loss considered along with the minimization of the installation cost of TCSCs. In this paper, Multi-Objective Particle Swarm optimization (MOPSO) and Multi-Objective Comprehensive Learning Particle Swarm optimizer (MOCLPSO) have been applied for the optimal allocation of single and multiple TCSCs. The best compromise solution from the set of pareto-optimal solutions is selected based on TOPSIS method. The IEEE 14 bus system and IEEE 118 bus system are taken as test systems to investigate the performance of the MOPSO and MOCLPSO techniques.

2

Problem Formulation

The objective of this paper is to find out the optimal location and parameters of TCSCs which simultaneously maximize the loadability of transmission lines and minimize the transmission loss in the system and minimize the cost of TCSCs, while satisfying the line flow limits and bus voltage limits. The objectives for placing the TCSCs are described below. The constraints considered in this problem are the power balance constraints, line flow constraints, bus voltage constraints and TCSC parameter operating range. 2.1 Objectives for the Problem 2.1.1 Loadability Maximization The objective function for loadability maximization is given as, Maximize where

λ

f1 = λ

(1)

is a scalar parameter representing the loading factor.

2.1.2 Loss Minimization The objective function for loss minimization is given as, Minimize

f2 =

NL

∑ (P k =1

LK

)

(2)

Application of MOPSO and MOCLPSO for the Optimal Placement of TCSCs

139

where,

PLK - real power loss of line k NL - number of transmission lines 2.1.3 Minimization of Installation Cost of TCSCs The objective function for minimization of installation cost of a TCSC is given as, Minimize

f3 = CTCSC

x

S

x

1000

(3)

Where,

CTCSC - cost of TCSC in US$ / kVAR S - operating range of the TCSC in MVAR The cost of TCSC is given as,

CTCSC = 0.0015S 2 − 0.713 S + 153.75

(4)

3 Application of MOPSO and MOCLPSO for the Multi-Objective Optimization Problem In this paper, two multi-objective optimization techniques namely Multi-Objective Particle Swarm Optimization (MOPSO) algorithm and Multi-Objective Comprehensive Learning Particle Swarm Optimizer (MOCLPSO) are employed for solving the multi-objective loadability maximization, loss minimization and minimization of installation cost of TCSCs problem. MOPSO is an extension of Particle Swarm Optimization (PSO) algorithm to handle multi-objective optimization problems [13]. MOPSO algorithm uses an external archive to record the non-dominated solutions obtained during the search process. MOCLPSO is an extension of Comprehensive Learning Particle Swarm Optimizer (CLPSO) to handle multiple objective optimization problems [14]. In CLPSO, the particle swarm learns from the global best (gbest) solution of the swarm, the particle’s personal best (pbest) solution, and the pbests of all other particles [15]. In this paper, pbest is selected randomly as in [16]. In this paper, gbest is selected by randomly choosing a particle from the non-dominated solutions. The crowding distance method of density estimation is employed in MOCLPSO to maintain the archive size when the archive reaches the maximum allowed capacity.

4 TOPSIS Method of Decision Making After solving the multi objective problem, the researcher is required to select a solution from the finite set by making compromises. In this paper, TOPSIS method of decision making is adopted to select the best solution. The basic principle of TOPSIS method is that the chosen points should have the shortest distance from the ideal solution and the farthest distance from the negative ideal solution. The researcher has to express the importance of criteria to determine the weights of the criteria.

140

S.T. Jaya Christa and P. Venkatesh

5 Numerical Results and Discussion The IEEE 14-bus power system and the IEEE 118-bus power system are used as the test systems. The numerical data of IEEE 14-bus system are taken from [17] and the numerical data of IEEE 118-bus system are taken from [7]. To successfully implement MOPSO and MOCLPSO, the values for the different parameters are to be determined correctly. In this paper, the inertia weight is linearly decreased from 0.9 to 0.1 over the iterations and the acceleration coefficients are taken as 2. For MOCLPSO, learning probability, Pc is set as 0.1 and elitism probability, Pm is chosen as 0.4. The archive size is maintained as 100 for both MOPSO and MOCLPSO. 5.1 IEEE 14 Bus System Generally, when the loadability increases, the total loss also increases. By optimally placing the TCSCs, the loadability can be increased and the loss can be minimized. Objective values obtained by MOPSO and MOCLPSO for the optimal placement of one TCSC are shown in Fig. 1, and Fig.2 respectively. The two extreme solutions of the pareto front obtained using MOPSO are f1 =1.000064, f 2 = 35.9167,

f 3 = 3.6991x105 and f1 =1.147815, f 2 = 49.996, f 3 = 6.6477x105. The two extreme solutions of the pareto front obtained using MOCLPSO are f1 =1.000064, f 2 = 35.7381, f 3 = 1.276x108 and f1 =1.147815, f 2 = 49.996, f 3 = 6.6477x105. From TOPSIS method, the best compromise solutions for the placement of one TCSC are selected among the pareto-optimal solutions obtained using MOPSO and MOCLPSO and are shown in Table 1. In this paper, the weight assigned to the loadability objective, w1 is 0.6, the weight assigned to the loss objective, w2 is 0.1 and the weight assigned to the cost objective,

w3 is 0.3.

Table 1. The best compromise solutions based on TOPSIS for placing 1 TCSC and 2 TCSCs

% Cost of Number of Optimization Loading Loss reduction TCSCs TCSCs Method Factor (MW) in loss (US$) 5 5.9434x10 MOPSO 1.1287 47.431 2.738 1 TCSC 5 MOCLPSO 1.1281 47.3769 2.738 5.935x10 MOPSO

1.1478 49.6745 1.894

2 TCSCs MOCLPSO 1.1576 50.9394 1.298

Lines Parameter of with TCSC (p.u.) TCSC 7-8 -0.14092 7-8 -0.14092 -0.14092 7 7-8 9.5296x10 3-4 -0.047153 7-8 -0.14092 1.4506x107 9-14 -0.048668

Application of MOPSO and MOCLPSO for the Optimal Placement of TCSCs

Fig. 1. Objective values obtained with MOPSO for the placement of one TCSC

Fig. 2. Objective values obtained using MOCLPSO for the placement of one TCSC

141

142

S.T. Jaya Christa and P. Venkatesh

The comparison of the number of fitness evaluations and computation time for MOPSO and MOCLPSO for the placement of one number of TCSC and two numbers of TCSCs are given in Table 2. The performance metric, spread is calculated to judge the performance of the algorithms and it is also presented in Table 2. The results prove that, MOCLPSO is able to obtain a more evenly distributed pareto front when compared to MOPSO. Table 2. Comparison of computation time and performance metrics for IEEE 14 bus system

Particulars No of fitness evaluations Average computation time for one trial (secs) Spread

MOPSO MOCLPSO 1 TCSC 2 TCSC 1 TCSC 2 TCSC 12,000 18,000 12,000 18,000 455 834 465 840 0.0701 0.0938 0.0075 0.0723

5.2 IEEE 118 Bus System A minimum of 5 TCSCs is considered for IEEE 118 bus system. The best compromise solutions selected using TOPSIS are presented in Table 3. Lower value of spread is obtained for MOCLPSO for this case also. The results obtained for the test systems prove that MOCLPSO is efficient in finding the optimal placement of TCSCs for this multi-objective optimization problem. Table 3. The best compromise solutions based on TOPSIS for placing five TCSCs in IEEE 118 bus system

Optimization Loading Method Factor

Loss % reduction (MW) in loss

Cost of TCSCs (US$)

MOPSO

1.188

158.183

0.0722

7.6132x108

MOCLPSO

1.188

158.057

0.1518

4.1563x108

Lines with TCSCs 75-118 106-107 104-105 76-118 105-106 75-118 106-107 104-105 76-118 105-106

Parameters of TCSCs (p.u.) -0.03848 -0.1464 -0.03024 -0.04352 -0.04376 -0.03848 -0.00183 -0.03024 -0.04352 -0.04376

6 Conclusion In this paper, the optimal placement of TCSCs for transmission system loadability maximization, transmission loss minimization and minimization of installation cost of TCSCs has been formulated as a multi-objective optimization problem. The thermal

Application of MOPSO and MOCLPSO for the Optimal Placement of TCSCs

143

limits of transmission lines and voltage limits of load buses are considered as security constraints. This problem is solved by using MOPSO and MOCLPSO. The proposed approach has been successfully tested on two IEEE test systems namely, IEEE 14 bus system and IEEE 118 bus system. The pareto optimal solutions obtained by MOPSO are compared with those obtained by MOCLPSO. From the performance metric evaluated for the pareto optimal solutions obtained by MOPSO and MOCLPSO, it is inferred that MOCLPSO performs well with a good spread of solutions along the pareto front. TOPSIS method has been adopted to ease the decision making process. Acknowledgement. The authors thank the managements of Thiagarajar College of Engineering and Mepco Schlenk Engineering College for their support. Simulation results are obtained by modifying the MOCLPSO matlab codes made available on the Web at http://www.ntu.edu.sg/home/EPNSugan. The authors express their sincere thanks to Prof.P.N. Suganthan, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore for providing the MOCLPSO matlab codes.

References 1. Hingorani, N.G.: High power electronics and Flexible AC Transmission System. IEEE Power Engineering Review 8(7), 3–4 (1988) 2. Sait, S.M., Youssef, H.: Iterative computer algorithms with applications in engineering: solving combinatorial optimization problems, 1st edn. IEEE Computer Society Press, California (1999) 3. Gerbex, S., Cherkaoui, R., Germond, A.J.: Optimal location of multi-type FACTS devices in a power system by means of genetic algorithms. IEEE Transactions on Power systems 16(3), 537–544 (2001) 4. Hao, J., Shi, L.B., Chen, C.: Optimising location of Unified Power Flow Controllers by means of improved evolutionary programming. IEE Proceedings on Generation, Transmission and Distribution 151(6), 705–712 (2004) 5. Mori, H., Maeda, Y.: Application of two-layered tabu search to optimal allocation of UPFC for maximizing transmission capability. In: IEEE International Symposium on Circuits and Systems (ISCAS), Island of Kos, Greece, pp. 1699–1702 (2006) 6. Rashed, G.I., Shaheen, H.I., Cheng, S.J.: Optimal Location and Parameter Settings of Multiple TCSCs for Increasing Power System Loadability Based on GA and PSO Techniques. In: Third International Conference on Natural Computation (ICNC), Haikou, Hainan, China, vol. 4, pp. 335–344 (2007) 7. Saravanan, M., Mary Raja Slochanal, S., Venkatesh, P., Prince Stephen Abraham, J.: Application of particle swarm optimization technique for optimal location of FACTS devices considering cost of installation and system loadability. Electric Power Systems Research 77(3-4), 276–283 (2007) 8. Deb, K.: Multi-objective optimization using evolutionary algorithms, 1st edn. John Wiley and Sons, New York (2001) 9. Deb, K.: Current trends in evolutionary multi-objective optimization. Int. J. Simul. Multidisci. Des. Optim. 1, 1–8 (2007) 10. Eghbal, M., Yorino, N., Yoshifumi Zoka El-Araby, E.E.: Application of evolutionary multi objective optimization algorithm to optimal VAr expansion and ATC enhancement problems. International Journal of Innovations in Energy Systems and Power 3(2), 6–11 (2008)

144

S.T. Jaya Christa and P. Venkatesh

11. Benabid, R., Boudour, M., Abido, M.A.: Optimal location and setting of SVC and TCSC devices using non-dominated sorting particle swarm optimization. Electric Power Systems Research 79(12), 1668–1677 (2009) 12. Hwang, C.L., Yoon, K.: Multiple attribute decision making: methods and applications. Springer, Berlin (1981) 13. Coello, C.A.C., Lechuga, M.S.: MOPSO: A proposal for multiple objective particle swarm optimization. In: IEEE World Congress on Computational Intelligence, pp. 1051–1056 (2003) 14. Huang, V.L., Suganthan, P.N., Liang, J.J.: Comprehensive learning particle swarm optimizer for solving multiobjective optimization problems. International Journal of Intelligent Systems 21, 209–226 (2006) 15. Liang, J.J., Qin, A.K., Suganthan, P.N., Baskar, S.: Evaluation of comprehensive learning particle swarm optimizer. In: Pal, N.R., Kasabov, N., Mudi, R.K., Pal, S., Parui, S.K. (eds.) ICONIP 2004. LNCS, vol. 3316, pp. 230–235. Springer, Heidelberg (2004) 16. Coello, C.A.C., Pulido, G.T., Lechuga, M.S.: Handling multiple objectives with particle swarm optimization. IEEE Trans. Evol. Comput. 8, 256–279 (2004) 17. The University of Washington: College of Engineering. Power systems test case archive, http://www.ee.washington.edu/research/pstca

Torsional Body Forces in a Kelvin-Voigt-Type Visco-Elastic Half Space P.C. Pal and Dinbandhu Mandal Department of Applied Mathematics Indian School of Mines, Dhanbad-826004 [email protected], [email protected]

Abstract. An analysis is presented to consider the problem of torsional body forces within a Kelvin-Voigt type visco- elastic material half space. Hankel & Fourier transform are used to calculate the disturbance for a body force located at a constant depth from the free surface. Numerical results are obtained and the natures are shown graphically. Keywords: Torsional body force, Visco elastic medium, Kelvin-Voigt material, Hankel Transform, Fourier cosine Transform.

1 Introduction The studies of propagation of torsional surface waves in anisotropic visco-elastic medium are of great importance, as they help in investigation to the structure of earth and in exploration of natural resources inside the earth as well. Such studies are also important in elastic medium due to their wide application in geophysics and to understand the cause of damage due to earthquake. In many industrial applications, the engineers are faced with stress analysis of three dimensional bodies in which geometry and loading involved are axisymmetric. For axisymmetric problems the point loads are generalized to ring loads. The determination of stresses and displacement in an isotropic, homogeneous elastic medium due to torsional oscillation has been a subject of considerable interest in solid mechanics and applied mathematics. In particular, torsional vibration of anisotropic visco-elastic material is largely used in measuring shear constants, shear velocities and shear elasticity of liquid. The problem of torsional body force in visco-elastic material is solved by many authors. Bhattacharya (1986) has studied the wave propagation in Random MagnetoThermo Visco-elastic Medium. Mukherjee and Sengupta (1991) have discussed the surface waves in thermo visco-elastic media of higher order. Manolis and Shaw (1996) have discussed the harmonic wave propagation through visco elastic heterogeneous media. Dey et (1996) have considered the propagation of torsional surface waves in visco-elastic medium. Drozdov (1997) has studied on a constitutive model for nonlinear visco elastic media. Lee and Wineman (1999) have discussed a model for non linear visco-elastic torsional response of an elastomeric bushing. Pal (2000, 2001) have discussed the effect of dynamic visco-elasticity on vertical and torsional vibrations of a half-space and the torsional body forces. Romeo (2001) has studied the P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 145–151, 2011. © Springer-Verlag Berlin Heidelberg 2011

146

P.C. Pal and D. Mandal

properties of wave propagator and reflectivity in linear visco-elastic media. Chen and Wang (2002) have discussed the torsional vibrations of elastic foundation of saturated media. Dey etl. (2003) have discussed the propagation of torsional surface waves in an elastic layer with void pores over an Elastic half space with void pores. Kristek and Moczo (2003) have discussed the Seismic wave propagation in visco elastic media with material discontinuities. Narain and Srivastava (2004) have discussed the magneto elastic torsional vibration of non-homogeneous Aeolotropic cylindrical shell of visco-elastic solid. Cerveny and Psencik (2006) have discussed the particle motion of plane waves in visco elastic anisotropic media. Borcherdt and Bielak (2010) have discussed the visco elastic waves in layered media. In this paper, we discuss the problem of torsional disturbance due to a decaying body forces within a visco-elastic medium of Kelvin-Voigt type material. Using the Hankel and Fourier cosine transforms, the solution is obtained for type of body forces located over a constant depth from the plane face. Numerical results are obtained for Kelvin-Voigt material and compared with elastic solid. Variations of disturbance are exhibited graphically for different value of α for elastic material and a constant value of depth from the surface.

2 Formulation of Problem Let (r,θ,z) be cylindrical polar coordinates. The origin is taken to any point of the boundary of the half space and z-axis is taken in the vertically downward directions. The material of the medium is taken to be visco elastic of Kelvin Voigt type material. Medium contains a body force which is torsional in nature and as a function r, z and t. For torsional disturbances the stress-strain relations as

τ rr =τ θθ =τ zz =τ rz =0

⎛ ⎞ ⎛ ⎞ ⎜ μ ⎟ ⎜ μ ⎟ ⎛ ∂v v ⎞ τ =2 ⎜ ⎟e = ⎜ ⎟⎜ - ⎟ 1 1 ⎝ ∂r r ⎠ ⎜ 1+ ⎟ ⎜ 1+ ⎟ ⎝ iωδ ⎠ ⎝ iωδ ⎠ rθ

rθ

⎛ ⎞ ⎛ ⎞ ⎜ μ ⎟ ⎜ μ ⎟ ∂v and ⎜ ⎟τ =⎜ ⎟ 1 1 ⎜ 1+ ⎟ ⎜ 1+ ⎟ ∂z ⎝ iωδ ⎠ ⎝ iωδ ⎠ θz

where μ = Lame constant,

ω =frequency parameter and δ =relaxation parameter.

τrθ and τθz are shear stresses in the solid. Here the motion is symmetrical about z-axis so v r = v z = 0 and v(r, z, t) (say), and is independent of θ and is circumferential component of displacement. The only non-zero equation of motion in terms of stresses is ∂τ rθ ∂r

where, Fθ is the body force.

+

2 r

τ rθ +

∂τ θz ∂z

∂ v 2

+ Fθ =ρ

∂t

2

.

(1)

Torsional Body Forces in a Kelvin-Voigt-Type Visco-Elastic Half Space

We take body force

147

Fθ in the form Fθ = F(r,z)e ,ω>0 -ωt

(2)

.

The equation of motion (1) in terms of displacement component v may be written as

⎡∂

k⎢

2

⎣ ∂r

v 2

+

1 ∂v

-

r ∂r

v r

2

∂ v⎤ 2

+

∂z

2

-ωt

⎦

1+

Let us take v (r, z, t ) = f (r, z )e ∂f 2

∂r

2

+

1 ∂f

2

, where,

1 iωδ

, ω > 0 and so final equation m ay be w ritten as +

r ∂r

∂t

μ

k=

− ωt

∂ v 2

⎥ + F(r,z)e =ρ

∂f ∂z

⎧1

−⎨

⎩r

+

2

ρw

2

k

⎫ F ⎬f + ⎭ k

=0

.

(3)

3 Method of Solution Eq. (3) will be solved using Hankel transform and Fourier cosine transform. The pair of transformation is defined as ∞

( f , F ) = ∫ ( f , F )rJ ( ξr ) dr 1

1

0

(f

2

2

, F2 ) =

π

(4)

∞

∫ ( f , F ) cos ( ηz ) dz . 1

1

0

where J1 ( ξr ) is Bessel unction of first kind and order one. Now applying Hankel transform to eq. (3), we have ∂ f1 2

∂z

2

⎡

− ⎢ξ + 2

ρω

⎣

⎤ ⎥f ⎦

2

1

k

+

F1 k

=0

.

(5)

The initial and boundary conditions of the problem require that: v(r, z, t) =

and the plane boundary is stress free so at

∂v ∂t

(r, z, t) at t = 0 .

∂F1

=0

∂z

on z=0. We may assume that

z → ∞.

Now applying Fourier cosine transform to (5), we have f 2 ( ξ, η ) =

F2

⎡ ⎣

k ⎢ξ + η + 2

2

ρω k

2

⎤ ⎥ ⎦

(6) ∂f1 ∂z

→0

148

P.C. Pal and D. Mandal

Hence, by inversion theorem on Hankel and Fourier transform, we have finally v(r, z, t) which is given by v(r, z, t) = f(r,z)e =

e

− ωt

k

− ωt

2

∞ ∞

∫∫ π

ξF2 ( ξ, η ) J 1 ( ξr ) cos ( ηz ) dξdη ρω ⎤ ⎡ ⎢ξ + η + k ⎥ ⎣ ⎦

.

(7)

2

2

0 0

2

4 Decaying Torsional Body Force To determine F2 in (7), we assume that the body force acts inside the half space at a depth z=h(>0). The body force is defined as Fθ = f ( r, z ) e

= p (r) δ (z − h ) e

− ωt

− ωt

.

(8)

i.e. body force is too large at z=h and occurs as p(r) behaves. Hence ∞

f1 ( ξ, z ) = p ( r ) δ ( z − h ) rJ1 ( ξr ) dr

∫ 0

and F2 ( ξ, η ) =

2

∞

∫ p ( ξ )δ ( z − h ) cos ( ηz ) dz π 1

0

2

=

where,

P1 ( ξ ) =

π

p1 ( ξ ) cos ( ηh )

∞

∫ rp ( r ) J ( ξr ) dr 1

1

0

Hence finally v ( r, z, t ) =

where, n = ξ + α 2

2

and α =

ρω

2

1 e

− ωt

π k

⎡π ∫ p ( ξ ) ⎢⎣ 2n e ∞

−n(z+h)

1

0

+

π

e

−n(z −h)

2n

⎤ ξJ ( ξr ) dξ . ⎥⎦ 1

2

2

.

k

5 Special Case Case 1. Let us assume that p(r) the radial dependence of applied force be defined as p(r) =

⎧Qr, r ≤ a ⎨ ⎩0, r > a

(9)

Torsional Body Forces in a Kelvin-Voigt-Type Visco-Elastic Half Space

149

Q and a are constants then a

p1 ( ξ ) = Qr J 1 ( ξr )dr

∫

2

(10)

0

=

Q ξ

⎡ ⎢⎣ −a

2

2

J 0 ( ξa ) +

aJ 1

ξ

( ξa ) ⎤⎥ ⎦

.

Hence from (9), the displacement v(r,z,t) is given as v ( r, z, t ) =

Qe

− ωt

∞

∫

2k

e

− n( z + h )

+e

−n(z −h)

⎡ ⎢⎣ −a

n

0

2

J 0 ( ξa ) +

2 ξ

⎤ ⎥⎦

aJ1 ( ξa ) J1 ( ξr ) dξ .

(11)

Case2. Let ⎧⎪Qr/ ( a -r 2

2

p(r)= ⎨

)

1/2

, ra

Q again a constant. This shows that the body force is in the form of angular displacement over a circular area of radius a and zero elsewhere. p1 ( ξ ) =Q

a

∫ 0

r J 1 ( ξr ) dr 2

(a

2

-r

2

)

1/2

If a=1, then according to Gradshteyn and Ryzhik [1996, pp. 977], we get p1 ( ξ ) =

Q ⎡ sinξ

⎤ ⎢⎣ ξ -cosξ ⎥⎦

ξ

Hence from (9), the displacement v(r,z,t) is given as v ( r, z, t ) =

Qe

-ωt

2k

∞

∫

e

-n ( z+h )

+e

-n ( z-h )

(12)

1

n

0

⎡ sinξ ⎤ ⎢⎣ ξ -cosξ ⎥⎦J ( ξr ) dξ .

6 Numerical Results and Discussions For Kelvin-Voigt material, we assume the visco elastic parameter α 2 = hence the torsional disturbances is given by

ρω2 and a=1 k

vke ωt = I (say) (on thesurface z = 0) Q

where, ∞

I=

∫

(ξ

cosh

0

(ξ

2

2

)

+α h

+α

2

2

)

2 ⎡ ⎤ ⎢⎣ − J ( ξ ) + ξ J ( ξ ) ⎥⎦ J ( ξr ) dξ . 0

1

1

(13)

for special case (1) and ∞

I=

∫ 0

for special case (2).

cosh

(ξ

2

2

( ξ +α ) 2

)

+α h ⎡ sinξ 2

⎤ ⎢⎣ ξ -cosξ ⎥⎦ J ( ξr ) dξ . 1

(14)

150

P.C. Pal and D. Mandal -3

2

x 10

r=0.9 r=0.7 r=0.5

1.8 1.6 1.4 1.2 [I]

1 0.8 0.6 0.4 0.2 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

[α ]

Fig. 1. Variation of torsional disturbance with visco elastic parameter (case1)

-3

3

x 10

r=0.9 r=0.7 r=0.5

2.5

[I]

2

1.5

1

0.5

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

[α ]

Fig. 2. Variation of torsional disturbance with visco elastic parameter (case2)

7 Conclusion The value of I for different values of α (visco elastic parameter) are evaluated numerically from the Eqs. (13) and (14) for a particular value of h=3.75(depth) and radial distance taken as r0

(6)

which did not take into account proper boundary conditions for t → 0 lim 4t dt a2t → 0,

t→0

−at

d

lim e a (1−e

)

t→0

→1

(7)

Hence the derived formula dt (t) = 4t dt a2t ,

lim dt (t) → 0,

lim dt (t) → 0

t→∞

t→0

(8)

To find the explicit form of dt (t) and at (t) the relation d

−at

4t dt a2t = e a (1−e

)

−1

(9)

and its first derivative d

−at

4 dt a2t = d e−at e a (1−e

)

(10)

should be employed. The first of them satisfies the boundary conditions  d  −at lim 4t dt a2t = lim e a (1−e ) − 1 = 0 t→0

t→0

  d −at lim 4t dt a2t = lim e a (1−e ) − 1 = at ⇔ lim dt = 0

t→∞

t→∞

t→∞

Combining (9) and (10), we arrive at the analytical expressions   d −at  d  2t e a (1−e ) − 1 −at d dt (t) = te−at e a (1−e ) , at (t) = d −at 8 at d e−at e a (1−e )

(11)

(12)

Tumor Growth in the Fractal Space-Time with Temporal Density

169

which satisfies the boundary conditions lim dt (t) = 1,

t→0

lim dt (t) = 0

t→∞

(13)

hence, the model proposed predicts in an early stage of the Gompertzian growth is a linear growth according to the equation y(t → 0) = 4t a2t ,

dt = 1

(14)

Similar boundary relations can be specified for the scaling factor at (t). They take the forms   1 dG(t) G(t) − G0 d lim = e a −1 = lim at (t) (15) = d = lim at (t), lim t→∞ t→∞ t→0 G0 t→0 dt G0 The limit at (t → 0) is undefined due to the fundamental requirement t > 0 in (4). From (12) we can calculate the mean temporal fractal density and the scaling factor for evolving Gompertzian systems at an arbitrary period of time by making use of the equations ft (t) =

te 1  ft (t), N t=t d

N=

te − td p

(16)

Here td and te denote the beginning and the end of period under consideration; N is the number of the time-points in the period (td , te ), whereas p is an interval (step) between two neighbouring time-points used in calculations. The main objective of the present study is experimental verification of the model proposed. To this aim, we determine the time-dependent temporal fractal density dt (t) and the scaling factor at (t) for the Flexner-Jobling rat’s tumor. We shall also be concerned with calculation of the mean values of dt (t) and at (t) for other kinds of rat’s tumors. 1.1

Methods

The proposed model is performed by making use of the formulae (12), the boundary conditions (15) and the experimental Gompertz curve. First, we employ the curve evaluated by Laird for Flexner-Jobling rat’s tumor whose growth is characterized by the parameters: a = 0.0490 ± 0.0063 day, b = 0.394 ± 0.066 day and G0 = 0.015g [10]. The generated plots of dt (t) and at (t) presented in Fig.1, and calculated the values of Gompertz function (1) and its derivative to determine the limits according to Eq.(15). All calculations were performed using MATLAB 7.0 software for symbolic calculations. The results are reported in Table 1. From the determined analytical expressions (12), we can calculate their mean values by taking advantage of the formula (15) and Gompertzian parameters evaluated by Laird [9] for different kinds of rat’s tumors: Flexner-Jobling, R3a7, R4c4 and a7R3. The values of those parameters, the calculated mean values of the tempoW −K ral fractal density dt (t) and scaling factor at (t) and the values of dt (t) W −K and at (t) obtained are presented in Table 2.

170

P. Paramanathan and R. Uthayakumar

Fig. 1. Plots of the temporal Fractal desnity dt (t) and the scaling factor at (t) for Flexner-Jobling rat’s tumor whose growth is characterized by the parameters: a=0.0490 day and d=0.394 day and G0 =0.015g. Plots of the scaling factor at (t) are presented for short and long periods.

Tumor Growth in the Fractal Space-Time with Temporal Density

171

Table 1. The limiting values of the quantities used in the experimental test of the model proposed Time [day]

0

0.0001

∞

20

[G(t) − G0 ]/G0 0 0 - 2886.14 dG(t)/dt 0.284 0.38403 0 G−1 0 undefined 0.43201 0.08 2886.14 at (t) 1 1.000001 1.96 0 dt (t)

Table 2. The limiting values of the quantities used in the experimental test of the model proposed Tumor

2

dt (t) dt (t) W −K

at (t) at (t) W −K

td

te

p

N

F −J 3.6732 D=0.0490(63) 0.653 r=0.394(66) 0.088

2.381 0.756 0.100

0.0357 200.24 2000.96

0.154 99.87 2035.18

0 50.5 107

50.5 107 157.5

0.1 0.1 0.1

505 565 505

a7R3 D=0.063(22) r=737(162)

3.492 0.755 0.535

2.381 1.283 0.756 187.52 0.100 20,435.06

0.154 99.87 2035.18

0 43.5 58.7

43.5 58.7 102.2

0.1 0.1 0.1

435 132 435

R4C4 D=0.078(12) r=540(20)

3.768 0.525 0.039

2.381 0.756 0.100

1.339 225.32 560.18

0.154 99.87 2035.18

0 35.2 76.0

35.2 76.0 111.2

0.1 0.1 0.1

352 408 352

R3a7 D=0.124(11) r=1.28(25)

3.616 0.622 0.019

2.381 0.756 0.100

1.392 253.2 642.35

0.154 99.87 2035.18

0 21.8 28.4

21.8 28.4 50.2

0.1 0.1 0.1

218 66 218

Results and Discussions

The performance of our calculations proved that formulae (12) applied to the Flexner-Jobling rat’s tumor growth satisfactorily describe their time-dependence in the period (0, ∞). The temporal fractal density of the Flexner-Jobling’s tumor increases from 1 for t = 0 to a maximal value 2.32 for t=20 and then decreases to zero (see Fig.1). As to the scaling factor at (t), its value for t = 0 is equal to Gompertzian parameter d = 0.758 and increases to value of 0.95 for t = 1, then it decreases to a minimal value of 0.5 for t = 20 and increases again to a maximal constant value of 0.9903. We conclude that the analytical formulae (12) describing the time-dependence of the temporal fractal density and scaling factor very well reproduce the growth of the Flexner-Jobling rat’s tumor in the spacetime with temporal fractal density. The possibility of mapping the Gompertz function, describing the tumor growth, onto the fractal density function confirms that Gompertzian growth is self-similar. These results indicate that at the first stage of the Flexner-Jobling rat’s tumor growth, which includes the period from zero to the inflection time

172

P. Paramanathan and R. Uthayakumar

1 ti = ln a

  b = 42.54 day a

(17)

different complex processes at microscopic level take place, which result in the specific time-change of the temporal fractal density and the scaling factor. If we assume that time is a continuous variable, it is clear that initiation of tumorigenesis cannot take place in the space-time with temporal fractal density dt = 0 as at this time-point an extrasystemic time characterizes dt = 1. An analysis of the mean fractal density presented in Table 2 reveals that they are consistent with those obtained by fitting procedure whereas the mean values of the scaling factor differ from those derived previously. The reasons for this discrepancy are unknown at present. The proposed method permits calculation of dt (t) and at (t) for an arbitrary period of time. It should be pointed out here that the results obtained for the latter seem to be unreliable - probably due to a low quality data used in Laird’s analysis [9].

3

Conclusion

The discussed results shows that the model proposed here is the most consistent and easy method for computing Fractal Density of tumor growth in space-time. We have proved that the analytical formulae describing the time-dependence of the temporal fractal density and scaling factor very well reproduce the growth of the Flexner-Jobling rat’s tumor in particular and growth of other rat’s tumors in general and the calculated mean values of temporal fractal density coincide very well with those obtained by fitting procedure.

References 1. Falconer, J.: Fractal Geometry-Mathematical Foundations and Applications. John Wiley and Sons, Chichester (2003) 2. Losa, G.A., Nonnenmacher, T.F.: Self-similarity and fractal irregularity in pathologic tissues. Modelling Pathology 9, 174–182 (1996) 3. Sedive, R.: Fractal tumors: their real and virtual images. Wien Klin Wochenschr 108, 547–551 (1996) 4. Peitgen, H.O., Jurgens, H.H., Saupe, D.: Chaos and Fractals. Springer, Heidelberg (1992) 5. Waliszewski, P., Molski, M., Konarski, J.: Collectivity and evolution of fractal dynamics during retinoid-induced differentiation of cancer cell population. Fractals 7, 139–149 (1999) 6. Waliszewski, P., Molski, M., Konarski, J.: On the modification of fractal self-space during cell differentiation or tumor progression. Fractals 8, 195–203 (2000) 7. Waliszewski, P., Konarski, J.: Neuronal differentiation and synapse formation in space-time with fractal dimension. Synapse 43, 252–258 (2002) 8. Waliszewski, P., Konarski, J.: The Gompertzian curve reveals fractal properties of tumor growth. Chaos Solitons & Fractals 16, 665–674 (2003) 9. Laird, A.K.: Dynamics of tumor growth. British Journal of Cancer 18, 490–502 (1964)

Tumor Growth in the Fractal Space-Time with Temporal Density

173

10. Molski, M., Konarski, J.: Tumor growth in the space-time with temporal fractal dimension. Chaos Solitons & Fractals 36(4), 811–818 (2008) 11. Bru, A., Casero, D., de Franciscis, S., Herero, M.A.: Fractal analysis of tumor growth. Math. and Comp. Model. 47, 546–559 (2008) 12. Xu, S., Feng, Z.: Analysis of a mathematical model for tumor growth under indirect effect of inhibitors with time delay in proliferation. Jour. of Math. Anal. and Appl. 374, 178–186 (2011)

The Use of Chance Constrained Fuzzy Goal Programming for Long-Range Land Allocation Planning in Agricultural System Bijay Baran Pal1,*, Durga Banerjee2, and Shyamal Sen3 1

Department of Mathematics, University of Kalyani Kalyani-741235, West Bengal, India Tel.: +91-33-25825439 [email protected] 2 Department of Mathematics, University of Kalyani Kalyani-741235, West Bengal, India [email protected] 3 Department of Mathematics, B.K.C. College Kolkata-700108, West Bengal, India [email protected] Abstract. This paper describes how the fuzzy goal programming (FGP) can be efficiently used for modelling and solving land allocation problems having chance constraints for optimal production of seasonal crops in agricultural system. In the proposed model, utilization of total cultivable land, different farming resources, achievement of the aspiration levels of production of seasonal crops are fuzzily described. The water supply as a productive resource and certain socio-economic constraints are described probabilistically in the decision making environment. In the solution process, achievement of the highest membership value (unity) of the membership goals defined for the fuzzy goals of the problem to the extent possible on the basis of the needs and desires of the decision maker (DM) is taken into account in the decision making horizon. The potential use of the approach is demonstrated by a case example of the Nadia District, West Bengal (W. B.), INDIA. Keywords: Agricultural Planning, Chance Constrained Programming, Fuzzy Programming, Fuzzy Goal Programming, Goal Programming.

1 Introduction The mathematical programming models to agricultural production planning have been widely used since Heady [1] in 1954 demonstrated the use of linear programming (LP) for land allocation to cropping plan in agricultural system. From the mid-’60s to ’80s of the last century, different LP models studied [2, 3] for farm planning has been surveyed by Glen in [4] in 1987. Since most of the farm planning problems are multiobjective in nature, the goal programming (GP) methodology in [5] as a prominent *

Corresponding author.

P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 174–186, 2011. © Springer-Verlag Berlin Heidelberg 2011

The Use of Chance Constrained FGP for Long-Range Land Allocation Planning

175

tool for multiobjective decision analysis has been efficiently used to land-use planning problems [6] in the past. Although, GP has been widely accepted as a promising tool for multiobjective decision making (MODM), the main weakness of the conventional GP methodology is that the aspiration levels of the goals need be stated precisely. To overcome the above difficulty of imprecise in nature of them, fuzzy programming (FP) approach in [7] to farm planning problems has been deeply studied [8] in the past. The FGP approach [9] as an extension of conventional GP to agricultural production planning problems has also been studied by Pal et al. [6,10] in the past. Now, in most of the real-world decision situations, the DMs are often faced with the problem of inexact data due to inherent uncertain in nature of the resource parameters involved with the problems. To deal with the probabilistically uncertain data, the field of stochastic programming (SP) has been studied [11] extensively and applied to various real-life problems [12, 13] in the past. The use of chance constrained programming (CCP) to fuzzy MODM problems has also been studied by Pal et al. [14] in the recent past. However, consideration of both the aspects of FP and SP for modeling and solving real-life decision problems has been realized in the recent years from the view point of occurrence of both the fuzzy and probabilistic data in the decision making environment. Although, fuzzy stochastic programming (FSP) approaches to chance constrained MODM problems have been investigated [15] by active researchers in the field, the extensive study in this area is at an early stage. Now, in the agricultural production planning context, it is worthy to mention that the sustainable supply of water depends solely on the amount of rainfall in all the seasons throughout a year. As such, water supply to meet various needs is very much stochastic in nature. Although, several modeling aspects of water supply system have been investigated [12] in the past, consideration of probabilistic parameters to the agricultural systems in fuzzy decision environment is yet to be circulated in the literature. In this article, utilization of total cultivable land, different farming resources, achievement of the aspiration levels of production of seasonal crops are fuzzily described. The water supply as a productive resource and certain socio-economic constraints are described probabilistically in the decision making environment. In the solution process, achievement of the highest membership value (unity) of the membership goals defined for the fuzzy goals of the problem to the extent possible on the basis of the needs and desires of the decision maker (DM) is taken into account in the decision making horizon. The potential use of the approach is demonstrated by a case example of the Nadia District, West Bengal (W. B.), INDIA. Now, the general chance constrained FGP formulation is presented in the Section 2.

2 Problem Formulation The general form of a fuzzy MODM problem with chance constrained can be stated as:

176

B.B. Pal, D. Banerjee, and S. Sen

Find X (x1,x2,…,xn) so as to satisfy

⎛> ⎞ ⎜~⎟ Z k (X) ⎜ ⎟ ⎜< ⎟ ⎝~⎠

gk ,

k = 1, 2,….,K.

(1)

subject to X ∈ S{X ∈ R n | Pr[H( X) ≥ b] ≥ p, X ≥ 0 , H, b ∈ R m },

(2)

where X is the vector of decision variables in the bounded feasible region S (≠Φ), and where ) and 1indicate the fuzziness of ≥ and ≤ restrictions, respectively, in the sense of Zimmermann [7], and where gk be the imprecise aspiration level of the k-th objective. Pr stands for probabilistically defined (linear / nonlinear ) constraints set H(X), b is a resource vector, and p (0< p 0) is the numerical weight associated with d −rk and it designates the weight of importance of achieving the aspired level of the k-th goal relative to other which are grouped at the r-th priority level and where w rk values are determined as [16]: − w rk

⎧ ( g −1g ) , for the defined μ k ( X) in (3) ⎪ k kl r =⎨ 1 ⎪⎩ ( g ku −g k ) r , for the defined μ k ( X) in (4)

where (g k − g kl ) r and (g ku − g k ) r are used to present g k − g kl and g ku − g k respectively, at the r-th priority level. Now, the FGP model formulation of the proposed problem is presented in the Section 3.

178

B.B. Pal, D. Banerjee, and S. Sen

3 FGP Model Formulation of the Problem The decision variables and different types of parameters involved with the problem are defined first in the following Section 3.1. 3.1 Definition of Decision Variables and Parameters (a) Decision Variable: lcs = Allocation of land for cultivating the crop c during the season s, c = 1,2, ..., C; s = 1, 2, ..., S. (b) Productive resource parameters: • Fuzzy resources: LAs = Total farming land (hectares (ha)) currently in use for cultivating the crops in the season s. MHs = Estimated total machine hours (in hrs.) required during the season s. MDs = Estimated total man-days (in days) required during the season s. = Estimated total amount of the fertilizer f (f = 1,2,…,F) (in quintals (qtls.)) Ff required during the planning year. RS = Estimated total amount of cash (in Rupees) required per annum for supply of the productive resources. • Probabilistic resource:

WSs = Total supply of water (in inch / ha) required during the season s. (c) Fuzzy aspiration levels: = Annual production level (in qtls.) of the crop c. Pc MP = Estimated total market value (in Rupees) of all the crops yield during the planning year. (d) Probabilistic aspiration levels: = Ratio of annual production of the i-th and j-th crop (i, j = 1, 2, ...,C; i ≠ j). Rij = Ratio of annual profits obtained from the i-th and the j-th crops (i, j=1,2,...,C; rij i ≠ j). (e) Crisp coefficients: MHcs = Average machine hours (in hrs.) required for tillage per ha of land for cultivating the crop c during the season s. MDcs = Man days (in days) required per ha of land for cultivating the crop c during the season s. Ffcs = Amount of the fertilizer f required per ha of land for cultivating the crop c during the season s. Pcs Acs MPcs

= Estimated production of the crop c per ha of land cultivated during the season s. = Average cost for purchasing seeds and different farm assisting materials per ha of land cultivated for the crop c during the season s. = Market price (Rupees / qtl.) at the time of harvest of the crop c cultivated during the season s.

The Use of Chance Constrained FGP for Long-Range Land Allocation Planning

179

(f) Random coefficients: Wcs = Estimated amount of water consumption (in inch) per ha of land for cultivating the crop c during the season s. 3.2 Description of Fuzzy Goals and Chance Constraints (a) Land utilization goal: The land utilization goal for cultivating the seasonal crops appears as: C

∑l

cs

c =1

< LA s , ~

s = 1, 2, . . . , S.

(b) Productive resource goals: • Machine-hour goals: An estimated number of machine hours is to be provided for cultivating the land in different seasons of the plan period. The fuzzy goals take the form: C

∑ MH

, ~> MH S

cs .l cs

c =1

s = 1, 2,..., S.

• Man-power requirement goals: A number of labors are to be employed through out the planning period to avoid the trouble with hiring of extra labors at the peak times. The fuzzy goals take the form: C

∑ MD

~> MD s . s = 1, 2,..., S.

cs .l cs

c =1

• Fertilizer requirement goals: To maintain the fertility of the soil, different types of fertilizer are to be used in different seasons in the plan period. The fuzzy goals take the form: C

∑F

fcs

c =1

.l cs ~ > Ff ,

f = 1, 2, ..., F; s = 1, 2,..., S.

(c) Cash expenditure goal: An estimated amount of money (in Rs.) is involved for the purpose of purchasing the seeds, fertilizers and other productive resources. The fuzzy goals take the form: S

C

s =1

c =1

∑ ∑A

cs

.l cs ~ > RS

(d) Production achievement goals: To meet the demand of agricultural products in society, a minimum achievement level of production of each type of the crops is needed. The fuzzy goals appear as: S

∑P

cs .l cs

s =1

~> Pc

,

c = 1, 2,..., C.

180

B.B. Pal, D. Banerjee, and S. Sen

(e) Profit goal: A certain level of profit from the farm is highly expected by the farm decision maker.

The fuzzy profit goal appears as:

S

C

s =1

c =1

∑ ∑ ( MP

cs

. P cs − A cs ) . l cs ~ > MP .

3.3 Description of Chance Constraints

The different chance constraints of the problem are presented in the following Sections. (a) Water-supply constraints: An estimated amount of water need be supplied to the soil for sustainable growth of the crop c cultivated during the season s. But, water-supply resources solely depends on rainfall and so probabilistic in nature. The water-supply constraints appear as: C

∑W l

Pr[

cs cs

≥ WSs ] ≥ ps , s =1,2,…,S.

c=1

where ps (0< ps 0.5.

236

P. Rajasekaran, R. Prabakaran, and R. Thanigaiselvan

4 Elman Backprop Neural Networks A Neural Network is an interconnection of processing nodes. In Neural Network method, the first step is data preparation and pre-processing [9]. Neural Networks are also named Universal Function Approximators. Most of the networks use single hidden layer. Elman Network is one of the recurrent networks. These networks are two layer back propagation networks, a feedback connection is taken from hidden layer output to input. This feedback connection enables them to learn, to recognize, to generate temporal and spatial patterns. Elman Network is an extension of two layer sigmoid architecture. Elman Network has tansig neurons in hidden layer and purelin neurons in output layer. This network differs from other conventional two layer networks in that the first layer has the recurrent connection. If two Elman Networks are given same inputs, weights and biases, their outputs differ because of the feedback states. The back propagation training function used here is trainbfg. Compared to other methods Elman Network needs more hidden neurons to learn a problem. This network is useful in signal processing and prediction. Output Y (t)

Hidden Z (t) Copy Z

Input X (t)

Hidden Z (t-1)

Fig. 2. Basic Structure of Elman Network

Least mean Square (LMS) algorithm is an example of supervised training. To reduce the Mean Square Error (MSE), LMS algorithm adjusts the weights and biases of the linear network. For every input, there exists an output which is compared to the target output. Error is the difference between the target output and the network output. Adaptive networks use LMS or Widrow Hoff learning algorithm. These equations form the basis for LMS algorithm. W (k+1) = W (k) + 2αe (k) pT (k) b (k+1) = b (k) + 2αe (k)

(10) (11)

where e is the error, bias b are vectors and α is the learning rate. If α is high, learning occurs quickly.

Implementation of Elman Backprop for Dynamic Power Management

237

5 Experimental Setup Simulation is performed for the proposed Elman Method based on the usage of PC/laptop by an under graduate student. Breakeven Time (Tb) Calculation: Breakeven time is defined as the transition time plus the ratio between energy latency and power saved. Steps to calculate Tb is as follows: 1.

Calculate Energy latency, which is the sum of wake up energy and shut down energy: El = Ew + Es

2.

(12)

Calculate the power saved, which is the difference when the system is in low power state and the active state: Ps = Pa - Plp

3.

(13)

where Ps is the power saved, Pa is active power and Plp is power in low power state Find transition time Tt by using (1).

5.1 Experimentation Laptop LCD is used as power manageable component. Tb is found as shown below. Energy consumed during turn on and turn off is 15.2J. Therefore, El = 15.2 + 15.2 = 30.4J When LCD is off, Plp = 17.1W, when it is on Pa = 21.1W. Therefore, Ps = 21.1 – 17.1 =4W Then, t = El

= 30.4

= 7.6s

Ps 4 Tb= t + Tt = 7.6 + 8 = 15.6s

≈ 16s

If the idle period is greater than 16s, it is better to turn off. Table 2. Power Consumption for LAPTOP

STATE

POWER CONSUMED (%)

SLEEP STAND - BY

0 0

IDLE ACTIVE

4 32

In the above table, power consumption is checked for every one hour. When a request arrives, when the system is in sleep state, the disk will go to active state directly.

238

P. Rajasekaran, R. Prabakaran, and R. Thanigaiselvan

Fig. 3. View of the Elman Backprop network

Fig. 4. Training parameters

Transition from higher power state to lower power state consumes negligible power. Thus the proposed method is able to predict idle period from past idle period history. In Matlab simulation we have to give the input and target data to the network. Select the neural network through which we have to train the neurons. Training function selected is traingdm and learning function is said to be learngdm. The performance function used is Mean Square Error (MSE) function. Number of layers used here is 2. Layer 1 has 10 neurons and transfer function is tansig. In layer 2, user can specify number of neurons and transfer function is same. This is how the proposed network looks like. To train the network, the maximum epoch 1000 is specified. Training is done with Gradient Descent back propagation with adaptive learning rate. The delay states are taken from the same network itself. By simulation we obtain the required output, error and delay states.

Implementation of Elman Backprop for Dynamic Power Management

239

Fig. 5. Performance plot for the network

Fig. 6. Training State

6 Results and Discussions The experiment is done with the proposed method and comparison is done with other DPM techniques. Performances are compared on the basis of Competitive Ratio (CR). It is a method to analyse online algorithms, where the performance of an online algorithm is compared to the performance of an optimal algorithm. Time out: It is a static technique. Hybrid Method: This method uses Moving Average, Time Delay Neural Network and Last Value as predictor. We assume maximum training epoch as 1000 for every training round during which laptop is running in full speed. Comparison chart for three DPM techniques are given below based on Competitive Ratio. Table 3. Comparison Chart Techniques Time out Hybrid Method Elman Model

CR 1.51 1.18 1.04

Time out produces average result in most of the cases and performs worst when the series have small idle periods by which it fails to save power. Hybrid method

240

P. Rajasekaran, R. Prabakaran, and R. Thanigaiselvan

performs well all cases but yields high Competitive Ratio. Elman Model is best among all DPM techniques, since it achieves most power saving.

7 Conclusion In portable devices workload characteristics vary according to usage pattern which causes the memory component to change. Therefore, it is not wise to predict the future idle period based on single technique. By estimating Central Tendency and Long Range Dependency we have proposed a more effective Elman Model to predict the future idle period. Simulation results show that our method can perform better than many other available DPM techniques.

References 1. Lee, W.-K., Lee, S.-W., Siew, W.-O.: Hybrid model for Dynamic Power Management. IEEE Transaction on Consumer Electronics 55(2), 650–655 (2009) 2. Greenwalt, P.: Modelling Power Management for Hard Disks. In: Proceedings of International Workshop Modelling, Analysis and Simulation for Computer and Telecommunication Systems, pp. 62–65 (1994) 3. Golding, R., Bosh, P., Wilkes, J.: Idleness Is Not Sloth. In: Proceeding of Winter USENIX Technical Conference, pp. 201–212 (1995) 4. Benini, L., Bogliolo, A., Paleologo, G.A., Micheli, G.D.: Policy Optimization For Dynamic Power Management. IEEE Transaction on Computer-aided Design of Integrated Circuits and System 18(6), 813–833 (1999) 5. Chung, E.Y., Benini, L., Bogliolo, A., Lu, Y.H., de Michelli, G.: Dynamic Power Management for Non-stationary Service Requests. IEEE Transaction on Computers 51(11), 1345–1361 (2002) 6. Qiu, Q., Pedram, M.: Dynamic Power management Based on Continuous Time Markov Decisoin Processes. In: Design Automation Conference, pp. 555–561 (1999) 7. Simunic, T., Benini, L., Glynn, P., de Micheli, G.: Dynamic Power Management of Laptop Hard Disk. In: IEEE Proceedings of Design Automation and Test Conference and Exhibition in Europe, p. 736 (2001) 8. Lu, Y.H., De Micheli, G.: Adaptive Hard Disk Power Management on Personal Computers. In: Proceeding of Great Lakes Symposium VLSI, pp. 50–53 (1999) 9. Qian, B., Rasheed, K.: Hurst Exponent and Financial Market Predictability. In: Proceedings of FEA – Financial Engineering and Applications, pp. 437–443 (2004) 10. Lee Giles, C., Lawrence, S., Tsoi, A.C.: Noisy Time Series Prediction using a Recurrent Neural Network and Grammatical Inference. Machine Learning 44, 161–183 (2001) 11. Karagiannis, T., Faloutsos, M., Riedi, R.H.: Long range dependence: Now you see it, now you don’t. In: IEEE Global Telecommunications Conference, vol. 3, pp. 2165–2169 (2002)

Comparison of Fuzzy and Neural Network Models to Diagnose Breast Cancer W. Abdul Hameed1 and M. Bagavandas2 1

Department of Mathematics, C. Abdul Hakeem College, Melvisharam, India [email protected] 2 School of Public Health, SRM University, Chennai-603203, India [email protected]

Abstract. The automatic diagnosis of breast cancer is an important, real-world medical problem. A major class of problems in Medical Science involves the diagnosis of disease, based upon various tests performed upon the patient. When several tests are involved, the ultimate diagnosis may be difficult to obtain, even for a medical expert. This has given rise, over the past few decades, to computerized diagnostic tools, intended to aid the Physician in making sense out of the confusion of data. This Paper carried out to generate and evaluate both fuzzy and neural network models to predict malignancy of breast tumor, using Wisconsin Diagnosis Breast Cancer Database (WDBC). Our objectives in this Paper are: (i) to compare the diagnostic performance of fuzzy and neural network models in distinction between malignance and benign patterns, (ii) to reduce the number of benign cases sent for biopsy using the best model as a supportive tool, and (iii) to validate the capability of each model to recognize new cases. Keywords: Breast cancer, neural network, fuzzy logic, learning vector quantization, fuzzy c-means algorithm.

1 Introduction Cancer, in all its dreaded forms, causes about 12 per cent of deaths throughout the world. In the developed countries, cancer is the second major cause of death, accounting for 21 per cent of all mortality. In the developing countries, cancer ranks third major cause of death accounting for 9.5 per cent of all deaths (ICMR, 2002). Cancer has become one of the ten main causes of death in India. As per the statistics, there are nearly 1.5 to 2 million cancer cases in the country at any given point of time. Over 7 lakh new cases of cancer and 3 lakh deaths occur annually due to cancer. Nearly 15 lakh patients require facilities for diagnosis, treatment and follow-up procedure (ICMR, 2001). Despite a great deal of public awareness and scientific research, breast cancer continues to be the most common cancer and the second largest cause of cancer deaths among women (Marshall E, 1993). In the last decade, several approaches to classification had been utilized in health care applications. A woman’s chances for long term survival are improved by early detection of the cancer, and early detection P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 241–248, 2011. © Springer-Verlag Berlin Heidelberg 2011

242

W. Abdul Hameed and M. Bagavandas

is enhanced by accurate diagnosis techniques. Most breast cancers are detected by the patient or by screening as a lump in the breast. The majority of breast lumps are benign. And therefore, it is binding to diagnose breast cancer, that is, to distinguish benign lumps from malignant ones. In order to diagnose whether the lump is benign or malignant, the Physician may use mammography, fine needle aspirate (FNA) with visual interpretation or surgical biopsy. The reported ability for accurate diagnosis of cancer when the disease is prevalent is between 68% - 79%, in case of mammography (Fletcher SW et al., 1993); 65% - 98% in case FNA technique is adopted (Giard RWM et al., 1992), and close to 100% if a surgical biopsy is undertaken. And from this it is clear that mammography lacks sensitivity; FNA sensitivity varies widely and surgical biopsy although accurate is invasive, time consuming, and expensive. The goal of the diagnostic aspect of this Paper is to develop a relatively objective system to diagnose FNA with an accuracy that is best achieved visually. In this Paper, it is compared how fuzzy and neural network models can diagnose whether the lump in the breast is cancerous or not, in the light of the patient’s medical data.

2 Material This Paper makes use of the Wisconsin Diagnosis Breast Cancer Database (WDBC) made publicly available at http://ftp.ics.uci.edu/pub/machine-learning-database/ breastcancer-wisconsin/. This data set is the result of efforts made at the University of Wisconsin Hospital for the diagnosis of breast tumor, solely based on the Fine Needle Aspirate (FNA) test. This test involves fluid extraction from a breast mass using a small gauge needle and then a visual inspection of the fluid under a microscope. The WDBC dataset consists of 699 samples. Each sample consists of visually assessed nuclear features of FNA taken from patient’s breast. Each sample has eleven attributes and each attribute has been assigned a 9-dimensional vector and is in the interval, 1 to 10 with value ‘1’ corresponding to a normal state and ‘10’ to the most abnormal state. Attribute ‘1’ is sample number and attribute ‘11’ designates whether the sample is benign or malignant. The attributes 2 to 10 are: clump thickness, uniformity of cell size, uniformity of cell shape, marginal adhesion, single epithelial cell size, bare nuclei, blend chromatin, normal nucleoli and mitosis. There had 16 samples that contained a single missing (i.e., unavailable) attribute value and had been removed from the database, setting apart the remaining 683 samples. Each of these 683 samples has one of two possible classes, namely benign or malignant. Out of 683 samples, 444 have been benign and the remaining 239 have been malignant, as given by WDBC dataset.

3 Neural Network Model Research on neural network dates back to the 1940s when McCulloch and Pitts found that the neuron can be modelled as a simple threshold device to perform logic function (McCulloch WS and Pitts W, 1943). The modern era of neural network research

Comparison of Fuzzy and Neural Network Models to Diagnose Breast Cancer

243

is commonly deemed to have commended with the publication of the Hopfield network in 1982. This Paper has been worked adopting vector quantization as a technique for breast cancer diagnosis. Learning vector quantization (LVQ) was developed by Kohonen (1986) and in the year 1990 (Kohonen T), who summarizes three versions of the algorithm. This is a supervised learning technique that can classify the input vectors based on vector quantization. The version of LVQ incorporated in this Paper is LVQ1 - Kohonen’s first version of Learning Vector Quantization (Kohonen T, 1986). Given an input vector xi to the network, the “input neuron” (i.e., the class or category) in LVQ1 is deemed to be a “winner” according to min d(xi, wj)=min || xi-wj||2

(1)

The set of input vectors have been denoted as {xi}, for i =1, 2,…,N, and the network synaptic input vectors has been denoted as {wj}, for j = 1, 2, …,m. Cwj has been the class (or category) that has been associated with the (weight) Voronoi vector wj, and Cxi as the class label of the input vector xi to the network. The weight vector wj has been adjusted in the following manner: If the class associated with the weight vector and the class label of the input are the same, that is, Cwj = Cxi, then wj(k + 1) = wj( k ) + μ(k) [xi - wj( k )]

(2)

where 0< μ(k)1, equation (9) could be minimized by Picard iteration (Bezdek, 1984): c

µ ij = 1 / ∑ { dij / dik ) 2/(m-1) } i=1,2, …, n

j=1,2, ….., c

k=1

n

n

vj = ∑ (µ ij )m xi / ∑ (µ ij )m i=1

j=1, 2, ….., c

i=1

5 Methods This Paper has compared two models namely, Neural Network model and Fuzzy model for breast cancer diagnosis. To determine the performance of these models in

246

W. Abdul Hameed and M. Bagavandas

practical usage, the database has been divided randomly into two separate sets, one for training and another for validation: (a) the training samples comprising 500 patients records (303 benign, 197 malignant) and (b) the validation samples comprising 183 patients records (141 benign, 42 malignant). Using the patients’ records in training sample the models have been trained by adjusting the weight values for interconnection limits for the neural network model and for estimating the membership function needed to establish the classification rules for fuzzy model. Then, the patients’ record in validation samples (n = 183) have been utilized to evaluate the generalizing ability of the above said models, separately. The best of these models has been compared in terms of accuracy, sensitivity, specificity, false positive and false negative.

6 Experiment and Results Out of 683 samples of Wisconsin Diagnosis Breast Cancer data set, 500 samples have been used to train the neural network using the Kohonen’s first version of Learning Vector Quantization (LVQ1) technique and the remaining 183 samples have been used to test the sample data. The learning rate parameter has been initialized to μ(k=1) = μ(1) = 0.1 and decreasing it by k at every training epoch. For example μ(2) = μ(1)/2, μ(3) = μ(2)/ 3, etc. To initialize the codebook vectors, this study uses one benign and one malignant as the first two samples. The associated classes for them are ‘1’ and ‘2’ respectively. The Matlab software has been used with the number of training epochs set to 35,000 to find the codebook vectors. If we calculate the minimum distance between each of the remaining 183 samples and the computed weight vectors of the codebook, we will know the class to which each of the remaining samples belongs. Table-1 presents the true positive (TP), false positive (FP), true negative (TN) and false negative (FN) results. Thus, having a priori known set of 444 benign and 239 malignant instances, the neural network has successfully identified 435 (96.67%) instances as negative and 224 (96.14%) instances as positive. According to these observations, Table-2 presents the sensitivity, specificity and efficiency of the neural network model using LVQ1 technique, as well as the predicted values of a positive/negative test results. As a second model we have applied fuzzy classification using fuzzy c-means algorithm to the same 683 samples of WDBC data set. Table-1 presents results that have been obtained using this model. Out of 444 benign and 239 malignant instances, the fuzzy model has successfully identified 436 (93.16%) instances as negative and 217 (96.44%) instances as positive. The same table also presents the true positive (TP), false positive (FP), true negative (TN) and false negative (FN) results. According to these observations, Table-2 shows the sensitivity, specificity and efficiency of the fuzzy model using fuzzy c-means algorithm. The table presents the predicted values of a positive/negative test results.

Comparison of Fuzzy and Neural Network Models to Diagnose Breast Cancer

247

Table 1. Comparative performance of the models Predicated instances Data

Training Data (500) Testing Data (183) Total Data (683)

Actual Instances

Group

Benign

303

Malignant

197

Benign

141

Malignant

42

Benign

444

Malignant

239

Neural Network

Fuzzy Logic

Positive

Negative

Positive

Negative

09 (FP) 182 (TP) 0 (FP) 42 (TP) 09 (FP) 224 (TP)

294 (TN) 15 (FN) 141 (TN) 0 (FN) 435 (TN) 15 (FN)

08 (FP) 175 (TP) 0 (FP) 42 (TP) 8 (FP) 217 (TP)

295 (TN) 22 (FN) 141 (TN) 0 (FN) 436 (TN) 22 (FN)

Table 2. Comparative results of the models Data

Measurements Sensitivity Specificity Efficiency Predictive Value (Positive) Predictive Value (Negative)

Neural Network 92.39 97.03 95.20 95.29 95.15

Fuzzy Logic 88.83 97.36 94.00 95.60 93.05

Testing Data (183)

Sensitivity Specificity Efficiency Predictive Value (Positive) Predictive Value (Negative)

100 100 100 100 100

100 100 100 100 100

Total Data (683)

Sensitivity Specificity Efficiency Predictive Value (Positive) Predictive Value (Negative)

93.72 97.97 96.49 96.14 96.67

90.80 98.20 95.60 96.44 93.16

Training Data (500)

7 Conclusion This Paper has compared two models namely neural Network model and fuzzy model for the diagnosis of breast cancer. The main aim of this Paper is to investigate which model obtains more reasonable specificity while keeping high sensitivity. The benefit is that the number of breast cancer patients for biopsy can be restricted. The seriousness of the ailment can easily be assessed. The output of the artificial neural network has yielded a perfect sensitivity of 93.72%, specificity of 97.97% and high efficiency of 96.49% for the total data set. The output of the fuzzy model has yielded a sensitivity of 90.8%, maximum specificity of 98.2% and efficiency of 95.6% for the total

248

W. Abdul Hameed and M. Bagavandas

data set, demonstrating that fuzzy model also gives similar result as neural network model differentiate a malignant from a benign tumor. The results of this comparative study suggest that the diagnostic performance of neural network model is relatively better than the fuzzy model. This Paper developed a diagnostic system that performs at or above an accuracy level in any procedure short of surgery. The results have also suggested that neural networks and fuzzy models are a potentially useful multivariate method for optimizing the diagnostic validity of laboratory data. The Physicians can combine this unique opportunity extended by fuzzy and neural network models with their expertise to detect the early stages of the disease.

References 1. Barras, J.S., Vigna, L.: Convergence of Kohonen’s Learning Vector Quantization. In: International Joint Conference on Neural Networks, San Diego, CA, vol. 3, pp. 17–20 (1990) 2. Bezdek, J.: Cluster validity with fuzzy sets. J. Cybern. (3), 58–71 (1974) 3. Bezdek, J.C.: A Convergence Theorem for the Fuzzy ISODATA Clustering Algorithms. IEEE Trans. Pattern Anal. Machine Intell. PAM 1-2(1), 1–8 (1980) 4. Bezdek, J.C., Ehrlich, R., Full, W.: FCM: the fuzzy c-means clustering algorithm. Computer & Geosciences 10, 191–203 (1984) 5. Fletcher, S.W., Black, W., Harrier, R., Rimer, B.K., Shapiro, S.: Report of the International workshop on screening for breast cancer. Journal of the National Cancer Institute 85, 1644–1656 (1993) 6. Giard, R.W.M., Hermann, J.: The value of aspiration cytologic examination of the breast: A statistical review of the medical literature. Cancer 69, 2104–2110 (1992) 7. ICMR, National Cancer Registry Programme, Consolidated report of the population based cancer registries,1997 Indian Council of Medical Research, New Delhi (2001) 8. ICMR, National Cancer Registry Programme, 1981-2001, An Overview. Indian Council of Medical Research, New Delhi (2002) 9. Kent, J.T., Mardia, K.V.: Spatial Classification Using Fuzzy Memberships Models. IEEE Trans. on PAMI 10(5), 659–671 (1988) 10. Kohonen, T.: Learning Vector Quantization for Pattern Recognition. Technical Report TKK-F-A601, Helsinki University of Technology, Finland (1986) 11. Kohonen, T.: Improved Version of Learning Vector Quantization. In: Proceedings of the International Joint Conference on Neural Networks, San Diego, CA, vol. 1, pp. 545–550 (1990) 12. Kolen, J.F., Hutcheson, T.: Reducing the time complexity of the fuzzy c-means algorithm. IEEE Trans. Fuzzy Syst. 10(2), 263–267 (2002) 13. Marshall, E.: Search for a Killer: Focus shifts from fret to hormones in special report on breast cancer. Science 259, 618–621 (1993) 14. McBratney, A.B., de Gruijter, J.J.: A Continuoum Approach to Soil Classification by Modified Fuzzy k-Means with Extra-grades. Journal of Soil Sciences 43, 159–175 (1992) 15. McCulloch, W.S., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. Bull. of mathematical Biophysics 5, 115–133 (1943) 16. Zadeh, L.A.: Fuzzy Sets, Information and Control, vol. 8, pp. 338–353 (1965)

A Secure Key Distribution Protocol for Multicast Communication P. Vijayakumar1, S. Bose1, A. Kannan2, and S. Siva Subramanian3 1

Department of Computer Science & Engineering, Anna University, Chennai-25 [email protected], [email protected] 2 Department of Information Science & Technology, Anna University, Chennai-25 [email protected] 3 Department of Mathematics, University College of Engineering Tindivanam, India [email protected]

Abstract. Providing efficient security method to support the distribution of multimedia multicast is a challenging issue, since the group membership in such applications requires dynamic key generation and updation which takes more computation time. Moreover, the key must be sent securely to the group members. In this paper, we propose a new Key Distribution Protocol that provides more security and also reduces computation complexity. To achieve higher level of security, we use Euler’s Totient Function and in the key distribution protocol. Therefore, it increases the key space while breaking the re-keying information. Two major operations in this scheme are joining and leaving operations for managing group memberships. An N-ary tree is used to reduce number of multiplications needed to perform the member leave operation. Using this tree, we reduce the computation time when compared with the existing key management schemes. Keywords: Multicast Communication, Key Distribution, GCD, One Way hash Function, Euler’s Totient Function, Computation Complexity.

1 Introduction Multimedia services, such as pay-per-view, videoconferences, some sporting event, audio and video broadcasting are based upon multicast communication where multimedia messages are sent to a group of members. In such a scenario, group can be either opened or closed with regard to senders. In a closed group, only registered members can send messages to this closed group. In contrast, data from any sender is forwarded to the group members in open groups. Group can be classified into static and dynamic groups. In static groups, membership of the group is predetermined and does not change during the communication. In dynamic groups, membership can change during multicast communication. Therefore, in dynamic group communication, members may join or depart from the service at any time. When a new member joins into the service, it is the responsibility of the Group Centre (GC) to disallow new members from having access to previous data. This provides backward secrecy in P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 249–257, 2011. © Springer-Verlag Berlin Heidelberg 2011

250

P. Vijayakumar et al.

a secure multimedia communication. Similarly, when an existing group member leaves from any group, he/she do not have access to future data. This achieves forward secrecy. GC also takes care of the job of distributing the Secret key and Group key to group members. In this paper, we propose a key Distribution scheme that reduces the computational complexity at the same time; it increases security by proving a large key space. The remainder of this paper is organised as follows: Section 2 provides the features of some of the related works. Section 3 discusses the proposed key distribution protocol and a detailed explanation of the proposed work. Section 4 shows the simulation results. Section 5 gives the concluding remarks and suggests few possible future enhancements.

2 Literature Survey There are many works on key management and key distribution that are present in the literature [1],[2],[12]. In most of the Key Management Schemes, different types of group users obtain a new distributed multicast key for every session update. Among the various works on key distribution, Maximum Distance Separable (MDS) [6] method focuses on error control coding techniques for distributing re-keying information. In MDS, the key is obtained based on the use of Erasure decoding functions [7] to compute session keys by the group members. Here, Group center generates n message symbols by sending the code words into an Erasure decoding function. Out of the n message symbols, the first message symbol is considered as a session key and the group members are not provided this particular key alone by the GC. Group members are given the (n-1) message symbols and they compute a code word for each of them. Each of the group members uses this code word and the remaining (n-1) message symbols to compute the session key. The main limitation of this scheme is that it increases both computation and storage complexity. The computational complexity is obtained by formulating lr+(n-1)m where lr is the size of r bit random number used in the scheme and m is the number of message symbols to be sent from the group center to group members. If lr=m=l, computation complexity is nl. The storage complexity is given by [log2L]+t bits for each member. L is number of levels of the Key tree. Hence Group Center has to store n ([log2L]+t ) bits. The Data Embedding Scheme proposed in [3, 4] is used to transmit rekeying message by embedding the rekeying information in multimedia data. In that scheme, the computation complexity is O(log n). The storage complexity also increases to the value of O(n) for the server machine and O(log n) for group members. This technique is used to update and maintain keys in secure multimedia multicast via media dependent channel. One of the limitations of this scheme is that a new key called embedding key has to be provided to the group members in addition to the original keys, which causes a lot of overhead. Key management using key graphs [11] has been proposed by Wong Gouda which consists of creation of secure group and basic key management graphs scheme using Star based, Tree based method. The limitation of this approach is that scalability is not achieved. A new group keying method that uses one-way functions [8] to compute a tree of keys, called the One-way Function Tree (OFT) algorithm has been proposed by David and Alan. In this method, the keys are

A Secure Key Distribution Protocol for Multicast Communication

251

computed up the tree, from the leaves to the root. This approach reduces re-keying broadcasts to only about log n keys. The major limitation of this approach is that it consumes more space. However, time complexity is more important than space complexity. In our work, we focused on reduction of time complexity. Wade Trappe and Jie Song proposed a Parametric One Way Function (POWF) [5] based binary tree key Management. Each node in the tree is assigned a Key Encrypting Key (KEK) and each user is assigned to a leaf is given the IKs of the nodes from the leaf to the root node in addition to the session key. These keys must be updated and distributed using top down or bottom up approach. The storage complexity is given by logan+2 keys for a group centre. The amount of storage needed by the individual user is given as S=aL+1-1/a-1Keys. Computation time is represented in terms of amount of multiplication required. The amount of multiplication needed to update the KEKs using bottom up approach is 1. Multiplication needed to update the KEKs using top down approach is Ctu= (a-1)logan(logan+1)/2.This complexity can be reduced substantially if the numbers of multiplications are reduced. Therefore, in this paper we propose a new N-ary tree based key Management Scheme using Euler’s Totient Function φ n and gcd φ n which provides more security by increasing the key space and reduces computation time by reducing the number of multiplications required in the existing approaches.

3 Key Distribution Protocol Based on Greatest Common Divisor and Euler’s Totient Function (Proposed Method) 3.1 GC Initialization Initially, the GC selects a large prime number P. This value, P helps in defining a multiplicative a group Zp* and a secure one-way hash function H(.). The defined function, H(.) is a hash function defined from where X and Y are nonidentity elements of Zp*. Since the function H(.) is a one way hash function, x is computationally difficult to determine from the given function Z = y x (mod p) and y. 3.2 Member Initial Join Whenever a new user i is authorized to join the multicast group for the first time, the GC sends it (using a secure unicast) a secret key Ki which is known only to the user Ui and GC. Ki is a random element in Zp*. Using this Ki the Sub Group Keys (SGK) or auxiliary Keys and a Group key Kg are given for that user ui which will be kept in the user ui database. 3.3 Rekeying Whenever some new members join or some old members leave the multicast group, the GC needs to distribute a new Group key to all the current members in a secure way with minimum computation time. When a new member joins into the service it is easy to communicate the new group key with the help of old group key. Since old

252

P. Vijayakumar et al.

group key is not known to the new user, the newly joining user can not view the past communication. This provides backward secrecy. Member Leave operation is completely different from member join operation. In member leave operation, when a member leaves from the group, the GC must avoid the use of old Group key/SGK to encrypt new Group key/SGK. Since old members, knows old GK/SGK, it is necessary to use each user’s secret key to perform re-keying information when a member departs from the services. In the existing key management approaches, this process increases GC’s computation time since aims only the security level. However, the security levels achieved in the existing works are not sufficient with the current computation facilities. Therefore this work focuses on increasing the security level as well as attempts to reduce the computation time. The GC executes the rekeying process in the following steps: 1.

GC defines a one way hash function h(ki,y) where ki is the users secret information, y is the users public information and computes its value as shown in equation (1). ,

2.

(1)

GC computes Euler’s Totient Function φ n [9] for the user ui using the function , as shown in equation (2). Next it can compute , for the user uj. Similarly it can compute Totient value for ‘n’ numbers of user if the message has to be sent to ‘n’ numbers of user. ,

3.

,

(2)

It defines a new function g(ki,y) which is obtained by appending , with the GCD (Greatest Common Divisor) [10] of , , , as shown in equation (3). If the total number of values produced by the func, tion is one, then by default its GCD value is considered as 1. If , , , , then , and , are said to be rela1= gcd tively prime. In such cases, the security level is equivalent to the security level provided in [5]. ,

4.

gcd

,

,

,

(3)

, , , with The purpose of concatenating the value of gcd , is to increase the security and each user can recover the original keying information. for the new GK to be GC computes the new keying information sent to the group members as shown below. ∏

5.

,

GC sends the newly computed GCD value and members.

,

(4) to the existing group

Upon receiving the GCD value and the encoded information from the GC, an authorized user u of the current group executes the following steps to obtain the new group key.

A Secure Key Distribution Protocol for Multicast Communication

1. 2. 3. 4.

253

where Ki is user’s secret key and Calculate the value , y is the old keying information which is known to all the existing users. , , Compute , . Append the received GCD value from GC in front of A legitimate user ui may decode the rekeying information to get the new group key by calculating the following value. ,

(5)

The proposed method uses the Euler’s Totient Function together with GCD function for improving security and computation time. By defining this function, it is possible to overcome the pitfall discussed in the existing key distribution protocol [5] [7] [11]. The key space in our proposed work is increased by ten times in comparison with the original key space indicated in the earlier approach. For instance, if the key values range from 1 to 10 (maximum size=10) in the existing work, the key values range from 1 to 100 (maximum size=100) in the proposed approach. This increases the key space from 101 to 102 for 8-bit keying information. Thus it provides more difficulty for an adversary to break the keying information. 3.4 Tree Based Approach Scalability can be achieved by employing the proposed approach in a key tree based key management scheme to update the GK and SGK. Fig.1. shows a key tree in which, the root is the group key, leaf nodes are individual keys, and the other nodes are auxiliary keys (SGK).

Fig. 1. Key Tree based Key management Scheme

In a key tree, the k-nodes and u-nodes are organized as a tree. Key star is a special key tree where tree degree equals group size [11]. In this paper we have discussed about N-ary based key tree (N=3) wherein the rekeying operation used for member leave case is alone considered. For Example, if a member M9 from the above figure leaves from the group, the keys on the path from his leaf node to the tree’s root should

254

P. Vijayakumar et al.

be changed. Hence, only the keys k 7, 9 , k 1, 9 will become invalid. Therefore, these keys must be updated. In order to update the keys, two approaches namely top-down and bottom-up are used in the members departure (Leave) operation. In the top-down approach, keys are updated from root node to leaf node. On the contrary, in the bottom-up approach, the keys are updated from leaf node to root node. When member M9 leaves from the group, GC will start to update the keys k 7, 9 , k 1, 9 using bottom-up approach. In top-down approach, the keys are updated in the order k 1, 9 , k 7, 9 . The number of multiplications required to perform the rekeying operation is high in top down approach than bottom up approach. This is evident from the complexities shown in equation (10) and (11). So it is a good choice to use bottom up approach in key tree based key management scheme. In binary tree based approach, three updating are required for a group with a size of 8 members. Only two keys are updated in N-ary tree for the group size 9. In this way, this N-ary tree approach reduces the computation time when the group has large number of group members. The working principle of the top-down approach process can be described as follows: When a member M9 leaves from the service, GC computes the Totient value for the remaining members of the group. For the computed Totient value it computes GCD value and it is sent to all the existing group members accordingly. For simplicity, GC chooses K1,9 (old Group Key) as y. Next, GC forms the message given in equation (6) and sends the message to all the existing members in order to update the new Group Key. ,

g k

,

,

, y

g k

,

, y

g k , y

g k , y

(6)

After the successful updation of Group key, GC will update K7, 8 using the formula, ,

,

g k , y

g k , y

(7)

In the bottom up approach, the updation of keys follows an organized procedure and the keys are updated using the formula, ,

∏

,

,

(8)

The next key to be updated is K1, 8. This is performed by using the formula, ,

,

g k

,

, y

g k

,

, y

g k

,

, y

(9)

After updating all the above keys successfully, data can be encrypted using the new Group Key K7, 8. Now the remaining members of the group can decrypt the data using the new Group key K7,8. In the N-ary tree based method, the number of multiplications needed to compute the re-keying information using the bottom up approach is given by the relation, 2

2

Where, B=the number of multiplications needed in bottom up approach, a= the number of children for each node of the tree, l = Level of the tree (0 …K), n= number of registered users.

(10)

A Secure Key Distribution Protocol for Multicast Communication

255

Similarly, the number of multiplications m needed for top down approach in the N--ary based method to compute th he re-keying information can be given by the relation, 1

((11)

4 Simulation Resultss The proposed method has been simulated in NS-2 for more than 500 users and we have analyzed the computation time with existing approaches to perform the rekeyying operation. The graphical reesult shown in Fig.2.is used to compare the key compuutation time of proposed meth hod with the existing methods. It compares the results obtained from GCD based keey distribution scheme with the binary tree-based, erassure encoding and some of the cryptographic methods (RC-4 and AES).

Fig. 2. GC Key Co omputation Time of various Key Distribution schemes

From this, it is observed that when the group size is 600, the key computation tiime is found to be 12µs in ourr proposed approach, which is better in comparison w with existing schemes. Moreoveer if the number of members who are joining and leavving increases the computation time t proportionately increases. However it is less in coomparison with the existing ap pproaches. Computation Time(µs)

15

Binary Tree

10

Erasure method gcd method

5

AES 0

RC-4

Group Size 100

2000

300

400

500

600

Fig. 3. User Key computation for getting the new key value

256

P. Vijayakumar et al.

The results shown in Fig.3.is used to compare the user’s key computation time of our proposed method with the existing methods. It compares the results obtained from GCD based key distribution scheme with existing approaches and it is observed that when the group size is 600, the key recovery time of a user found to be 2µs in our proposed approach, which is better in comparison with existing schemes.

5 Conclusion In this paper, a new N-ary tree based key distribution protocol for n bit numbers as the key value has been proposed for providing effective security in multicast communications. The major advantages of the work are provision of large key space to improve the security. In order to do that we introduced Euler’s Totient function and GCD function to achieve higher level of security. But it will increase the computation time. To overcome this, we introduced the N-ary tree based approach that reduces the computation time (computation complexity), because it takes less number of multiplications when a key is updated. Further extensions to this work are to devise techniques to reduce the storage complexity which is the amount of storage required to store the key related information, both in GC and group members’ area. Rekeying cost also plays a vital role in performing the rekeying for batch join and batch leave operations. Our next work is to device some algorithms to reduce rekeying cost for batch join and batch leave operations.

References 1. Li, M., Poovendran, R., McGrew, D.A.: Minimizing Center Key Storage in Hybrid OneWay Function based Group Key Management with Communication Constraints: Elsevier. Information Processing Letters, 191–198 (2004) 2. Lee, P.P.C., Lui, J.C.S., Yau, D.K.Y.: Distributed Collaborative Key Agreement Protocols for Dynamic Peer Groups. In: Proceedings of the IEEE International Conference on Network Protocols, p. 322 (2002) 3. Trappe, W., Song, J., Poovendran, R., Ray Liu, K.J.: Key Distribution for Secure Multimedia Multicasts via Data Embedding. In: IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 3, pp. 1449–1452 (2001) 4. Poovendran, R., Baras, J.S.: An Iinformation-Theoretic Approach for Design and Analysis of Rooted-Tree-Based Multicast Key Management Schemes. IEEE Transactions on Information Theory 47, 2824–2834 (2001) 5. Trappe, W., Song, J., Poovendran, R., Ray Liu, K.J.: Key Management and Distribution for Secure Multimedia Multicast. IEEE Transactions on Multimedia 5(4), 544–557 (2003) 6. Blaum, M., Bruck, J., Vardy, A.: MDS Array Codes with Independent Parity Symbols. IEEE Transactions on Information Theory 42(2), 529–542 (1996) 7. Xu, L., Huang, C.: Computation-Efficient Multicast Key Distribution. IEEE Transactions on Parallel and Distributed Systems 19(5), 577–587 (2008) 8. David, A., McGrew, Sherman, A.T.: Key Establishment in Large Dynamic Groups using One-Way Function Trees. Cryptographic Technologies Group, TIS Labs at Network Associates (1998)

A Secure Key Distribution Protocol for Multicast Communication

257

9. Apostol, T.M.: Introduction to Analytic Number Theory,Springer International Students Edition, vol. 1, pp. 25–28 (1998) 10. Trappe, W., Washington, L.C.: Introduction to Cryptography with Coding Theory, 2nd edn., pp. 66–68. Pearson Education, London (2007) 11. Wong, C., Gouda, M., Lam, S.: Secure Group Communications using Key Graphs. IEEE/ACM Transactions on Networking 8, 16–30 (2000) 12. Zhou, Y., Zhu, X., Fang, Y.: MABS: Multicast Authentication Based on Batch Signature. IEEE Transactions on Mobile Computing 9, 982–993 (2010)

Security-Enhanced Visual Cryptography Schemes Based on Recursion Thomas Monoth and P. Babu Anto Department of Information Technology Kannur University Kannur-670567, Kerala, India [email protected]

Abstract. In this paper, we propose a security-enhanced method for the visual cryptography schemes using recursion. Visual cryptography is interesting because decryption can be done with no prior knowledge of cryptography and can be performed without any cryptographic computations. The proposed method, the secret image is encoded into shares and subshares in a recursive way. By using recursion for the visual cryptography schemes, the security and reliability of the secret image can be improved than in the case of existing visual cryptography schemes. Keywords: visual cryptography, recursive visual cryptography, visual secret sharing, visual cryptography schemes.

1 Introduction Cryptography is the study of mathematical techniques related to the aspects of information security [1]. The cryptography techniques provide information security, but no reliability. That is, having only one copy of this information means that, if this copy is destroyed there is no way to retrieve it. Replicating the important information will give greater chance to intruders to access it. Thus, there is a great need to handle information in a secure and reliable way. In such situations, secret sharing is of great relevance. An extension of secret sharing scheme is visual cryptography [2]. Visual cryptography (VC) is a kind of secret image sharing scheme that uses the human visual system to perform the decryption computations.Recursive hiding of secrets in visual cryptography and recursive threshold visual cryptography proposed by A. Parakh and S. Kak [3,4]. When Recursive threshold visual cryptography is used in network application, network load is reduced.

2 Visual Cryptography Visual cryptography, proposed by Naor and Shamir [2], is one of the cryptographic methods to share secret images (SI). In visual cryptography scheme (VCS)[5], the SI is broken up into n pieces, which individually yield no information about the SI. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 258–265, 2011. © Springer-Verlag Berlin Heidelberg 2011

Security-Enhanced Visual Cryptography Schemes Based on Recursion

259

These pieces of the SI, called shares or shadows, may then be distributed among a group of n participants/dealers. By combining any k (k ≤ n) of these shares, the SI can be recovered, but combining less than k of them will not reveal any information about the SI. The scheme is perfectly secure and very easy to implement. To decode the encrypted information, i.e., to get the original information back, the shares are stacked and the SI pops out. VCS is interesting because decryption can be done with no prior knowledge of cryptography and can be performed without any cryptographic computations. 2.1 The Existing Model The existing model for black-and-white visual cryptography schemes are defined by Naor and Shamir [2],[6],[7]. In this model, both the original secret image and the share images contain only black and white pixels. The formal definition for blackand-white visual cryptography schemes is: Definition 1: A solution to the k-out-of-n visua1 cryptography scheme consists of two collections of n x m Boolean matrices C0 and C1. To share a white pixel, the dealer randomly chooses one of the matrices in C0 and to share a black pixel, the dealer randomly chooses one of the matrices in C1. The chosen matrix defines the color of the m subpixels in each one of the n transparencies. The solution is considered valid if the following three conditions are met [8][9]: 1). For any S ∈ C0, the OR m-vector V of any k of the n rows in S satisfies H (V) ≤ d – α .m. 2). For any S ∈ C1, the OR m-vector V of any k of the n rows in S satisfies H (V) ≥ d. 3). For any set {r1, r2, . . . , rt} ⊂ {1,2,. . .,n} with t bp and ap < bp. When ap = bp, distance to be measured based on the following conditions: apl = bpl, apl > bpl and apl < bpl. The following illustrations explains these categories and the corresponding distance calculations. case ’A’: (when apl = bpl) p1: 001110, a[1] = 303. Here ap = 3 and apl = 3. p2: 001110, b[1] = 303. Here bp = 3 and bpl = 3. In this case, there is no dissimilarity between the run-lengths of the above two patterns. Now, the two pointers ’ap’ and ’bp’ are to be moved to the next runlength of 1s in the corresponding patterns. case ’B’: (when apl > bpl) p1: 001110, a[1] = 303. Here ap = 3 and apl = 3 p2: 00110, b[1] = 302. Here bp = 3 and bpl = 2 In this case, the pointer ’ap’ has to be moved to the position ap + bpl and the length of ’apl’ has to be decreased correspondingly, that is apl = apl − bpl. The pointer ’bp’ has to be moved to the next run-length of 1s. case ’C’: (when apl < bpl) p1: 001110, a[1] = 203. Here ap = 3 and apl = 3 p2: 0011110, b[1] = 304. Here bp = 3 and bpl = 4 In this case, ’ap’ has to be moved to the next run-length of 1s and ’bp’ has to be moved to the position bp + apl and the value of ’bpl’ has to be decreased correspondingly,that is bpl = bpl − apl . When ap > bp, distance to be measured based on the following conditions: (A) the position of ’ap’ = the last position of continuous 1s of ’b’ (B) the position of ’ap’ > the last position of continuous 1s of ’b’ (C) the position of ’ap’ < the last position of continuous 1s of ’b’ case ’A’: p1: 00001, a[1] = 501. Here ap = 5 p2: 011110, b[1] = 204. Here bp =2 and bpl = 4 In the above, there is dissimilarity between the run-length of the patterns from the starting position of ’bp’ to a position before ’ap’. This dissimilarity is calculated by ”distance = distance + (ap − bp)”. The value of ’bpl’ is decreased by ap − bp and the position of ’bp’ is moved to the position of ’ap’.

276

T. Kathirvalavakumar and R. Palaniappan

case ’B’: p1: 0001, a[1] = 401. Here ap = 4 p2: 0110, b[1] = 202. Here bp = 2 and bpl = 2 In the above case, the dissimilarity is the length of continuous 1s from ’bp’, that is ’bpl’. The pointer ’bp’ has to be moved to the starting position of next continuous 1s. case ’C’: p1: 001, a[1] = 301. Here ap = 3 p2: 011110, b[1] = 204. Here bp = 2 and bpl = 4 In the above case, there is dissimilarity between the run-length of the patterns from the starting position of ’bp’ to a position before ’ap’. The length of ’bpl’ has to be decreased by ap − bp and the pointer ’bp’ has to be moved to the position of ’ap’. When ap < bp, distance to be measured based on the following conditions: (A) the position of ’bp’ = the last position of continuous 1s of ’a’ (B) the position of ’bp’ > the last position of continuous 1s of ’a’ (C) the position of ’bp’ < the last position of continuous 1s of ’a’ case ’A’: p1: 011110, a[1] = 204. Here ap = 2 and apl = 4 p2: 00001, b[1] = 501. Here bp =5 In the above, there is dissimilarity between the run-length of the patterns from the starting position of ’ap’ to a position before ’bp’. This dissimilarity is calculated by ”distance = distance + (bp − ap)”. The value of ’apl’ is decreased by bp − ap and the position of ’ap’ is moved to the position of ’bp’. case ’B’: p1: 01110, a[1] = 203. Here ap = 2 and apl = 3 p2: 00001, b[1] = 501. Here bp = 5 In the above case, the dissimilarity is the length of continuous 1s from ’ap’, that is ’apl’. The pointer ’ap’ has to be moved to the starting position of next continuous 1s. case ’C’: p1: 011110, a[1] = 204. Here ap = 2 and apl = 4 p2: 001, b[1] = 301. Here bp = 3

Modified Run-Length Encoding Method and Distance Algorithm

277

In the above case, there is dissimilarity between the run-length of the patterns from the starting position of ’ap’ to a position before ’bp’. The length of ’apl’ has to be decreased by bp − ap and the pointer ’ap’ has to be moved to the position of ’bp’.

4

Results and Discussion

The proposed modified RLE and the corresponding distance measure algorithms are demonstrated by classifying the Handwritten Digit Data set comprising of 10 digits ( 0 - 9 ). We have taken 10000 patterns each with 192 attributes. Among these, 6670 patterns are used for training and 3330 patterns are used for testing, with equal proportion of patterns in each digit. The above data set is classified using the following steps: Step 1: All training patterns from the data file are read one by one and the patterns are encoded using the proposed algorithm and stored in an array. Step 2: Read a test pattern from the file and encode it using the proposed algorithm and store it in an array. Step 3: Classify the encoded test pattern using the K-Nearest Neighbor (KNN) algorithm. Proposed distance measure algorithm is used to find the pattern proximity of the data in the encoded form. Step 4: Step 2 and Step 3 are repeated for all the test patterns. The above procedure is repeated for different K values (1 - 20) of KNN for the Handwritten Digit Data. The classification accuracy(CA) obtained for runlength encoded binary data of Babu et al.[3] using KNN algorithm is matched with the proposed procedure. It has been found that the CA of proposed procedure with modified RLE is same as Babu et al.[3]. Fig. 1 shows the CA obtained for different K values of the proposed procedure. The best CA obtained is 92.52% for k=7. The processing time of the proposed procedure is comparatively lesser than Babu et al.[3]. when executed in the Intel Core2Duo 2.66GHz system. The time requirement for KNN classification using the proposed modified RLE and the Run-length encoded CDR of Babu et al.[3]. are shown in Fig. 2. The average memory requirement of the proposed modified RLE and the CDR of Babu et al.[3] for each digit of the Handwritten Digit Data are shown in Fig. 3. It is found that, a significant amount of data size has been reduced by virtue of the proposed encoding technique. Handwritten Digit Data set needs totally 1280640 number of memory locations (6670 x 192), but the proposed encoded representation occupies only 139054 number of memory locations(Run dimension). It has also been observed that the memory requirement(Run dimension) of the proposed

278

T. Kathirvalavakumar and R. Palaniappan

93

92.5 CA (%) 92

Classification Accuracy (% )

91.5

91

90.5

90

89.5

89

88.5

88 0

5

10

15

20

K - Value

Fig. 1. Classification Accuracy

52 Proposed

CDR

51.5

51

Time (Sec)

50.5

50

49.5

49

48.5

48

47.5 0

5

10

15

20

K Value

Fig. 2. Time Comparision - Proposed vs. CDR Data

60

CDR PROPOSED

Average Run Dimension

50

40

30

20

10

0 1

2

3

4

5

6

7

8

9

10

CLASS

Fig. 3. Memory Requirement - Proposed vs. CDR Data

encoded data is approximately 50% of the memory requirement(Run dimension) of the run-length encoded CDR of Babu et al.,[3]. In Table 1 we have shown the Run dimension (memory requirement) for the proposed RLE and CDR of Babu et al., [3] with six different possible binary patterns.

Modified Run-Length Encoding Method and Distance Algorithm

279

Table 1. Comparison of Proposed RLE and the CDR in terms of Run Dimension Sl. No. Pattern CDR Run Proposed RLE Run Run (16 Bits) String Run String Dimension Dimension CDR Proposed 1 111000111 {3,3,5,2,3} {103,705,1403} 5 3 1100111 2 010101100 {0,1,1,1,1, {201,401,602, 12 6 1101000 2,2,2,1,1,3} 1002,1301,1701} 3 101010101 {1,1,1,1,1, {101,301,501, 16 9 0101010 1,1,1,1,1, 701,901,1101, 1,1,1,1,1,1} 1301,1501,1701} 4 010101010 {0,1,1,1,1, {201,401,601, 17 8 1010101 1,1,1,1,1,1 801,1001,1201, 1,1,1,1,1,1} 1401,1601} 5 000000000 {0,16} {1701} 2 1 0000000 6 111111111 {16} {116} 1 1 1111111

5

Conclusion

An efficient modified run-length encoding technique for classifying binary data is proposed. In this method, only the run-lengths of 1’s of a pattern is considered. Pattern proximity is computed over the run-length of encoded data using the proposed distance algorithm for classifying binary data. Proposed encoding method and corresponding distance algorithm are used to classify handwritten digit data set. KNN algorithm is used for classification. The tabulated values and figures show the efficiency of the proposed method in terms of memory and time.

References 1. Akimov, A., Kolesnikov, A., Franti, P.: Lossless compression of map contours by context tree modelling of chain codes. Pattern Recognition 40, 944–952 (2007) 2. Al-laham, M., Emary, I.: Comparative study between various algorithms of data compression techniques. IJCSNS, International Journal of Computer Science and Network Security 7(4), 284–290 (2007) 3. Babu, T.R., Murty, N., Agarwal, V.K.: Classification of run-length encoded binary data. Pattern Recognition 40, 321–323 (2007) 4. Fung, B.C.M.: Hierarchical document clustering using frequent item-sets, Thesis submitted in Simon Fraser University (1999), http://citeseer.ist.psu.edu/581906.html 5. Gilbert, H.: Data and image compression tool and techniques, pp. 68–343. Wiley, Chichester (1996) 6. Hsu, H.W., Zwaarico, E.: Automatic synthesis of compression techniques for heterogeneous files. Software-Practice and Experience 25(10), 1097–1116 (1995)

280

T. Kathirvalavakumar and R. Palaniappan

7. Jones, C.B.: An efficient coding system for long source sequence. IEEE Trans. Information Theory 27, 280–291 (1981) 8. Kim, W.-J., Kim, S.-D., Radha, H.: 3D binary morphological operations using run-length representation. Signal Processing: Image Communication 23, 442–450 (2008) 9. Kim, W., Kim, S., Kim, K.: Fast algorithm for binary dilation and erosion using run-length encoding. ETRI Journal 27, 814–817 (2005) 10. Langdon, G.G., Rissannen, J.J.: A simple general binary source code. IEEE Trans. Information Theory 29, 858–867 (1982) 11. Makinen, V., Navano, G., Ukkonen, E.: Approximate matching of run-length compressed strings. Algorithmica 35(4), 347–369 (2003) 12. Liang, J., Piper, J., Tang, J.Y.: Erosion and Dilation of binary images by arbitrary structuring elements using interval coding. Pattern Recognition Letters 9, 201–209 (1989) 13. Ryan, O.: Runlength-based Processing Methods for Low Bit-depth Images. IEEE Transactions on Image Processing 18, 2048–2058 (2009) 14. Sedgewick, R.: Algorithms, 2nd edn. Addision-Wesly, Reading (1988) 15. Storer, J.A.: Data compression: Methods and Theory. Computer Science Press, Rockville (1988) 16. Zahir, S., Naqvi, M.: A Near Minimum Sparse Pattern Coding Based Scheme for Binary Image Compression. In: IEEE-ICIP 2005, Genoa, Italy (September 2005) 17. Zhang, T., Ramakrishnan, R., Linvy, M.: BIRCH: An efficient data clustering method for very large databases. In: Proceedings of ACM SIGMOD, International Conference on Management of Data. ACM Press, New york (1996)

A Fast Fingerprint Image Alignment Algorithms Using K-Means and Fuzzy C-Means Clustering Based Image Rotation Technique P. Jaganathan and M. Rajinikannan Department of Computer Applications, P.S.N.A. College of Engineering and Technology, Dindigul, Tamilnadu-624622, India [email protected], [email protected]

Abstract. The Fingerprint recognition system involves several steps. In such recognition systems, the orientation of the fingerprint image has influence on fingerprint image enhancement phase, minutia detection phase and minutia matching phase of the system. The fingerprint image rotation, translation, and registration are the commonly used techniques, to minimize the error in all these stages of fingerprint recognition. Generally, image-processing techniques such as rotation, translation and registration will consume more time and hence impact the overall performance of the system. In this work, we proposed fuzzy c-means and k-mean clustering based fingerprint image rotation algorithm to improve the performance of the fingerprint recognition system. This rotation algorithm can be applied as a pre-processing step before minutia detection and minutia matching phase of the system. Hence, the result will be better detection of minutia as well as better matching with improved performance in terms of time. Keywords: k-means clustering, fuzzy c-means clustering, Fingerprint Image Enhancement, Fingerprint Image Registration, Rotation, Alignment, Minutia Detection.

1 Introduction Fingerprint recognition is part of the larger field of Biometrics. Other techniques of biometrics include face recognition, voice recognition, hand geometry, retinal scan, ear surface and so on. 1.1 The Fingerprint Image and Minutia A fingerprint is the feature pattern of one finger. Fingerprints are believed to be unique across individuals, and across fingers of the same individual [10]. Fingerprintbased personal identification has been used for a very long time [7]. It is believed that each person has his own fingerprints with the permanent uniqueness. Hence, fingerprints have being used for identification and forensic investigation for a long P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 281–288, 2011. © Springer-Verlag Berlin Heidelberg 2011

282

P. Jaganathan and M. Rajinikannan

time. Minutia points occur either at a ridge ending or a ridge bifurcation. A ridge ending is defined as the point where the ridge ends abruptly and the ridge bifurcation is the point where the ridge splits into two or more branches. A fingerprint expert is often able to correctly identify the minutia by using various visual clues such as local ridge orientation, ridge continuity, ridge tendency, etc., as long as the ridge and furrow structures are not corrupted completely [1].

Fig. 1. The Ridge Endings and Bifurcation

Fig. 2. The Characteristic of a Minutia

The above diagram shows the clear view of a minutia and its characteristic attributes. 1.2 The Importance of Fingerprint Image Alignment Algorithms A typical fingerprint recognition system, have several types of algorithms in each stage. We can broadly reduce these stages of algorithms as Fingerprint Image Enhancement (Ridge and Valley Enhancement) Algorithms, Minutia Detection Algorithms and Minutia Matching Algorithms. Fingerprint Image alignment algorithm is the most important step, since the results of Minutia extraction Algorithms and the Minutia matching Algorithms are very much depend on the clarity and orientation of the fingerprint image. Point pattern matching is generally intractable due to the lack of knowledge about the correspondence between two point’s sets. To address this problem, Jain et al. proposed alignment-based minutia matching [4], [5]. Two sets of minutia are first aligned using corresponding ridges to find a reference minutia pair, one from the input fingerprint and another from the template fingerprint, and then all the other minutia of both images are converted with respect to the reference minutia. Jiang et al. also proposed an improved method [6], using minutia together with its neighbors to find the best alignment. In general, minutia matcher chooses any two minutias’s as a reference minutia pair and then matches their associated ridges first. If the ridges match well [11], two fingerprint images are aligned and matching is conducted for all remaining minutia. In any of this matching policy, the fingerprint image alignment will play significant role and will have influence on accuracy. In the following sections, the previous work on fingerprint alignment and the proposed algorithms are discussed.

A Fast Fingerprint Image Alignment Algorithms

283

2 Previous Works 2.1 Registration Using Mutual Information The Registration based method called ‘Automatic Image Registration’ by Kateryna Artyushkova, University of New Mexico, has been used for comparing the performance of the proposed fuzzy c-means and k-means based image alignment algorithm. Registration or alignment is a process through which the correct transformation is determined. It uses mutual information as the similarity measure and aligns images by maximizing mutual information between them under different transformations [8]. Mutual information describes the uncertainty in estimating the orientation field at a certain location given another orientation field. The more similar or correlated the orientation fields are, the more mutual information they have. MI measures the statistical dependency between two random variables. The physical meaning of MI is the reduction in entropy of Y given X. Viola et al. proposed [12] that registration could be achieved by maximization of mutual information. In this paper, we apply MI to fingerprint registration. Here we use MI between template and input’s direction features to align the fingerprints. In this registration algorithm, we assume that distortion is not very large. 2.2 The Proposed Fingerprint Image Alignment Algorithms 2.3 Pre-processing the Input Fingerprint Image Initially, the input fingerprint image is pre–processed by changing the intensity value to 0 for the pixel, whose intensity value is less than 32. Now, only the ridges and valleys of the fingerprint image are considered. Except ridges and valleys the other parts of the fingerprint image pixels intensity values are changed to 0. Finally the binary image will be constructed. The co-ordinates of (x,y) the one valued pixels are considered as two attributes and passed as Input to the K-Means algorithm. 2.4 K-Means Algorithm K-Means algorithm is very popular for data clustering. Generally, K-Means algorithm is used in several iterations to cluster the data since the result is very much depend on the initial guess of the cluster centres. The Algorithm goes like this 1. Start iteration 2. Select k Center in the problem space (it can be random). 3. Partition the data into k clusters by grouping points that are closest to that k centers. 4. Use the mean of these k clusters to find new centers. 5. Repeat steps 3 and 4 until centers do not change, for N iterations. 6. Among the N results, find the result with minimum distance. 7. Display the results corresponding to that minimum distance. 8. Stop iterations.

284

P. Jaganathan and M. Rajinikannan

2.5 Fuzzy C-Means Algorithm Fuzzy c-means (FCM) is a method of clustering which allows one piece of data to belong to two or more clusters. This method was developed by Dunn [3] in 1973 and improved by J.C. Bezdek [2] in 1981 is frequently used in pattern recognition. Fuzzy c-means (FCM) is a data clustering technique wherein each data point belongs to a cluster to some degree that is specified by a membership grade. The Fuzzy c-means algorithm starts with an initial guess for the cluster centers. The initial guess for these cluster centers is most likely incorrect. By iteratively updating the cluster centers and the membership grades for each data point, Fuzzy c-means algorithm iteratively moves the cluster centers to the right location within a data set. This iteration is based on minimizing an objective function that represents the distance from any given data point to a cluster center weighted by that data point's membership grade. The Algorithm is composed of the following steps. 1. 2.

Initialize U=[uij] matrix, U(0) At k-step: calculate the centers vectors C(k)=[ c j ] with U(k)

3. 4.

Update U(k) , U(k+1) If || U(k+1) - U(k)||< ε then STOP; otherwise return to step 2.

Finally, the above algorithm gives the two cluster points C1 and C2. After getting the cluster points C1 and C2 the following steps will be followed to rotate the image to 90 degrees. 2.6 K-Means and Fuzzy C-Means Algorithm for Fingerprint Image Rotation Let us consider the pixels of the two-dimensional finger print image as the plotted three data points of x, y, Gray level. 1. The fingerprint image pixels are assumed as data points in 2D space.

2. The data points were clustered in to two groups using k-means and fuzzy c means clustering algorithm. The green line showing the cluster boundary. 3. The points C1 and C2 are the centers of the two clusters.

A Fast Fingerprint Image Alignment Algorithms

285

4. A line connecting C1 and C2 will be almost equal to the inclination of the fingerprint image. 5. The inclination angle θ can be measured from the base of the image. θ = atan ( (x1-x2) / (y1-y2 ) ) (in radians) θ = θ *(180/pi) (in degree) if θ < 0 θ = 90+ θ else θ = - ( 90- θ) end 6. Now rotating the image by angle θ. 7. The direction of rotation can be decided with respect to the location of the point C1 in the top two quadrants of the four Quadrants. 8. This will finally give the well aligned image.

3 Implementation Results and Analysis The following diagram shows the implemented model of the fingerprint alignment system

Read Input Fingerprint Image

Enhance the Fingerprint Image

Estimate the Inclination angle using k-means and fuzzy c-Means

Rotate the image by angle θ

Fig. 3. The fingerprint alignment system

3.1 Fingerprint Database Used for Evaluation A fingerprint database from the FVC2000 [9] (Fingerprint Verification Competition 2000) is used to test the experiment performance. FVC2000 was the First International Competition for Fingerprint Verification Algorithms. This initiative is organized by D. Maio, D. Maltoni, R. Cappelli from Biometric Systems Lab (University of Bologna), J. L. Wayman from the U.S. National Biometric Test Center (San Jose State University) and A. K. Jain from the Pattern Recognition and Image Processing Laboratory of Michigan State University. Sets of few selected images from the above database were used to evaluate the performance of the algorithms. 3.2 Sample Set of Results The following outputs show some of the inputs as well as the corresponding outputs. In the second column images, the angle of rotation estimated by the k-means based algorithm also given below the image.

286

P. Jaganathan and M. Rajinikannan

103_1.TIF

103_2.TIF

106_3.TIF

105_2.TIF

102_3.TIF

Input Image

Vertically Aligned Image

k-means : θ = 26.5651 FCMeans : θ = 26.9395

k-means : θ = -14.0362 FCMeans : θ = -13.4486

k-means : θ = -33.2936 FCMeans : θ = -33.6901

k-means : θ = -6.9343 FCMeans : θ = -7.1250

k-means : θ = 21.9672 FCMeans : θ = 23.0026 Fig. 4. The Results of Rotation

The following tables show the performance of the algorithms in terms of time. The registration based image alignment algorithm will use two images. One will be considered as reference image to which that other image has to be aligned. The registration-based method has been used for comparing the performance of the proposed fuzzy c-means and k-means based image alignment algorithm. The results of all the three algorithms are shown in the following table.

A Fast Fingerprint Image Alignment Algorithms

287

Table 1. The average time taken for Alignment

Time Taken for Rotation (in seconds)

1

102_3.TIF

18

0.92

Fuzzy C-means based Algorithm 1.84

2

105_2.TIF

17

0.97

1.25

3

106_3.TIF

18

0.98

3.01

4

103_2.TIF

18

0.93

3.13

5

103_1.TIF Average

17 17.6

0.93 0.946

3.56 2.558

Sl. No

Fingerprint Image

Registration Based Algorithm

k-means based Algrithm

Table 1 shows the average time taken (in seconds) by the registration based method, proposed K-Means Algorithm and Fuzzy C-Means Algorithm. The result shows that the average time taken by the proposed K-Means Algorithm of fingerprint alignment is approximately 0.946 seconds, whereas the Fuzzy C-Means Algorithm takes 2.558 seconds and the registration based method takes 17.6 seconds. The following chart compares the average time taken for alignment of the fingerprint image.

Time(in seconds)

Time Taken for Alignment 20 18 16 14 12 10 8 6 4 2 0

17.6

0.946 Registration Based Algorithm

k-means based Algorithm Method

2.558

C-means based Algorithm

Fig. 5. The average time taken for Alignment

288

P. Jaganathan and M. Rajinikannan

4 Conclusion and Future Work We have successfully implemented and evaluated the proposed k-means and fuzzy cmeans based fingerprint image alignment algorithm under Matlab 6.5 on Intel Core 2 Duo processor and RAM size is 2 GB DDR2. Its performance is compared with normal registration based alignment algorithm. The arrived results were significant and more comparable. It is clear that the performance of k-means algorithm is better than that of Registration based and fuzzy c-means algorithms. If we use the aligned fingerprint image for the detection of minutia and minutia matching, then we may expect better accuracy in recognition. Future works may evaluate the difference in recognition with and without the proposed fingerprint image alignment phase.

References 1. Hong, L., Wan, Y., Jain, A.K.: Fingerprint image enhancement: Algorithm and performance evaluation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 777– 789 (1998) 2. Bezdek, J.C.: Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York (1981) 3. Dunn, J.C.: A Fuzzy Relative of the ISODATA Process and its use in Detecting Compact Well – Separated Clusters. Journal of Cybernetics 3, 32–57 (1973) 4. Jain, A.K., Hong, L., Pankanti, S., Bolle, R.: An Identity-authentication system using fingerprints. Proc. IEEE 85(9), 1365–1388 (1997) 5. Jain, A.K., Hong, L., Bolle, R.: On-Line fingerprint verification. IEEE Trans. Pattern Anal. Mach. Intell. 19(4), 302–314 (1997) 6. Jiang, X., Yau, W.: Fingerprint minutia matching based on the local and global structures. In: Proc. 15 th Int. Conf. Pattern Recognition, Barcelona, Spain, vol. 2, pp. 1042–1045 (September 2000) 7. Lee, H.C., Gaensslen, R.E. (eds.): Advances in Fingerprint Technology. Elsevier, New York (1991) 8. Liu, L., Jiang, T., Yang, J., Zhu, C.: Fingerprint Registration by Maximization of Mutual Information. IEEE Transactions on Image Processing 15, 1100–1110 (2006) 9. Maio, D., Maltoni, D., Cappelli, R., Wayman, J.L., Jain, A.K.: FVC2000: Fingerprint Verification Competition. In: 15th ICPR International Conference on Pattern Recognition, Spain, September 3-7 (2000), http://bias.csr.unibo.it/fvc2000/ 10. Pankanti, S., Prahakar, S., Jain, A.K.: On the individuality of fingerprints. IEEE Trans. Pattern and Mach. Intell. 24, 1010–1025 (2002) 11. Singh, R., Shah, U., Gupta, V.: Fingerprint Recognition, Student project, Department of Computer Science and Engineering, Indian Institute of Technology, Kanpur, India (November 2009) 12. Viola, P.: Alignment by Maximization of Mutual Information. Ph. D thesis, M.I.T. Artificial Intelligence Laboratory (1995)

A Neuro Approach to Solve Lorenz System J. Abdul Samath1, P. Ambika Gayathri1, and A. Ayisha Begum2 1

Department of Computer Applications, Sri Ramakrishna Institute of Technology, Coimbatore-641010, Tamilnadu, India [email protected], [email protected] 2 Department of Information Technology, VLB Janakiammal College of Engineering and Technology, Coimbatore-641008, Tamilnadu, India [email protected]

Abstract. In this paper, Neural Network algorithm is used to solve Lorenz System. The solution obtained using neural network is compared with Runge-Kutta Butcher (RK Butcher) method and it is found that neural network algorithm is efficient than RK method. Keywords: Lorenz System, Runge-Kutta Butcher method, Neural Network.

1 Introduction The earth’s atmosphere is approximately a system of Rayleigh-Benard heat convection that is a layer of radially flowing fluid bounded by two boundaries of different temperature: the warm crust and the chilling upper stratosphere. In 1963, Edward Lorenz attempted to model this convection occurring in the earth’s atmosphere as a simple set of three first-order non-linear ordinary differential equations, called the Lorenz system. Like the weather, this system is highly sensitive to its initial conditions. Due to the non-linearity of the equations, an exact form of the solution cannot be calculated. Instead, numerical methods, such as Euler’s and Runge-Kutta methods, are employed to calculate approximations to the system solution through iteration. Morris Bader [2, 3] introduced the RK-Butcher algorithms for finding the truncation error estimates and intrinsic accuracies and the early detection of stiffness in coupled differential equations that arises in theoretical chemistry problems. Most recently, Murugesan et al [1] and Park et al [4] applied the RK-Butcher algorithm for an industrial robot arm control problem and optimal control of linear singular systems. Many methods have been developed so far for solving differential equations. Some of them produce a solution in the form of an array that contains the value of the solution at a selected group of points. Others use basis-functions to represent the solution in analytic form and transform the original problem usually to system of algebraic equations. The solution of a linear system of equations is mapped onto the architecture of a Hopfield neural network. The minimization of the network’s energy function provides the solution to the system of equations [5, 6, and 7]. P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 289–296, 2011. © Springer-Verlag Berlin Heidelberg 2011

290

J. Abdul Samath, P. Ambika Gayathri, and A. Ayisha Begum

The dynamic of a physical system is usually determined by solving a differential equation. It normally proceeds by reducing the differential equation to a familiar form with known solutions using suitable coordinate transformations [8, 9]. In nontrivial cases however, finding the appropriate transformations is difficult and numerical methods are used. A number of algorithms (e.g., Runge-Kutta, finite difference, etc.) are available for calculating the solution accurately and efficiently [10]. Neural network or simply neural nets are computing systems, which can be trained to learn a complex relationship between two or many variables or data sets. Having the structures similar to their biological counterparts, neural networks are representational and computational models processing information in a parallel distributed fashion composed of interconnecting simple processing nodes. In this paper, Neural Network algorithm is used to solve Lorenz System. The solution obtained using neural network is compared with Runge-Kutta Buthcer(RK Butcher) method and it is found that neural network algorithm is efficient than RK Butcher method.

2 Statement of the Problem The Lorenz model is a dissipative system, with some property of the flow, such as total energy is conserved. It consists of three first-order nonlinear ordinary differential equations with no exact solution.

.

.

.

. .

(1)

.

where x,y,z are respectively convective velocity, the temperature differences between descending and ascending flow and the mean convective heat flow. The parameter Pr, R and b are kept constant within integration, but they can be changed to form a family of solutions of the dynamical system defined by the differential equations. The particular values chosen by Lorenz were Pr=10, R=28, and b=8/3, which result in nonperiodic or chaotic solutions.

3 Runge - Kutta Butcher Solution RK Butcher algorithm is normally considered as sixth order, since it requires six functions evaluation. Even though it seems to be sixth order method, it is only a fifth order method. The accuracy of this algorithm is considered better than other algorithms. The system of non-linear differential equation (1) is solved as follows: 1

90 1

7 1

90

71

32 3

12 4

35 5

32 3

12 4

35 5

7 6 76

A Neuro Approach to Solve Lorenz System

291

Where , ,

,

, ,

, ,

,

,

, ,

8 2

, ,

,

2 3

, ,

1

,

9 16 1 9 16 1 2

3

,

2 2

3, 1

3

12 8

3,

2 2

3 , 4 ,

, 2

1

,

2 ,

,

, 4

,

,

3 4

,

4 2

,

3

,

1

9

4

16

12

3

12

4

8

5

7 3

1

2

2

12

3

12

4

,

8

, 5

,

7 3

1

2

2

12

3

12

4

8

5

7

4 Neural Network Solution In this approach, new feedforward neural network is used to change the trail solution of Eq (1) to the neural network solution of (1). The trial solution is expressed as the difference of two terms as below. Vi ,Ij a

,

-

(2)

The first term satisfies the TCs and contains no adjustable parameters. The second correspond to the term employs a feedforward neural network and parameters weights of the neural architecture. Consider a multi layer perception with n input units, one hidden layer with n sigmoidal units and a linear output unit. The extension to the case of more than one hidden layer can be obtained accordingly. For a given input vector, the output of the network is Nij = ∑ where zi =∑ hidden unit i,

i)

(zi)

, denotes the weight from the input unit j to the ij)tj + denotes the weight from the hidden i to the output, denotes the

bias of the hidden unit i and

is the sigmoidal transfer function.

292

J. Abdul Samath, P. Ambika Gayathri, and A. Ayisha Begum

The error quantity to be minimized is given by E =∑ ,

i,

j)a

-

ij

(t, (

i,

j)a))

2

.

(3)

The neural network is trained till the error function (3) becomes zero. Whenever E becomes zero, the trail solution (2) becomes the neural network solution of Eq (1). 4.1 Structure of the FFNN The architecture consists of n input units, one hidden layer with n sigmoidal units and a linear output. Each neuron produces its output by computing the inner product of its input and its appropriate weight vector.

Fig. 1. Neural Network Architecture

During the training, the weights and biases of the network are iteratively adjusted by Nguyen and Widrow rule. The neural network architecture is given in Fig. 1 for computing Nij. The neural network algorithm was implemented in MATLAB on a PC, CPU 1.7 GHz for the neuro computing approach. Neural Network Algorithm Step 1: Feed the input vector tj Step 2: Initialize randomized weight matrix wij and bias ui. Step 3: Compute =∑ Step 4: Pass zi into n sigmoidal functions. Step 5: Initialize the weight vector vi from the hidden unit to output unit. Step 6: Calculate Nij=∑ Step 7: Compute purelin function (Nij) Step 8: Repeat the neural network training until the following error function E=∑ ,

i,

j) a -

ij

(t,(

i,

j) a))

2

= 0.

A Neuro Approach to Solve Lorenz System

293

5 Results and Discussion MATLAB Neural Network Toolbox has been used to solve the Lorenz system where Pr and b is set to be 10 and 8/3 respectively for which the initial condition is given by x(0)=-15.8, y(0)=-17.48 and z(0)=35.64. The methods were examined in time range [0, 1] with two time steps ∆t=0.01 and ∆t=0.001. 5.1 Non-chaotic Solution For R=23.5, we determine the accuracy of NN, and RK Butcher with time steps ∆t=0.01 and ∆t=0.001. The results are presented in Table1-3 and the corresponding graph in Fig. 2-4. Table 1. x-direction difference between all methods for R=23.5 x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

NN0.01 − NN0.001 0.00E+00 5.64E-02 2.23E-02 5.05E-03 7.37E-03 6.76E-04 1.27E-02 1.13E-02 1.94E-02 2.85E-02

Absolute error for x

RKButcher0.01 − RKButcher0.001 0.00E+00 1.92E-01 7.95E-02 1.24E-02 2.03E-02 2.33E-03 4.39E-02 3.44E-02 7.00E-02 9.68E-02

1.00E+00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.00E-01

Absolute error

t

NN

1.00E-02

RK Butcher

1.00E-03

1.00E-04 Time, t

Fig. 2. Absolute error of all methods for x

Table 2. y-direction difference between all methods for R=23.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

NN0.01 − NN0.001 0.00E+00 4.52E-02 1.60E-02 1.04E-02 2.44E-03 8.88E-03 2.02E-02 1.05E-02 4.02E-02 2.15E-02 3.91E-03

y RKButcher0.01 − RKButcher0.001 0.00E+00 1.52E-01 5.35E-02 3.14E-02 3.69E-03 3.41E-02 6.64E-02 4.31E-02 1.38E-01 6.83-02 8.39E-03

Absolute error for y 1.00E+00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.00E-01 Absolute error

t

NN RK Butcher 1.00E-02

1.00E-03 Time,t

Fig. 3. Absolute error of all methods for y

294

J. Abdul Samath, P. Ambika Gayathri, and A. Ayisha Begum

Table 3. z-direction difference between all methods for R=23.5

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

3.25E-02 2.17E-02 2.19E-02 1.88E-02 6.75E-03 3.81E-02 1.79E-03 3.34E-02 4.13E-02

1.15E-01 7.52E-02 7.30E-02 5.63E-02 3.18E-02 1.28E-01 1.83E-03 1.18E-01 1.40E-01

t

Absolute error for z 1.00E+00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.00E-01 Absolute error

0 0.1

NN0.01 − NN0.001 0.00E+00 5.78E-02

z RKButcher0.01 − RKButcher0.001 0.00E+00 2.07E-01

NN RK Butcher

1.00E-02

1.00E-03 Time, t

Fig. 4. Absolute error of all methods for z

5.2 Chaotic Solutions

For a chaotic system, R=28, it can be seen that once again NN shows a better accuracy as compared to the other two methods for simple time steps ∆t=0.01 and ∆t=0.001. This can be observed from Table 4-6 and Fig 5-7 below: Table 4. x-direction difference between all methods for R=28

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

NN0.01 − NN0.001 0.00E+00 4.53E-02 2.76E-02 1.61E-03 2.97E-03 2.19E-03 4.75E-03 2.86E-02 4.47E-02 7.33E-02 3.85E-02

Absolute error for x 1.00E+00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.00E-01

Absolute error

t

x RKButcher0.01 − RKButcher0.001 0.00E+00 1.53E-01 9.73E-02 8.93E-03 5.94E-03 3.37E-04 2.57E-02 9.64E-02 1.62E-01 2.50E-01 1.29E-01

NN

1.00E-02

RK Butcher

1.00E-03

1.00E-04 Time, t

Fig. 5. Absolute error of all methods for x

A Neuro Approach to Solve Lorenz System

295

Table 5. y-direction difference between all methods for R=28

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

NN0.01 − NN0.001 0.00E+00 5.48E-02 6.86E-03 4.84E-03 7.05E-04 2.26E-03 1.62E-02 9.18E-03 1.28E-01 3.86E-02 2.13E-02

y RKButcher0.01 − RKButcher0.001 0.00E+00 1.84E-01 2.33E-02 1.44E-02 2.69E-03 1.81E-02 6.72E-02 2.07E-02 4.33E-01 1.24E-01 6.60E-02

Absolute error for y 1.00E+00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.00E-01 Absolute error

t

NN

1.00E-02

RK Butcher

1.00E-03

1.00E-04 Time,t

Fig. 6. Absolute error of all methods for y

Table 6. z-direction difference between all methods for R=28

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

NN0.01 − NN0.001 0.00E+00 4.62E-02 4.71E-02 3.09E-02 2.42E-02 2.08E-02 1.33E-02 6.82E-02 2.24E-03 8.21E-02 6.61E-02

Absolute error for z 1.00E+00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.00E-01 Absolute error

t

z RKButcher0.01 − RKButcher0.001 0.00E+00 1.64E-01 1.65E-01 1.08E-01 8.23E-02 6.58E-02 2.77E-02 2.48E-01 1.09E-02 2.87E-01 2.28E-01

NN RK Butcher

1.00E-02

1.00E-03 Time, t

Fig. 7. Absolute error of all methods for z

References 1. Murugesan, K., Sekar, S., Murugesh, V., Park, J.Y.: Numerical solution of an industrial robot arm control problem using RK-Butcher algorithm. International Journal of Computer Applications in Technology 19, 132–138 (2004) 2. Bader, M.: A comparative study of new truncation error estimates and intrinsic accuracies of some higher order Runge-Kutta algorithms. Comput. Chem. 11, 121–124 (1987) 3. Bader, M.: A new technique for the early detection of stiffness in coupled differential equations and application to standard Runge-Kutta algorithms. Theor. Chem. Acc. 99, 215–219 (1998) 4. Park, J.Y., Evans, D.J., Murugesan, K., Sekar, S., Murugesh, V.: Optimal control of singular systems using the RK -Butcher algorithm. Intern. J. Comp. Math. 81(2), 239–249 (2004)

296

J. Abdul Samath, P. Ambika Gayathri, and A. Ayisha Begum

5. Lee, H., Kang, I.: Neural algorithms for solving differential equations. J. Comput. Phys. 91, 110–117 (1990) 6. Yentis, R., Zaghoul, M.E.: VLSI implementation of locally connected Neural Network for Solving Partial Differential Equations. IEEE Trans. Circuits Syst. I 43(8), 687–690 (1996) 7. Wang, L., Mendel, J.M.: Structured Trainable Networks for Matrix Algebra. In: IEEE Int. Joint Conf. Neural Networks, vol. 2, pp. 125–128 (1990) 8. Joes, G., Freeman, I.: Theoretical Physics. Dover Publication, New York (1986) 9. Arfken, G., Weber, H.: Mathematical Methods for Physicists, 4th edn. Academic Press, New York (1995) 10. Press, W., Flannery, S., Teakolesky, S., Vetterling, W.: Numerical Recipies: The Art of Scientific Computing. Cambridge University Press, New York (1986)

Research on LBS-Based Context-Aware Platform for Weather Information Solution Jae-gu Song and Seoksoo Kim* Department of Multimedia, Hannam University, 133 Ojeong-dong, Daedeok-gu, Daejeon-city, Korea [email protected], [email protected]

Abstract. Mobile solutions using LBS (Location based Service) technology have significantly affected development of various application programs. Particularly, application technology using GPS-based location information and load map is essential to smartphone application. In this study, we designed a context-aware processing module using location and environment information in order to apply weather information to LBS-based services. This research is expected to provide various application services according to personal weather environments. Keywords: LBS, Context-aware, weather information, smartphone appliciation.

1 Introduction LBS-based service is a technology of providing various application services to users utilizing location information obtained from GPS or mobile communication network. The service is of growing interest as major application in the mobile communication industry, promoted by rapid technology development and fast distribution of PDA, smartphones, notebooks, and so on [1, 2]. LBS is a service of providing varied information related with user location through wireless communication, mainly used for effectively managing people or vehicles on the move. More recently, the service is used for locating those who need care and protection such as children and the elderly or tracking sex offenders. LBS involves previous technologies such as GIS (geographical information system), GPS (global positioning system), and telematics, applied to a vide range of areas. Basic technologies for providing LBS services include positioning technology, LBS platform technology for location-based services, and various LBS application technologies [3, 4]. In this study, we suggest LBS application technology, which is a solution technology for location-based services. The service is divided into a general consumer service and a corporate service while we carried out research on services for general consumers utilizing common information services. *

Corresponding author.

P. Balasubramaniam (Ed.): ICLICC 2011, CCIS 140, pp. 297–302, 2011. © Springer-Verlag Berlin Heidelberg 2011

298

J.-g. Song and S. Kim

2 Background Research 2.1 LBS Application Technology LBS offers an additional application service by combining location information of a user on the move with other information. Effective application service requires a mobile device, a wireless network, positioning technology, a solution for data processing, and content/application for providing additional services. More recently, smartphones support Wi-Fi based wireless services, making possible more complicated services to be offered [5]. Generally, the architecture of LBS platform is as follows, depicted in Figure 1. The system includes Positioning, Location Managing, Location based Function, Profile Management, Authentication and security, Location based billing, and Information roaming between carriers.

Fig. 1. Referential architecture of LBS Platform

2.2 Context-Aware Technology Context-awareness is software which is adaptive to places and surrounding people/objects as well as can accept changes of objects over time [6]. Such contextawareness technology is considered essential to processing data in various application service areas. Major research efforts are being made in areas of advertisement, shopping, and user preferences [7]. Context awareness technology should take into consideration the following elements.

Research on LBS-Based Context-Aware Platform for Weather Information Solution

299

Table 1. Categories of context Primary context Location(where) Identity(who)

Secondary context Person’s identity (age, phone number, address) Entity’s location (e.g. what object are near, what activity is occurring near the entity)

Time(when) Activity(what) Table 2. Another categories of Context External (physical) context Measured by h/w sensors (e.g. location, light, sound, movement, touch, temperature)

Internal (logical) context Specified by users or captured monitoring the user’s interaction (e.g. user’s goal, tasks, user’s emotional state)

In this study, we designed a context information processing module using primary context, weather information, and location information (based on the Web and GPS).

3 LBS-Based Context-Aware Platform for Weather Information Solution In this research, we designed a system of providing weather information to users by obtaining mobile location information. A user can employ a social network to send his context information and receive more accurate real-time weather information. Figure 2 shows the structure of LBS-based context-aware platform for weather information solution. The information of the system is largely composed of Weather information, Location information, Smartphone apps, and User information. The information is divided into user-based application service and system-based application through context information matching. User-based application is a method of responding to information frequently changing through user feedback in case patterns are not defined. For example, an alarm application program, combining information of destination and weather, provides users the estimated time required and receives feedback from users for application when a user needs to decide time to leave for an appointment. The program combines location and map information using GPS with sensor data in order to accurately find out locations and calculate the time required through weather and vehicle information. Then, the system asks user to confirm the time to depart.

300

J.-g. Song and S. Kim

Fig. 2. LBS-based Context-Aware Platform for Weather Information Solution