A Wavelet Tour of Signal Processing: The Sparse Way [3 ed.] 0123743702, 9780123743701

Table of contents :
Cover Page
A Wavelet Tour of Signal Processing: The Sparse Way (Third edition)
Copyright Page
9780123743701
Dedication
Contents
Preface to the Sparse Edition
Notations
1 Sparse Representations
1.1 Computational Harmonic Analysis
1.1.1 The Fourier Kingdom
1.1.2 Wavelet Bases
1.2 Approximation and Processing in Bases
1.2.1 Sampling with Linear Approximations
1.2.2 Sparse Nonlinear Approximations
1.2.3 Compression
1.2.4 Denoising
1.3 Time-Frequency Dictionaries
1.3.1 Heisenberg Uncertainty
1.3.2 Windowed Fourier Transform
1.3.3 Continuous Wavelet Transform
1.3.4 Time-Frequency Orthonormal Bases
1.4 Sparsity in Redundant Dictionaries
1.4.1 Frame Analysis and Synthesis
1.4.2 Ideal Dictionary Approximations
1.4.3 Pursuit in Dictionaries
1.5 Inverse Problems
1.5.1 Diagonal Inverse Estimation
1.5.2 Super-resolution and Compressive Sensing
1.6 Travel Guide
1.6.1 Reproducible Computational Science
1.6.2 Book Road Map
2 The Fourier Kingdom
2.1 Linear Time-Invariant Filtering
2.1.1 Impulse Response
2.1.2 Transfer Functions
2.2 Fourier Integrals
2.2.1 Fourier Transform in L^1(mathbb{R})
2.2.2 Fourier Transform in L^2(mathbb{R})
2.2.3 Examples
2.3 Properties
2.3.1 Regularity and Decay
2.3.2 Uncertainty Principle
2.3.3 Total Variation
2.4 Two-Dimensional Fourier Transform
2.5 Exercises
3 Discrete Revolution
3.1 Sampling Analog Signals
3.1.1 Shannon-Whittaker Sampling Theorem
3.1.2 Aliasing
3.1.3 General Sampling and Linear Analog Conversions
3.2 Discrete Time-Invariant Filters
3.2.1 Impulse Response and Transfer Function
3.2.2 Fourier Series
3.3 Finite Signals
3.3.1 Circular Convolutions
3.3.2 Discrete Fourier Transform
3.3.3 Fast Fourier Transform
3.3.4 Fast Convolutions
3.4 Discrete Image Processing
3.4.1 Two-Dimensional Sampling Theorems
3.4.2 Discrete Image Filtering
3.4.3 Circular Convolutions and Fourier Basis
3.5 Exercises
4 Time Meets Frequency
4.1 Time-Frequency Atoms
4.2 Windowed Fourier Transform
4.2.1 Completeness and Stability
4.2.2 Choice of Window
4.2.3 Discrete Windowed Fourier Transform
4.3 Wavelet Transforms
4.3.1 Real Wavelets
4.3.2 Analytic Wavelets
4.3.3 Discrete Wavelets
4.4 Time-Frequency Geometry of Instantaneous Frequencies
4.4.1 Analytic Instantaneous Frequency
4.4.2 Windowed Fourier Ridges
4.4.3 Wavelet Ridges
4.5 QuadraticTime-Frequency Energy
4.5.1 Wigner-Ville Distribution
4.5.2 Interferences and Positivity
4.5.3 Cohen\222s Class
4.5.4 Discrete Wigner-Ville Computations
4.6 Exercises
5 Frames
5.1 Frames and Riesz Bases
5.1.1 Stable Analysis and Synthesis Operators
5.1.2 Dual Frame and Pseudo Inverse
5.1.3 Dual-Frame Analysis and Synthesis Computations
5.1.4 Frame Projector and Reproducing Kernel
5.1.5 Translation-Invariant Frames
5.2 Translation-Invariant Dyadic Wavelet Transform
5.2.1 Dyadic Wavelet Design
5.2.2 Algorithme \340 Trous
5.3 Subsampled Wavelet Frames
5.4 Windowed Fourier Frames
5.4.1 Tight Frames
5.4.2 General Frames
5.5 Multiscale Directional Frames for Images
5.5.1 Directional Wavelet Frames
5.5.2 Curvelet Frames
5.6 Exercises
6 Wavelet Zoom
6.1 Lipschitz Regularity
6.1.1 Lipschitz Definition and Fourier Analysis
6.1.2 Wavelet Vanishing Moments
6.1.3 Regularity Measurements with Wavelets
6.2 Wavelet Transform Modulus Maxima
6.2.1 Detection of Singularities
6.2.2 Dyadic Maxima Representation
6.3 Multiscale Edge Detection
6.3.1 Wavelet Maxima for Images
6.3.2 Fast Multiscale Edge Computations
6.4 Multifractals
6.4.1 Fractal Sets and Self-Similar Functions
6.4.2 Singularity Spectrum
6.4.3 Fractal Noises
6.5 Exercises
7 Wavelet Bases
7.1 Orthogonal Wavelet Bases
7.1.1 Multiresolution Approximations
7.1.2 Scaling Function
7.1.3 Conjugate Mirror Filters
7.1.4 In Which Orthogonal Wavelets Finally Arrive
7.2 Classes of Wavelet Bases
7.2.1 Choosing aWavelet
7.2.2 Shannon, Meyer, Haar, and Battle-Lemari\351 Wavelets
7.2.3 Daubechies Compactly Supported Wavelets
7.3 Wavelets and Filter Banks
7.3.1 Fast Orthogonal Wavelet Transform
7.3.2 Perfect Reconstruction Filter Banks
7.3.3 Biorthogonal Bases of ell^2(mathbb{Z})
7.4 Biorthogonal Wavelet Bases
7.4.1 Construction of Biorthogonal Wavelet Bases
7.4.2 Biorthogonal Wavelet Design
7.4.3 Compactly Supported Biorthogonal Wavelets
7.5 Wavelet Bases on an Interval
7.5.1 Periodic Wavelets
7.5.2 Folded Wavelets
7.5.3 Boundary Wavelets
7.6 Multiscale Interpolations
7.6.1 Interpolation and Sampling Theorems
7.6.2 Interpolation Wavelet Basis
7.7 Separable Wavelet Bases
7.7.1 Separable Multiresolutions
7.7.2 Two-Dimensional Wavelet Bases
7.7.3 Fast Two-Dimensional Wavelet Transform
7.7.4 Wavelet Bases in Higher Dimensions
7.8 Lifting Wavelets
7.8.1 Biorthogonal Bases over Nonstationary Grids
7.8.2 Lifting Scheme
7.8.3 Quincunx Wavelet Bases
7.8.4 Wavelets on Bounded Domains and Surfaces
7.8.5 Faster Wavelet Transform with Lifting
7.9 Exercises
8 Wavelet Packet and Local Cosine Bases
8.1 Wavelet Packets
8.1.1 Wavelet PacketTree
8.1.2 Time-Frequency Localization
8.1.3 Particular Wavelet Packet Bases
8.1.4 Wavelet Packet Filter Banks
8.2 Image Wavelet Packets
8.2.1 Wavelet Packet Quad-Tree
8.2.2 Separable Filter Banks
8.3 Block Transforms
8.3.1 Block Bases
8.3.2 Cosine Bases
8.3.3 Discrete Cosine Bases
8.3.4 Fast Discrete Cosine Transforms
8.4 Lapped Orthogonal Transforms
8.4.1 Lapped Projectors
8.4.2 Lapped Orthogonal Bases
8.4.3 Local Cosine Bases
8.4.4 Discrete Lapped Transforms
8.5 Local Cosine Trees
8.5.1 Binary Tree of Cosine Bases
8.5.2 Tree of Discrete Bases
8.5.3 Image Cosine Quad-Tree
8.6 Exercises
9 Approximations in Bases
9.1 Linear Approximations
9.1.1 Sampling and Approximation Error
9.1.2 Linear Fourier Approximations
9.1.3 Multiresolution Approximation Errors with Wavelets
9.1.4 Karhunen-Loeve Approximations
9.2 Nonlinear Approximations
9.2.1 Nonlinear Approximation Error
9.2.2 Wavelet Adaptive Grids
9.2.3 Approximations in Besov and Bounded Variation Spaces
9.3 Sparse Image Representations
9.3.1 Wavelet Image Approximations
9.3.2 Geometric Image Models and Adaptive Triangulations
9.3.3 Curvelet Approximations
9.4 Exercises
10 Compression
10.1 Transform Coding
10.1.1 Compression State of the Art
10.1.2 Compression in Orthonormal Bases
10.2 Distortion Rate of Quantization
10.2.1 Entropy Coding
10.2.2 Scalar Quantization
10.3 High Bit Rate Compression
10.3.1 Bit Allocation
10.3.2 Optimal Basis and Karhunen-Loeve
10.3.3 Transparent Audio Code
10.4 Sparse Signal Compression
10.4.1 Distortion Rate and Wavelet Image Coding
10.4.2 Embedded Transform Coding
10.5 Image-Compression Standards
10.5.1 JPEG Block Cosine Coding
10.5.2 JPEG-2000 Wavelet Coding
10.6 Exercises
11 Denoising
11.1 Estimation with Additive Noise
11.1.1 Bayes Estimation
11.1.2 Minimax Estimation
11.2 Diagonal Estimation in a Basis
11.2.1 Diagonal Estimation with Oracles
11.2.2 Thresholding Estimation
11.2.3 Thresholding Improvements
11.3 Thresholding Sparse Representations
11.3.1 Wavelet Thresholding
11.3.2 Wavelet and Curvelet Image Denoising
11.3.3 Audio Denoising by Time-Frequency Thresholding
11.4 Nondiagonal Block Thresholding
11.4.1 Block Thresholding in Bases and Frames
11.4.2 Wavelet Block Thresholding
11.4.3 Time-Frequency Audio Block Thresholding
11.5 Denoising Minimax Optimality
11.5.1 Linear Diagonal Minimax Estimation
11.5.2 Thresholding Optimality over Orthosymmetric Sets
11.5.3 Nearly Minimax with Wavelet Estimation
11.6 Exercises
12 Sparsity in Redundant Dictionaries
12.1 Ideal Sparse Processing in Dictionaries
12.1.1 Best M-Term Approximations
12.1.2 Compression by Support Coding
12.1.3 Denoising by Support Selection in a Dictionary
12.2 Dictionaries of Orthonormal Bases
12.2.1 Approximation, Compression, and Denoising in a Best Basis
12.2.2 Fast Best-Basis Search in Tree Dictionaries
12.2.3 Wavelet Packet and Local Cosine Best Bases
12.2.4 Bandlets for Geometric Image Regularity
12.3 Greedy Matching Pursuits
12.3.1 Matching Pursuit
12.3.2 Orthogonal Matching Pursuit
12.3.3 Gabor Dictionaries
12.3.4 Coherent Matching Pursuit Denoising
12.4 l^1 Pursuits
12.4.1 Basis Pursuit
12.4.2 l1 Lagrangian Pursuit
12.4.3 Computations of l1 Minimizations
12.4.4 Sparse Synthesis versus Analysis and Total Variation Regularization
12.5 Pursuit Recovery
12.5.1 Stability and Incoherence
12.5.2 Support Recovery with Matching Pursuit
12.5.3 Support Recovery with l1 Pursuits
12.6 Multichannel Signals
12.6.1 Approximation and Denoising by Thresholding in Bases
12.6.2 Multichannel Pursuits
12.7 Learning Dictionaries
12.8 Exercises
13 Inverse Problems
13.1 Linear Inverse Estimation
13.1.1 Quadratic and Tikhonov Regularizations
13.1.2 Singular Value Decompositions
13.2 Thresholding Estimators for Inverse Problems
13.2.1 Thresholding in Bases of Almost Singular Vectors
13.2.2 Thresholding Deconvolutions
13.3 Super-resolution
13.3.1 Sparse Super-resolution Estimation
13.3.2 Sparse Spike Deconvolution
13.3.3 Recovery of Missing Data
13.4 Compressive Sensing
13.4.1 Incoherence with Random Measurements
13.4.2 Approximations with Compressive Sensing
13.4.3 Compressive Sensing Applications
13.5 Blind Source Separation
13.5.1 Blind Mixing Matrix Estimation
13.5.2 Source Separation
13.6 Exercises
APPENDIX Mathematical Complements
A.1 FUNCTIONS AND INTEGRATION
A.2 BANACH AND HILBERT SPACES
A.3 BASES OF HILBERT SPACES
A.4 LINEAR OPERATORS
A.5 SEPARABLE SPACES AND BASES
A.6 RANDOM VECTORS AND COVARIANCE OPERATORS
A.7 DIRACS
Bibliography
Books
Articles
Index
A, B
C
D
E, F
G, H, I
J, K, L, M
N, O, P
Q, R, S
T
U, V, W
Z

Citation preview

h

1

!!

A

of

Tour

Wavelet

Processing

Signal

The Sparse

Way

Mallat

Stephane

with

contributions

from

Gabriel Peyre

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\302\267 BOSTON

NEWYORK SAN

FRANCISCO Academic

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Contents

to

Preface

the

xv

Edition

Sparse

xix

Notations

CHAPTER

1 1.1

1.2

1.3

Representations

Sparse

\316\271

Harmonic

Computational

1.1.1

The

1.1.2

Wavelet Bases

Fourier

1

Analysis

2

Kingdom

2

and Processing in Bases Approximation 1.2.1 Sampling with Linear Approximations 1.2.2 Sparse Nonlinear Approximations

1.2.3

Compression

1.2.4

Denoising

5 7

8 11

11 14

Dictionaries

Time-Frequency

15

Uncertainty

1.3.1

Heisenberg

1.3.2

Windowed

1.33

Continuous

Fourier

Transform

16

Transform

17

Wavelet

Time-Frequency Orthonormal Bases Sparsity in Redundant Dictionaries 1.4.1 Frame Analysis and Synthesis 1.4.2 Ideal Dictionary Approximations 1.3.4

1.4

1.5

1.6

1.4.3

Pursuit

Inverse

Problems

1.5.1

Diagonal

1.5.2

Super-resolution

2

The

2.1

Linear

2.2

Fourier

26

Inverse Estimation and

27

Compressive

Sensing

28 30

Science

30

30 33

Kingdom

Time-Invariant

33

Filtering

2.1.1

Impulse

Response

33

2.1.2

Transfer

Functions

35

Fourier

Integrals

2.2.1

Fourier Transform in

2.2.2

Fourier

2.2.3 2.3

23 24

Reproducible Computational Book Road Map

1.6.2

21

Dictionaries

Guide

Travel

1.6.1

CHAPTER

in

19 21

35

L1(R)

in L2(R)

Transform

35 38 40

Examples

42

Properties

2.31

Regularity

2.32

Uncertainty

and

Decay

Principle

42 43

vN

viii

Contents

2.3.3

Two-Dimensional

2.5

Exercises

CHAPTER 3 3.1

Discrete

General

59

59

and

Linear Analog

and Transfer

Response

76

3.33

3.3.4

Fast

76

78

Convolutions

Discrete Image

79 80

Processing

3.4.1

Two-Dimensional

3.4.2

Discrete

Sampling Theorems

Meets

4.2

Windowed

4.4

4.5

4.6

85

89

Fourier Transform

92

Discrete

Wavelet

Transforms

4.3.1

RealWavelets Discrete Wavelets

4.4.2

Windowed

4.4.3

Wavelet Wigner-Ville

Geometry Instantaneous Fourier

of Instantaneous Frequency Ridges

112 ... 115 Frequencies 115 118 129

Ridges

Energy

4.5.3

4.5.4

Discrete

Wigner-Ville

134 136

Distribution

Interferences and Cohen's Class

Exercises

101

107

Wavelets

Time-Frequency

Quadratic

Transform

103

Analytic

Analytic

98 Fourier

102

4.3.3 4.4.1

94

Stability

Windowed

4.3.2

Time-Frequency

83

89

4.2.3

4.5.2

Basis

Atoms

4.2.2

4.5.1

Fourier

Frequency

and Completeness Choice of Window

4.2.1

80 82

Filtering

Image

Circular Convolutionsand Exercises Time

70 70

75

Discrete Fourier Transform Fast Fourier Transform

Time-Frequency

65

72

Convolutions

Circular

4.1

4.3

Function

Finite Signals

3.4.3

4

Conversions ..

Filters

Series

Fourier

33.2

CHAPTER

59

61 Sampling

Impulse

3.31

3.5

Theorem

Sampling

Time-Invariant

Discrete

3.2.1 3.2.2

3.4

Revolution

Aliasing

3.1.3

3.3

51

Analog Signals Shannon-Whittaker

3.1.2 3.2

Fourier Transform

55

Sampling

3.1.1

46

Variation

Total

2.4

Positivity

140

145 Computations

149

151

Contents

CHAPTER

5

Frames

5.1

Frames

5.2

5.1.1

Stable

5.1.2 5.1.3

Dual Frame

5.1 4

Frame

5.1.5

Translation-Invariant

5.2.2

5.4

CHAPTER

Computations...

161

Kernel

166

Dyadic Dyadic Wavelet Design

Wavelet

Algorithme

178 181

Frames

183

Tight Frames

184

Frames

General

for

188

Images

189

Frames

194 201

Zoom

205 205

Regularity

Lipschitz Definition

6.1.2 6.1.3 Wavelet

Wavelet Vanishing Moments Regularity Measurements with Wavelets Transform Modulus Maxima

6.2.1

Detection

6.2.2

Dyadic

Multiscale

Maxima

6.5

Exercises

7

Wavelet

Sets

and

211 218

218 224 230 230

Images

239

Computations

Self-Similar

242 242

Functions

246

Spectrum

Singularity

205 208

Representation

254 259

Bases Wavelet

263 263

Bases

264

MultiresolutionApproximations

7.1.2

Scaling

7.1.3

Conjugate Mirror Filters

7.1.4

In

Classes

ofWavelet Bases

7.2.1

Analysis

Fractal Noises

Orthogonal

7.1.1

Fourier

Detection

Edge

Fractal

and

of Singularities

Wavelet Maxima for Fast Multiscale Edge Multifractals

6.4.3

7.2

175

Frames

6.1.1

6.4.1 6.4.2

7.1

172

aTrous

6.3.1 6.32

CHAPTER

170

Exercises

Lipschitz

6A

168 Transform

Wavelet

Multiscale Directional Frames Directional Wavelet 5.5.1 Curvelet Frames 5.5.2

6.1

6.3

and Synthesis and Reproducing Frames

Subsampled Fourier

6

6.2

Projector

Windowed

5.4.2

5.6

155 159

Analysis

Wavelet

5.4.1

5.5

and

Dual-Frame

Synthesis Operators Pseudo Inverse

and

Analysis

Translation-Invariant 5.2.1

5.3

155 155

Bases

Riesz

and

Which

Choosing

267

Function

Orthogonal aWavelet

Wavelets

270 Finally

Arrive

278

284 284

\317\207

Contents

Shannon, Meyer, Haar, and Battle-Lemarie Daubechies Compactly Supported

7.2.2 7.2.3 7.3

7.5

Fast

Perfect Reconstruction 7.32 7.3.3 Biorthogonal Bases of l2(Z) Bases Biorthogonal Wavelet Construction of Biorthogonal 7.4.1 7.4.2 Biorthogonal Wavelet Design

7.4.3

Compactly

Wavelet

Bases

7.5.1

Periodic

7.5.2

Folded Wavelets

7.5.3 7.6

Separable Two-Dimensional

7.7.3 7.7.4

Fast Two-Dimensional Wavelet Bases

Lifting

Sampling

333 338

338

Wavelet in

Transform

350

Biorthogonal Bases over Lifting Scheme Quincunx

Nonstationary

359 with

and

Wavelet

Wavelet

and

Packet

Local

Cosine

Time-Frequency

8.1.3 8.1.4

Particular Wavelet Packet Wavelet Packet Filter Banks

Image Wavelet Wavelet Separable

377

383

Localization Bases

Packet Quad-Tree Filter

388 393 395

Packets

Banks

Block Transforms

395 399 400

Block Bases Cosine

Bases

377

Packet Tree

8.1.2

8.3.4

367

377

Packets

Wavelet

8.33

361 370

8.1.1

8.3.2

Surfaces

Lifting

Exercises

8.1

350

352

Domains

Transform

Wavelet

Grids

Bases

Wavelet

Faster

8.3.1

348

Dimensions

Higher

7.8.5

8.2.2

340 346

Bases

Wavelet

Wavelets on Bounded

8.2.1

328

Basis

Wavelet

7.8.4

8

8.3

328 Theorems

Wavelets

7.8.3

8.2

322

Multiresolutions

7.7.2

7.8.1 7.8.2

CHAPTER

320

and

Interpolation Separable Wavelet Bases

7.7.1

7.9

318

Wavelets

Interpolation

313

317

Wavelets

Boundary

308

311 Wavelets

Interval

7.6.2

7.8

308 Bases

Wavelet

Multiscale Interpolations

7.6.1 7.7

298 302 306

Biorthogonal

Supported

on an

289

298 Transform Filter Banks

Wavelet

Orthogonal

.

292

and Filter Banks

Wavelets

7.3.1 7.4

Wavelets Wavelets

Bases

Discrete Cosine Bases Fast Discrete Cosine Transforms

401 403 406 407

Contents

8.4

8.5

Orthogonal

Lapped Projectors

8.4.2

Lapped

8.4.3

Local Cosine

8.4.4

Discrete

Orthogonal

8.5.1

8.6 CHAPTER

419

422 426

Binary 8.5.2 Tree of Discrete Cosine 8.5.3 Image Exercises Approximations

9.1

Linear

426 429

Bases

Cosine

of

Bases

429

Quad-Tree

432

in Bases

435 435

Approximations

911 9.12

Sampling and Approximation Linear Fourier Approximations

9.1.3

Multiresolution

435

Error

438

Approximation

Errors

442

Wavelets

with

9.1.4

446

Karhunen-LoeveApproximations

Nonlinear

9.2.2 92.3

Nonlinear Approximation Error Wavelet Adaptive Grids in Besov and Bounded Approximations

451

Variation

459

455

Spaces

463

Representations

Image

Sparse

450

Approximations

92.1

9.3

416

Bases

Lapped Transforms Tree

9

9.2

410 Bases

Trees

Cosine

Local

410

Transforms

Lapped

8.4.1

931

Wavelet

9.32

Geometric

464

Approximations

Image

Image

and

Models

Adaptive

471

Triangulations

933 9.4 CHAPTER

10

Curvelet

476

Approximations

Exercises

478

Compression

48i

10.1

Transform

482

10.2

10.1.1 Compression State of theArt 10.1.2 Compression in Orthonormal Bases Distortion Rate of Quantization

10.2.1

Entropy

485

10.2.2

Scalar Quantization

10.3

High

Bit

10.3.1 10.32 10.3.3

10.4

481

Coding

Sparse

Rate Bit

483

485

Coding

493 496

Compression

496

Allocation

Optimal Basis and Karhunen-Loeve Audio Code Transparent Signal

498

501 506

Compression

10.4.1

Distortion

Rate

10.4.2

Embedded

Transform

and

Wavelet Coding

Image

Coding

506 516

xii

Contents

10.5

10.5.2

Wavelet

JPEG-2000

523

Coding

Exercises

531

11

Denoising

535

11.1

Estimation with Additive Noise 11.1.1 Bayes Estimation

535

11.1.2

544

10.6

11.2

11.3

Wavelet

12.2

568 Thresholding

Block

Audio

DenoisingMinimax

..571 575

Frames

575

581

Thresholding

582 585

Optimality

11.5.1 11.5.2

Linear Diagonal Minimax Estimation Thresholding Optimality over

587

Orthosymmetric Sets

590

11.5.3

Nearly

Minimax with

Estimation

Wavelet

Sparse

611

in Dictionaries

Processing

12.1.1

Best

12.1.2

Compression by

M-Term

612

Approximations

614

Coding

Support

12.1.3 Denoising by Support Selection Dictionaries of Orthonormal Bases

Compression,

Approximation,

in a Best

611

Dictionaries

in Redundant

Sparsity Ideal

in a Dictionary

621

622

Basis

12.2.3

12.2.4

Bandlets

Greedy

Matching

12.3.1

Matching Pursuit

for

6l6

and Denoising

Fast Best-Basis Search in Tree Dictionaries Wavelet Packet and Local Cosine Best Bases

12.2.2

595

606

Exercises

12.2.1

12.3

563 Denoising

Block Thresholding in Bases and Thresholding Wavelet Block Thresholding

11.4.3 Time-Frequency

12.1

562

Thresholding

Block

11.4.2

12

558

Sparse Representations

11.33 Nondiagonal 11.4.1

11.6

548

552

Wavelet and Curvelet Image Audio Denoising by Time-Frequency

11.32

11.5

548 Oracles

Improvements

Thresholding

Thresholding

11.3.1 11.4

536

Estimation

Minimax

Diagonal Estimation in a Basis 11.2.1 Diagonal Estimation with 11.2.2 Estimation Thresholding

11.2.3

CHAPTER

519

JPEG Block Cosine Coding

10.5.1

CHAPTER

519

Standards

Image-Compression

Geometric

Image

Regularity

Orthogonal

12.33

Gabor Dictionaries

12.3.4

Coherent

Matching

Matching

626 631 642

Pursuits

12.32

623

642 648

Pursuit

650 Pursuit

Denoising

655

Contents

12.4

l1

659

Pursuits

12.4.1

Basis Pursuit

12.4.2

l1 Lagrangian

659 664

Pursuit

12.4.3 Computations of l1 Minimizations 12.4.4 Sparse Synthesis versus Analysis Variation

12.5

Pursuit

Total

673

Regularization

677

Recovery

677

Stability and Incoherence

12.5.1

12.5.2 12.5.3 12.6

668 and

Support Support

Multichannel

Recovery

with

Recovery

with

Matching l1 Pursuits

679 684

Pursuit

688

Signals

12.6.1

and Denoisingby

Approximation

Thresholding

689

in Bases

690

12.6.2 Multichannel Pursuits

CHAPTER

12.7

Learning

12.8

Exercises

13

Inverse

131

Linear

Problems

Thresholding

13.2.2

Sparse

Super-resolution

13.3.2

Sparse

Spike

Recovery Compressive Sensing

Blind

Source

of

719 722

Deconvolution Missing

Data

728 729 Sensing

735

742 744

Separation

Blind Mixing Matrix Source Separation

Estimation

745

751 752

Exercises

Mathematical

703

713

Estimation

with Random Measurements with Compressive Approximations Sensing Applications Compressive

13.5.2

...

709

Incoherence

13.5.1

702 703

713

13.3.1

13.4.3

APPENDIX

700 700

Regularizations

Super-resolution

13.4.2

13.6

Tikhonov

Deconvolutions

Thresholding

13.4.1

135

and

Singular Value Decompositions Estimators for Inverse Problems in Bases of Almost Singular Vectors Thresholding

13.3.3

13.4

699

Estimation

Quadratic

13.2.1 13.3

696

Inverse

13.11 I3.I.2

132

693

Dictionaries

Complements

753

Bibliography

765

Index

795

I

the

to

Preface

Edition

Sparse

communities and but find striking resemblances between scientific in conferences We interact and through articles, and we move while a global from individual contributions. Some of emerges trajectory

cannot

help

of

schools

together us like

fish.

to be at

swim in

the center

multiple

school, others prefer to wander To avoid dying by starvation

of the

in front.

directions

around,

and

a few

in a progressively

scientific needs also to move on. community is still much alive it went beyond because Computational analysis very wavelets. Writing such a book is about decoding the trajectory of the school and that have been uncovered on the way. Wavelets are no longer gathering the pearls the central the edition's title. It is original topic, despite previous just an important as the Fourier transform is. Sparse representation and processing are now at tool,

and

narrower

domain, a

specialized harmonic

the core. In the decompositions,

were focused on building time-frequency the uncertainty barrier, and hoping to discover the ultimate bases, orthogonal way came the construction of wavelet

researchers

1980s,

many

trying

to avoid

the representation. Along which opened new perspectives

with physicists and collaborations with Xlets became a popular sport with and Connections with approximations compression applications. and also became more apparent. The search for has taken over, sparsity sparsity where orthonormal bases are replaced by redundant leading to new grounds mathematicians.

Designing

through

bases

orthogonal noise-reduction

of waveforms.

dictionaries

During these last seven years, a lot of naiveness, some bandlets, with

that

learn

that

algorithms

to

Bernard, Jerome in three months in real time, operate

Christophe

time to

is a

mathematics

Semiconductor processing.

algorithms

trial-and-error

and data

good engineering should as

to

opposed

promising of

process.

industrial

Sparsity

it brings

by increasingly

required

With

world.

I cofounded a start-up mathematics, Erwan Le Pennec. It took us some robust

produce

three years we were used Yet, we survived perspectives. innovations for signal

the

offers amazing computational power and often do not scale easily and mathematics

development It is

and

Kalifa,

with

ideas

communications.Although

is not a luxury.

more

major source

technology

ad hoc

However, the

new

for writing

having

because

the industrial

encountered

I also and

decreases

beauty, sophisticated

flexibility. accelerates

computations,

mathematical

memory,

understanding

information-processing

devices.

New

Additions

Putting sparsity adding sections. representations

in

at

the

Chapters

redundant

of the book implied rewriting many parts 13 are new. They introduce sparse and inverse problems, dictionaries, super-resolution, center 12

and

and

and

XV

xvi

to the

Preface

Edition

Sparse

compressive sensing.

is

Here

a

small

in

of new elements

catalog

this

third

edition:

and tomography

\342\226\240 Radon

transform

\342\226\240 Lifting

for wavelets

\342\226\240 JPEG-2000 \342\226\240 Block

on surfaces, bounded domains,

fast

computations

compression

image

for denoising

thresholding

\342\226\240 Geometric

and

with

representations

adaptive triangulations, curvelets,

and

bandlets \342\226\240 Sparse \342\226\240 Noise

reduction

\342\226\240 Exact

recovery

\342\226\240 Dictionary

of sparse

dictionaries

approximation supports in dictionaries and processing

representations

signal

in redundant

algorithms

learning

\342\226\240 Inverse

and super-resolution

problems

\342\226\240 Compressive \342\226\240 Source

model selection

with

\342\226\240 Multichannel

dictionaries with pursuit

in redundant

approximations

sensing separation

Teaching

This book is intended of teaching

as

courses in

a graduate-level electrical

textbook. and

engineering

Its evolution is also the result mathematics. A new applied

software for reproducible solutions, experimentations, exercise software teaching material such as slides with figures and MATLAB classes of http://wavelet-tour.com. More exercises have been added at the end of each chapter, ordered by level of difficulty Level1 exercises are direct applications of the course. Level2 exercises includes some technical derivation exercises. Level4 requires more thinking. Level3 are projects at the interface of research that are possible topics for a final course in the More exercises and projects can be found project or independent study.

website

provides

with together for numerical

website.

Sparse

Course Programs

Fourier

The

approximations It

introduces

and

and

analog-to-digital

conversion through

a common ground for provide and basic signal representations

all

courses

reviews

the way.

linear

sampling

(Chapters

2 and

important

afterward. algorithmic tools needed Many trajectories and teach sparse signal The following processing. that can orient a course's structure with elements that

explore topics

transform

3).

mathematical

are then possible to list notes several can be covered along

Sparse

of linear

\342\226\240 Lipschitz

7) and nonlinear

of linear

\342\226\240 Properties

basis

wavelet

6)

wavelet

\342\226\240 Linear

and

representations: wavelet and windowed

time-frequency

\342\226\240 Time-frequency

9)

(Chapter

approximations

compression (Chapter 10) nonlinear diagonal denoising (Chapter

\342\226\240 Image

Edition

9)

(Chapter

regularity bases (Chapter

\342\226\240 Wavelet

Sparse

and applications: and nonlinear approximations in bases and wavelet coefficients decay (Chapter

Sparse

bases

with

representations

\342\226\240 Principles

the

to

Preface

11)

Fourier

ridges

for

audio

processing

(Chapter 4) \342\226\240 Local

cosine

\342\226\240 Linear

and

\342\226\240 Audio \342\226\240 Audio

bases (Chapter 8) nonlinear approximations

Sparse

(Chapter

9)

11) thresholding (Chapter in redundant time-frequency bases or pursuit algorithms (Chapter 12)

and

\342\226\240 Bayes \342\226\240 Wavelet \342\226\240 Linear

and linear versus

minimax

versus

7) (Chapter and nonlinear approximations

nonlinear estimations

(Chapter

bases

in

bases

(Chapter

9)

estimation

\342\226\240 Thresholding \342\226\240 Minimax

dictionaries

denoising

optimality selection for

\342\226\240 Compressive

sensing

Sparse compression

11) (Chapter 11) (Chapter

denoising in redundant 13) (Chapter information

and

dictionaries

(Chapter

theory:

(Chapter 7) 9) approximations in bases (Chapter \342\226\240 and sparse transform codes in bases Compression (Chapter \342\226\240 in redundant dictionaries (Chapter 12) Compression \342\226\240 13) sensing Compressive (Chapter \342\226\240 Source 13) separation (Chapter orthonormal

\342\226\240 Wavelet \342\226\240 Linear

and

bases

nonlinear

Dictionary representations and

\342\226\240 Frames \342\226\240 Linear

and

and

inverse

Riesz bases (Chapter 5) nonlinear approximations

problems:

in

bases

(Chapter

9)

dictionary approximations (Chapter 12) \342\226\240 Pursuit and dictionary incoherence (Chapter 12) algorithms \342\226\240 Linear and thresholding inverse estimators (Chapter 13) \342\226\240 and source separation (Chapter 13) Super-resolution \342\226\240 Ideal

redundant

\342\226\240 Compressive

with

estimation:

signal

\342\226\240 Model

bases

10)

and block

denoising

\342\226\240 Compression

best

(Chapter

compression

in

sensing

(Chapter

13)

10)

12)

11)

xvii

xviii

to the

Preface

Edition

Sparse

Geometric sparse

processing:

lines and

4) ridges (Chapter 5) (Chapter \342\226\240 Multiscale with wavelet maxima (Chapter edge representations \342\226\240 in bases (Chapter 9) Sparse approximation supports \342\226\240 with and bandlets curvelets, geometric Approximations regularity, and 12) \342\226\240 and geometric bit budget (Chapters signal compression Sparse \342\226\240 Exact of 12) recovery sparse approximation supports (Chapter \342\226\240 Time-frequency

spectral

Riesz bases

and

\342\226\240 Frames

\342\226\240 Super-resolution

6)

9

(Chapters 10

and

12)

13)

(Chapter

ACKNOWLEDGMENTS

Some things do ones who were, grateful to I spent

the last

few

\"startup.\"Pressure

blend

of

remain,

Bajcsy

what

new editions, for me important Yves Meyer. with three brilliant

with

change

and

Ruzena

Jerome

Bernard,

not

and years

Kalifa, and Erwan means stress, despite all of us could provide,

Le

in particular

Pennec\342\200\224in

and

kind

a pressure

very good moments. The which

brought

the As

references.

left by

traces

always,

I am

the

deeply

colleagues\342\200\224Christophe

cooker

called a

resulting

sauce

new flavors

was a

to our personalities.

thankful to them for the ones I got, some of which I am still discovering. This new edition is the result of a collaboration with Gabriel Peyre, who made these not only possible, but also very interesting to do. I thank him for his changes I am

remarkable work

and

help.

Stephane

Mallat

Notations

(f,g)

Inner

11/11

Euclidean

||/111

L1 or

f[n]

or Hubert

norm

space

l1 norm

L00 norm

||/1|oo

f[n]

(A.6)

product

= 0(g[n])

Order

=

Small

o(g[n])

f[n] ~g[n]

Equivalent

\320\233>

A

\316\266*

\\x\\

Largest Smallest

(x)+

max

\316\267 mod

N

of: to:

finite is much

=

f[n]

= 0

|gj

and g[n] = 0(f[n])

0(g[n]) \320\222

of

conjugate

integer integer

such that/[/z] ^Kg[n]

lim\342\200\236-+oo

bigger than

Complex

lx}

exists K

of: there

order

\316\266 e \320\241

^ \317\207 \316\267 \316\267 ^\317\207

(\317\207, \316\237)

of the integer

Remainder

of

division

Sets N

Positive integers including 0

\320\252

Integers

R

Real numbers

R+

Positive

N

\316\267 modulo

numbers

real

numbers

\320\241

Complex

|A |

Number of elements

in a set

A

Signals

fit)

Continuous

f[n]

Discrete

time

signal

signal distribution

\316\264 (t)

Dirac

\316\264[\316\267]

Discrete

l[a,b]

Indicator

Dirac of

(A. 30) (3.32)

a function

that is

I in [a,

b]

and

0 outside

Spaces continuous

Co

Uniformly

Cp

p times

C00

Infinitely

differentiable

W5(R)

Sobolev5

times

continuously

functions (7.207) differentiable functions

LP(R)

functions functions (9.8) oo Finite energy functions / \\f(t) |2 dt < + Functions such that / \\f(t)\317\210 dt

vector (6.51)

Gradient

f*g(t)

Continuous time

f*g[n]

Discrete

convolution

(333)

f\302\256g[n]

Circular

convolution

(373)

convolution

(2.2)

Transforms /(\317\211)

Fourier

/\342\204\226]

Discrete

Sf(u,

s)

Psfiu, Wf(u,

\316\276)

s)

Pwfiu^)

transform (2.6), (3.39) transform

Fourier

Short-time windowed (4.12) Spectrogram

Wavelet transform Scalogram

(3.49)

Fourier

transform

(4.11)

(4.31)

(4.55)

Wigner-Ville distribution

\316\241\316\275\316\257(\316\274,\316\276)

(4.120)

Probability

X

Random

E{X}

Expected

\320\251\320\245)

Entropy (10.4)

\320\250\320\245)

Differential

entropy

Cov(XbX2)

Covariance

(A.22)

variable

value

F[n]

Random vector

RP\\k\\

Autocovariance

of

(10.20)

a stationary

process (A. 26)

CHAPTER

Representations

Sparse

\316\271

in a

overwhelming to find than

carry

Signals

more

difficult

representation

sparse

amounts of data in which relevant information is often a needle in a haystack. is faster and Processing simpler where few coefficients reveal the information we are

signals over by decomposing called a But the search for elementary family dictionary. the Holy Grail of an ideal sparse transform adapted to all signals is a hopeless quest. The of wavelet orthogonal bases and local time-frequency dictionaries has discovery the door to a of new transforms. huge jungle opened Adapting sparse representations to signal and deriving efficient is therefore a properties, processing operators, survival necessary strategy. An orthogonal of minimum size that can yield a sparse basis is a dictionary if designed representation to concentrate the signal over a set of few vectors. This energy set gives a geometric Efficient and noisesignal description. signal compression reduction algorithms are then implemented with diagonal operators computed with fast algorithms. But this is not always optimal. In natural languages, a richer dictionary helps to build shorter and more precise sentences. Similarly, dictionaries of vectors that are larger than are needed bases to build of But is difficult and choosing sparse representations complex signals. in more redundant dictionaries algorithms. requires complex Sparse representations can improve and noise reduction, but also the recognition, pattern compression, resolution of new inverse problems. This includes superresolution,source separation, for. Such

looking

representations can

and

first

the main

1.1

N^

implemented

oscillatory

to

sparse

106, and with

0(N

orientation

HARMONIC

and wavelet

over

a path size

sparse book representation,

a sense of

It gives

COMPUTATIONAL

Fourier signals

is a

chapter

ideas.

constructed

sensing.

compressive

This

be

in a

chosen

waveforms

bases

are

the

waveforms representations.

thus can only logN)

for

the

providing

choosing

a path

story

line and

to travel.

ANALYSIS

starting point. They decompose journey's that reveal many signal and provide properties Discretized signals often have a very large

be

operations

processed

by fast

and memories.

Fourier

algorithms, typically and

wavelet

transforms

2

1

CHAPTER

illustrate the fast

Representations

Sparse

strong connection

algorithms.

Fourier

The

1.1.1

The Fourier

Kingdom is everywhere

transform

nalizes time-invariant

convolution

in physics and mathematics It rules over linear operators.

the building blocks of which \320\266\320\265 frequency Fourier analysis represents any finite function energy

processing,

/(0 The amplitude

/(\317\211)

\342\200\224

/

sinusoidal

each

of

=

of

sinusoidal

more

(1.1)

f(co)ei(0tdco.

wave

-00 /+00

The

as a sum

etcot

is equal

to its

correlation with /,

transform:

Fourier

called

also

signal

operators.

filtering

fit)

it diago-

because time-invariant

ela)t:

waves

when

tools and

mathematical

well-structured

between

fit), the

regular

faster

the

decay

(1.2)

f(t)e-iu>tdt. of the

sinusoidal

wave

amplitude

|/(\317\211)|

\317\211 increases.

frequency When fit)

is defined only on an interval, transform say [0, 1], then the Fourier in a Fourier a decomposition orthonormal basis {et27rmt}mez of L2[0, 1]. If fit) is uniformly transform coefficients also have a fast regular, then its Fourier when the 2\321\202\320\263\321\202 so it can be increases, decay frequency easily approximated with few Fourier coefficients. The Fourier transform therefore defines a low-frequency of functions. regular sparse representation uniformly Over discrete signals, the Fourier transform is a decomposition in a discrete Fourier of which has properties similar to a basis CN, {et27Tkn/N}o^k8m)gm \316\247\316\273\317\204=

for

(W,gm}

meT.

an orthogonal

yields

AT =

with

{meT :

projection

estimator (1.9)

\\(X,gm)\\^T}.

meAr

The set A^ is

the

coefficients,

of

estimate

1.2(b) shows the estimated | {X, ty,n\\ ^ T, that can be

in

A^ shown

1.1(b).

Figure

support of /. : \\(f,gm)\\^T). e {m \320\223

an approximation support

approximation

optimal

Figure

an

The

At =

set At

approximation to

compared

the

optimal

close to

of noisy-wavelet approximation support

in Figure 1.2(d)

estimation

is hopefully

It

from

wavelet

in regular At has considerably reduced the noise regions while keeping the sharpness of edges coefficients. This estimation is by preserving large-wavelet with a translation-invariant that this estimator over averages improved procedure several translated wavelet an bases. Thresholding wavelet coefficients implements which the data X with a kernel that on the averages adaptive smoothing, depends coefficients

in

estimated

of the

signal /. that for Gaussian white noise of variance \317\2032, = \316\244 a risk of the order of to ~F\\\\2} choosing yields ||2,up E{\\\\f \\\\f -fAr cry21ogeN a loge N factor. This result shows that the estimated At does spectacular support risk is small if the nearly as well as the optimal unknown support A^. The resulting regularity

Donoho and

Johnstone

original proved

and representation is sparse precise. The set A^ in Figure \"looks\" different from the A^ in Figure 1.2(b) 1.1(b) This indicates that some prior information because it has more isolated points. on the geometry of At could be used to improve the estimation. For audio noisein sparse representations reduction, thresholding estimators are applied provided by Similar isolated time-frequency coefficients produce a highly bases. time-frequency \"musical noise.\" Musical noise is removed with a block that thresholding annoying the of the estimated and avoids isolated At regularizes leaving geometry support wavelet estimators. points. Block thresholding also improves If IF is a Gaussian in B, then noise and signals in \316\230 have a sparse representation 11 that estimators can a minimax risk. thresholding Chapter proves produce nearly In particular, wavelet thresholding estimators have a nearly minimax risk for large classes of piecewise smooth signals, variation images. bounded including

1.3 Motivated

TIME-FREQUENCY by

decomposing

DICTIONARIES

mechanics, signals over dictionaries

quantum

in 1946 of

the

physicist

elementary

Gabor

waveforms

[267]

proposed

which he

called

1.3

atoms that have a minimal

time-frequency

By showing that

such

are

decompositions

in

spread

a time-frequency

related to

closely

15

Dictionaries

Time-Frequency

our

plane. of

perception

in speech and music recordings, sounds, they exhibit important structures demonstrated the importance of localized time-frequency Gabor signal as sums of sounds, processing. large classes of signals have sparse Beyond decompositions atoms selected from dictionaries. The time-frequency appropriate key issue is to understand how to construct dictionaries with time-frequency atoms to adapted signal properties.

and that

1.3.1

Heisenberg

A time-frequency

=

Uncertainty

V=

dictionary

localization

are

defined

the frequency

localization

=

and

\316\254\317\211 and \316\276(2\317\204\317\204\316\2231 \317\211\\\317\206\316\216(\317\211)\\

J

Fourier

norm

by

= /

and

The

time

in

dt.

\320\263\\\321\204\321\203{\320\263)\\2\321\2011\320\263 \\\320\263-\320\270\\2\\\321\204\321\203(\320\263)\\2 ojy

u=f Similarly,

of waveforms of unit and frequency. The time

is

composed {\321\204\321\203}\321\203\320\265\321\202

which have a narrow localization \320\270 of \321\204\321\203 and its spread around u,

1, || ||\317\206\316\263

Parseval

of

spread

defined

are

by

\321\204\321\203

=

\342\200\224 \316\231 \316\254\317\211. \\\317\211 \316\276\\ \\\317\206\316\216{\317\211)\\

(2\317\200) \317\203\317\211 y \317\211,\321\203

formula ,+\316\277\316\277

(/,

shows

that

(/,

\342\226\240 = /(\316\257)\317\210*(\316\257)\316\233 \320\244\321\203) -\316\257

plane

FIGURE

mostly

depends \321\204\321\203)

on the

hence for (t, nonnegligible This is illustrated rectangle at,y \316\247\317\203\317\211>\316\263. It can be interpreted as a \"quantum (t, \317\211). , and

are (\317\211) \317\206 \321\203

size

I

1.3

Heisenberg

box representing

an

atom

\317\206\316\263.

/(\317\211)\317\206*(\317\211)\316\254\317\211(1.10)

values f(t) in a \317\211)

and

rectangle

by Figure 1.3 of information

/(\317\211), where

centered

in

this

and

\317\206\316\216(\316\257)

at (u, \316\276), of

time-frequency

\"over an

elementary

CHAPTER 1 Sparse

cell. The

resolution

Representations

(see 2) that this principle theorem proves Chapter that limits the joint time-frequency resolution:

uncertainty

rectangle has a minimum surface

1 -

(1.11)

0\"t,y \302\260Oj,y

of time-frequency atoms can thus be thought of as with resolution cells a time width having time-frequency plane at, y and a frequency width \317\203\317\211,\316\216 which with a surface than one-half. but larger may vary Windowed Fourier and wavelet transforms are two important examples. a dictionary

Constructing

covering the

A

is constructed by dictionary norm ||g|| = 1, centered

Fourier

windowed

Transform

Fourier

Windowed

1.3.2

a time window g(t),

V=\\gu^(t)=g{t-u)^t\\ l The

is translated by

atomg^

in time \320\270

and by

of \320\270 and

.This \316\276

spread ofgu^ is independent

to a Heisenberg as

shown

at

J

transform

Fourier

Sf(u,\302\243)

(u. \316\276)\302\243\316\210

The time-and-frequency that each atomg^ corresponds \316\247 of its position (\316\267,\316\276), \317\203\317\211 independent in frequency. \316\276

means

=

/ on each

projects

image

in

redundant It is (\316\267,\316\276).

frequency

coefficients

and

one-dimensional

represents

thus necessary that

represent

gu^\\

\317\206

dt.

f(t)g(t-u)e

(f,gi

atom

dictionary

It can be interpreted as a Fourier transform of / at the frequency the window g(t \342\200\224 of u. This windowed u) in the neighborhood is highly

and frequency

= 0:

1.4.

by Figure

The windowed

has a size

that

rectangle

at t

time

in

translating

of unit

signals

to understand how to the signal efficiently

(1.12) localized \316\276,

by a

select

time-frequency

many

fewer

\320\226\321\202\320\234 ]|\317\203.

\302\261\316\276(\317\211)\\ ^L ]|\317\203.

IW)I

I&^WI

FIGURE

1.4

boxes (\"Heisenberg Time-frequency windowed Fourier atoms.

rectangles\")

representing

the energy

by

transform

Fourier

spread

of

two

time-

1.3

listening to music, we 4 shows that Chapter

When time.

in

These

components

spectral

maxima in this

are local

evolution

the

of

signal

high-amplitude

on the

the

modifying

time-frequency

sound duration of the

geometry

or

audio

ridge support in

frequency.

A windowed

the

and

time

same

not include

Fourier transform decomposes resolution. It is thus frequency

structures

by changing the

different

having

and others

in time

localized

and

time

very

as long

effective

time-frequency in frequency.

localized

that have as the signal does

waveforms

over

signals

resolutions, some being very Wavelets address this issue

resolution.

frequency

Continuous Wavelet Transform

1.3.3 In

/ with a geometry that depends spectral components. Modifying the

are implemented by

transpositions time

time-frequency

frequency that varies

that on time u. (\320\270) frequencies \316\276 depend and characterized by ridge which points, define a Ridge points plane. time-frequency

\320\233 of

support

approximation

spectral

have a

/ creates

of

line

at Sf(u,%) are detected

coefficients

Fourier

windowed

a

that

sounds

perceive

17

Dictionaries

Time-Frequency

reflection

at high

long

Instead of emitting pulses waveforms

Such

layers.

geophysical

spaced

that the

Morlet knew

seismology,

duration that is too

of

frequencies waveforms

separate

are

called

he thought

duration,

equal

waveforms

to

sent the

have a underground of fine, closely

returns

in

wavelets

of sending

geophysics.

shorter

These waveforms were obtained by scaling the mother in of this transform. Although Grossmann was working in he Morlet's some ideas that were close recognized physics, approach

at

high

hence

wavelet,

frequencies. the name

theoretical to his own work

on coherent

states.

quantum

and Grossmann reactivated a Gabor, theoretical which led to physics and signal processing, the formalization of the continuous wavelet transform [288].These ideas were not in new to mathematicians harmonic or to vision working totally analysis, computer researchers multiscale of image processing. It was thus only the beginning studying a rapid catalysis that brought scientists with different together backgrounds. very A wavelet is constructed from a mother wavelet of zero average \317\210 dictionary Nearly

forty years

Morlet

after

between

fundamental collaboration

is dilated

which

with a scale

continuous

The

of

/

on

the

wavelet transform

corresponding

wavelet

Wf(u, s) = (/, It represents

=

one-dimensional

\302\267

\316\221=\316\250 (\342\200\224)}

off

at

scale

any

5 and

position

(1-13)

\320\270 is the

projection

atom: =

\321\204\320\270,5) /

signals

0,

translated by u:

s, and

parameter

v = Uu,s(t)

=

ijj(t)dt

/

by

/(f)

highly

-\320\263\320\244*[

redundant

)dt.

(1.14)

time-scale images in (u,

s).

18

1

CHAPTER

Representations

Sparse

Resolution

Varying Time-Frequency As

opposed

resolution

Fourier wavelet

windowed The changes. to

that

atoms, \\fju^s

have

wavelets

has

a time

a time-frequency

support

centered at

\320\270 and

us choose a wavelet whose Fourier transform is (\317\211) \317\210 \317\210 centered at \316\267. The Fourier transform \321\204\320\2518 (\317\211) frequency interval in a positive is dilated at by 1/5 and thus is localized frequency interval centered = its size is scaled by 1/5. In the time-frequency the Heisenberg box of \316\276\316\267/s; plane, a wavelet atom is therefore a rectangle centered at (u, \316\267/s),with time and \321\204\320\270^ to 5 and 1/5. When 5 varies, the time and widths, frequency respectively, proportional width of this time-frequency resolution cell changes, but its area remains frequency to 5. Let

proportional

in a positive

nonzero

constant, as

illustrated

by Figure

Large-amplitude

1.5.

coefficients

wavelet

can

detect

and measure

short

high-

at high they have a narrow time localization At low frequencies their time resolution is lower, but frequencies. they have a better resolution. This modification of time and frequency resolution is adapted frequency to represent sounds with sharp attacks, or radar signals having a frequency that may

because

variations

frequency

vary quickly Multiscale

at high frequencies. Zooming

also to analyze the scaling evolution of transients adapted now that \317\210 is real. Since it has a zero zooming procedures across scales. Suppose a wavelet coefficient Wf(u, s) measures the variation average, off in a neighborhood of \320\270 that has a size proportional to 5. Sharp create large-amplitude signal transitions

A wavelet

dictionary is

with

wavelet

coefficients.

\317\211

\316\267_

s

\316\267_

FIGURE

1.5

Heisenberg time-frequency decreases, the time support interval

that

is shifted

boxes

of two

is reduced

wavelets, ipu,s but the frequency

toward high frequencies.

and

When ipu0,s0\302\267 spread

increases

the scale 5

and covers an

1.3

have

invariance characterized scaling by Lipschitz the pointwise regularity of/ to the asymptotic decay transform when 5 goes to zero. amplitude \\Wf(u,s)\\ maxima of the wavelet transform across by following the local

Signal singularities

exponents. Chapter of the wavelet

specific

6 relates

detected

Singularities are

19

Dictionaries

Time-Frequency

scales.

wavelet local maxima

In images,

of image

variations which

from

of this

varying

local

maxima

This multiscale

sizes.

in computer vision

recognition

the

indicate

defines

of edges,

position

are reconstructed.

At

are

sharp

of

support different

/

scales,

of image structures

contours

provides

support

which

approximation

scale-space

approximations

image

precise

the geometry of

intensity.

It

edge detection is particularly

for pattern

effective

[146].

transform not only locates isolated capability of the wavelet can also characterize more singular signals having complex multifractal nonisolated was the first to recognize the existence Mandelbrot [41] singularities. in most corners of nature. Scaling of multifractals one part of a multifractal produces a signal that is statistically similar to the whole. This self-similarity appears in the continuous wavelet which modifies the analyzing scale. From transform, global measurements of the wavelet transform 6 measures the decay, Chapter of multifractals. This is particularly important in analyzing their distribution singularity in in and multifractal models or financial time series. testing properties physics

The

zooming

but

events,

1.3.4

Time-Frequency Orthonormal

Bases

of time-frequency atoms remove all redundancy and define orthonormal is an of the basis representations. example time-frequency a wavelet with scales 5 = 2J and translating it by basis obtained by scaling \317\210 dyadic In the the resolution 2Jn, which is written Heisenberg time-frequency plane, {fjj4n. box of ifjj^n is a dilation box of \317\210. by 2^ and translation by 2\320\253of the Heisenberg A wavelet orthonormal is thus a subdictionary of the continuous wavelet transform in which a perfect tiling of the time-frequency illustrated dictionary, yields plane bases

Orthonormal

A wavelet

stable

Figure 1.6.

One can corresponding

bases

other orthonormal tilings of the time-frequency

construct to

many

different

of

atoms,

time-frequency

Wavelet

plane.

are

bases

Packet

Wavelet

two

that

Each

frequency

split

8,

time-

with

in intervals

Bases

Wavelet bases divide Coifman, Meyer, and bases

and local

packet

important examples constructed in Chapter and the time axis, respectively, frequency atoms that split the frequency of varying sizes.

cosine

the

frequency

Wickerhauser

axis [182]

into intervals have

the frequency axis in intervals interval is covered by the

wavelet packet functions as shown by Figure 1.7.

translated

in

time,

of

generalized of

bandwidth

Heisenberg

in order

1

octave

bandwidth.

this construction with that may be adjusted. boxes

time-frequency

to cover the

whole

plane,

of

CHAPTER 1 Sparse

Representations

+ 1,\321\200(*) \320\244\321\203

0/>wW

FIGURE

1.6

The time-frequency

boxes

of a

wavelet basis

a tiling of the

define

plane.

time-frequency

\317\211 \320\272

-\342\226\272\316\257

FIGURE

1.7

A wavelet

packet

is obtained

As

for

by

basis divides

Different

frequency

For images, a filter sizes that can be Local

filters

Cosine

that

axis

in separate

packets

the frequency

split

intervals

covering

coefficients

wavelet-packet

wavelets,

conjugate mirror

the frequency the wavelet

in time

translating

are

obtained

axis in

of varying

each frequency

several

with

sizes.

bank of

a filter

frequency

A1

interval.

intervals.

wavelet segmentations correspond to different packet divides the image frequency support in squares of bank

bases. dyadic

adjusted.

Bases

Local cosine orthonormal bases of the frequency axis. The time

the time axis instead intervals [\316\261\317\201,\316\261\317\201+\316\27 The local cosine bases of Malvar [368] are obtained smooth windows by designing that cover each interval and them by cosine gp(t) [\320\260\321\200, \320\260\321\200+\\], by multiplying functions cos(f t + \317\206) of different This is another idea that has been frequencies. yet in studied and mathematics. Malvar's signal independently physics, processing, original construction was for discrete signals. At the same the physicist Wilson time, [486] was designing a local cosine basis, with smooth windows of infinite support, are

axis

constructed

is segmented

by dividing in successive

1.4

\320\236

\320\260\321\200-\\ ap

ap + l

-^

FIGURE

1.8

A

cosine

basis divides the into frequency.

local

windows

to

the

analyze

and

rediscovered

properties generalized

different views of the that opened

time

axis

\342\226\272

gp{t) and

with smooth windows

of quantum coherent by the harmonic analysts same bases brought to

properties

Dictionaries

Redundant

in

Sparsity

bases

Malvar

states.

and

Coifman

light

these

translates

and

mathematical

also [181].These

were

Meyer

algorithmic

new applications.

+ \317\206) translates the Fourier transformgp(\317\211) of gpif) by the time-frequency box of the modulated window is therefore of gp translated by box equal to the time-frequency gp(t) cos(f\302\243 + \317\206) to \316\276 along Figure 1.8 shows the time-frequency tiling frequencies. corresponding such a local cosine cosine basis. For images, a two-dimensional basis is constructed in squares of varying sizes. by dividing the image support by cos(f t frequencies,

A multiplication

\302\261 Over \316\276.

natural

snow

large dictionaries are needed evolve with usage. Eskimos have whereas a single word is typically dictionaries of vectors are signal

they

quality,

Similarly,

large

of

representations

Suppose signal

Frame that

/.

complex

signals.

by choosing

approximation 1.4.1

DICTIONARIES

languages, and

sentences,

a

IN REDUNDANT

SPARSITY

1.4 In

positive

a

\316\234 dictionary

ideas with short different words to describe in a Parisian dictionary. to construct sparse

refine

sufficient needed

computing vectors

and

optimizing

is much

more

a signal difficult.

and Synthesis nas been selected to approximate sparse family of vectors {\321\204\321\200}\321\200\320\265\\ can be recovered as an orthogonal approximation projection in Analysis

An

the best

However,

to

eight

22

1

CHAPTER

the space

Representations

Sparse

vectors. We

by these

\320\243\320\264 generated

then

face

one

following two

of the

problems.

must be projection f\\ of / in \320\243\320\264 dual-synthesis problem, the orthogonal from dictionary coefficients, an by analysis {(/, \321\204\321\200)}\321\200\320\265\320\220,provided in a signal transform {(/, \321\204\321\200)}\321\200\320\265\321\202 is calculated operator. This is the case when some large dictionary and a subset of inner products are selected. Such inner a threshold or local maxima above products may correspond to coefficients

In a

1.

computed

values.

In a

2.

problem, the

dual-analysis

computed on a family

vectors

when sparse representation algorithms

products. This is the case The

frame

for

theory

energy

gives

stable operators. A family ^ A > 0 such that exists \320\222

with

This

problem {\321\204\321\200}\321\200\320\265\320\220vectors

select

as opposed

dictionaries.

equivalence is a

frame

conditions to solve of the space V it

{\321\204\321\200}\321\200\320\265\\

A\\\\h\\\\2^

V/zeV,

appears

to inner

which compute

algorithms,

pursuit

redundant

in highly

approximation supports

must be

of f\\

coefficients

decomposition

selected

of

both

problems

generates

if there

^\\{\320\232\321\204\321\200)\\2^\320\222\\\\\320\272\\\\2.

meA

The

representation

a modification of a dual frame

of

is stable since any perturbation of frame coefficients implies on h. Chapter 5 proves that the existence magnitude that solves both the dual-synthesis and dual-analysis {\321\204\321\200}\321\200\320\265\320\220 similar

problems: =

\316\221 \316\243

and

TV-1.

intervals

the signal

approximating

by

distance, h=N~x,

the sampling which gives

with an

obtained

signal

sampling at

a uniform

and

variation is calculated sum,

not

Signals

Let fy[n]

over

is

0(|\317\211|_1)

For

variation.

evaluated

the

in

derivative

= Therefore (\317\211) \320\263 (\317\211). \317\211/

is/'

-00 /+00

which

its generalized

of fjor

derivative

Properties

total

discrete

derivative by a finite the

replacing

filter,

averaging

The

difference

(2.58) by a Riemann

integral

II/vIIf = ^I/vW-/v[^-i]|.

(2.60)

\316\267

If

tip are

the abscissa

the

of

extrema

local

II/vIIf

=

of /v,

then

-fN\\np\\\\. \316\243\\\320\234\320\277\321\200+\\\\ \316\241

The

total

||/v

thus measures

variation with

accordance

is bounded || \321\203

the

total

amplitude

that the discrete (2.58), a constant by independent we say

of the

of

of /. In variation if

oscillations

a bounded signal the resolution N. has

Gibbs Oscillations Filtering

total

= If / transfer function is \317\206\316\276 \302\267 1[-\316\276,\316\276]

filter whose

L2(R) norm:lim^+oo (2.26) imply that \\\\f-M2

which

show

that have an

a signal with a low-pass filter can create oscillations variation. Let /\316\276 the filtered be signal obtained =/+\317\206\316\276

has /\316\276

suPieM 1/(0

=/

and the 1[-\316\276,\316\276]

Plancherel

infinite

low-pass to / in formula

1 1 f+0\302\260 \316\223 \342\200\224 |/(\317\211)|2 0, a box splines family and thus is a stable sampling.

and

Response

Impulse

(7.20) -

m

is odd,

the

satisfies (\317\211) \317\206

of the ns)}nez

{\317\2065(\316\257

then

resulting defines a

FILTERS

TIME-INVARIANT

DISCRETE

3,2

a support

have \317\206

condition

sampling

1

\316\257-\316\257\316\265\317\211\\ /8\316\257\316\267(\317\211/2)\\\316\234+1 \321\207

= 1 and is even, then \316\265 and \317\206(\316\257) are symmetric

Riesz

for

continuous.

and

of degree \317\206 times

[ns,

\\JS of spline functions sampling with a space of degree differentiable and equal to a polynomial in U5 are piecewise \316\267 e Z. When m = 1,functions

by spline

generalized

\342\200\224

m

are

Transfer

Function

most generally are based on timealgorithms signal-processing linear operators [51, 55].The time invariance is limited to translations on the sampling interval is normalized 5=1, grid. To simplify notation, the sampling and we denote f[n] the sample values. A linear discrete operator L is time-invariant if an input f[n], delayed an output also delayed e Z,fp[n] hyp produces =f[n \342\200\224p], discrete

invariant

hyp:

Lfp[n]=Lf[n-p]. Response

Impulse

We denote

by

Dirac

the discrete \316\264[\316\267]

Any signal f[n] can

be

as a

decomposed

sum of shifted

Diracs:

+ 00

fM=

j^f[p]8[n-p]. p=-GO

Let

L8[n]

implies

= h[n] be

the

discrete

impulse

response.

Linearity

and time

invariance

that

+ 00 if

W =

\316\243f[p\\

p=-oo

h[n-p\\

=/*\320\271[\321\217].

(3.33)

3.2

linear

A discrete

time-invariant

and

Causality

that

also

may

with

number

Convolutions number

Stability L is causal

if Lf[p] depends only on the values of f[n] for \316\267 ^p. = < if formula (3.33) implies that 0 \316\267 0. h[n] stable if any bounded input signal f[n] produces a bounded output

sufficient that this sufficient

h ell(Z)

a finite

response (FIR) be calculated with a finite a recursive equation (3.45).

impulse

filters.

|Z/Ml^sup|/[\"]|

it is

with

Since

Lf[n].

signal

with a discrete

computed

the sum (333) is calculated

of operations. These are called finite with infinite impulse response filters if they can be rewritten of operations

A discrete filter The convolution The filter is

is thus

operator

has a finite support,

If h[n]

convolution.

Filters

Time-Invariant

Discrete

\316\243

I^M I < + 00, \316\243^\316\223-\316\277\316\277

condition is also

|/\320\263\320\234|'

that h ell(Z). One can the filter is stable if and

means

which

Thus,

necessary.

verify

only if

3.6).

(Exercise

Transfer Function plays a

transform

Fourier

The

because discrete

operators

fundamental role in analyzing

+ 00 Lem[n\\=

is a Fourier

eigenvalue

time-invariant

+00

=

\316\243 e\"Mn-P)h[p}

(3-34)

h[p\\^~iap-

e\302\260>n

J^

p=-oo

p=-oo

The

discrete

= \320\265\321\216\320\277 are eigenvectors: \316\262\317\211[\316\267]

waves

sinusoidal

series +00 (3.35)

\316\233(\317\211)= \316\243 h[p]e~i)]0. Continuously functions with a differentiable expansion is less than \316\265, in [ \342\200\224 in L2[ \342\200\224 included are dense there exists \317\206 such that \317\200, \317\200, \317\200] \317\200]; thus, support \342\200\224 The uniform that there exists N for which ||\316\261\317\206\\\\ ^\316\265/2. pointwise convergence proves Since

is continuously \317\206

Poisson

\342\200\224 IS]\\[ (\317\211) \317\206 (\317\211)

sup

\317\211\316\225[ \342\200\2247\320\223,7\320\223]

:2'

which implies that f77

1 WSn-\320\244\320\223

It follows that

\"\" \\\316\236\316\235(\317\211)-\317\206(\317\211)\\2\316\254\317\211\317\204 TX\"\" 2\317\200 27Tj-7r,\342\200\224\"/ J-\317\200

4

by the

44)

(2.29).

filter

of

a recursive

equation

\316\234

=

f[n] = o can

is

\317\204\317\204\316\267

solution

is a

which

J2^kf[n-k]

bofo.

=

ei\302\253m dw [\316\276 J-\316\276

ideal analog

the

computes

_L

\316\232

with

components

function

and 0o)|2

\320\273 mi

^101O8l0Wr \342\226\240 The polynomial

large

which gives the

exponentp,

=

|^(\317\211)|

Table 4.1

a support

asymptotic

decay

of

\\g(co)\\

for

frequencies:

gives

the

restricted

values to

of these [-1/2,

three

1/2]

(4.26)

0(\317\211^-1).

parameters

for

[293]. Figure 4.5

several

windows

shows the

graph

g having of these

windows. To interpret the three let us consider the spectrogram of frequency parameters, ~ = a frequency tone f(t) = exp(7f00\302\267 If \316\224\317\211 is small, then has \316\276\316\277)\\2 \\Sf(u, \316\276)\\2 |g(f = = \302\261 concentrated near \316\276 at \316\276 lobes of g create \"shadows\" \317\2110, \316\2760.The side \316\2760 energy which can be neglected if A is also small.

100

4.1

Table

Meets

Time

4

CHAPTER

Frequency

Parameters

Frequency

of

Five

Windows

g

At)

Name

1

Rectangle

O54 +

Hamming

Gaussian

exp^-18f2)

Hanning + 0.5

0.42

Blackman Note:

v.46cos(27rt)

cos2

Supports

cos(27Tf)

are restricted to [-1/2,1

(77-f)

+ 0.08

cos(47ii)

/2]. The windows are

normalized

of

four

windows

If the frequency much higher energy g(o) that

\342\200\224 attenuates \316\276)

has |\302\243(\317\211)|

g

with

tone at

these a rapid

P

0.89

-13db

0

1.36

-43db

0

1.55

-55db -32db

0

1.44

1.68

-58db

2

so t hatg(0)

=

2

lbut\\\\g\\\\

f

1

Gaussian

Hamming

Graphs

A

\316\224\317\211

supports

that are

is embedded different

[-1/2,1/2].

in a

that

signal

frequencies, components rapidly

has

other

the tone can

when

decay, and Theorem 2.5 proves

still

components be

\342\200\224 increases. \\\317\211\316\276\\

that

this

decay

of if

detected This

means

depends

on

4.2

the regularity

of

is typically satisfied by windows

(4.26)

Property

g.

101

Transform

Fourier

Windowed

that arep

times

differentiable.

ideas as We consider discrete the same symmetric discrete

atoms

the

of period

of period

defined

TV

|| g\\\\

=

is chosen to be a 1. Discrete windowed

33.

Fourier

Fourier transform

is

\342\200\224 /\342\200\224\320\2632\321\202\320\263\321\202(\320\272 1) \342\200\224 I

exp

-1]

(\320\2632\321\202\321\2021\320\277^ ( \342\200\224j\316\2237

m] exP

transform (DFT) ofgmj

windowed

discrete

~

=gin

=g[k gm,\342\204\226] The

windowgM

in Section

described

by

Fourier

discrete

The

norm

unit

with

gm,lM The

TV.

follow

transform

Fourier

discretization signals

signal

are

fast computation of the windowed of the Fourier transform

and

discretization

The

Fourier Transform

Windowed

Discrete

4.2.3

a signal

of

/ of period

TV

is

\342\200\224 \316\2572\317\200\316\231\316\267\\

f[n] \316\243(

exp

g[n-m]

m].

This

total

of

Inverse

^ m < N,

\320\236 (TV2

log2

TV)

I] is

Sf[m,

with

is performed

TV

calculated

FFT

for

procedures

4.3 is

Figure

operations.

^

^

n=0

For each 0

(

/ < TV of size TV,

0 ^

'J

(4.27)

,

with

a DFT

of f[n]g[n

and

therefore

requires

computed with this

a

algorithm.

Transform

Theorem 4.3 Theorem

4.1.

Theorem

4.3.

If

/ is a signal f[n]

=

1

-

formula

reconstruction

the

discretizes

of

^-^

period

TV,

and the energy

conservation of

then

^-^ l]g[n-

\316\243\316\243Sf[m,

m] exp

m=0 1=0

/\316\2712\317\200\316\231\316\267\\ 1 ( \342\200\224\342\200\224

^

'

(4.28)

and

N-lN-l

N-l

n=0

1=0

m=0

is proved by applying the Parseval and Plancherel formulas of the in as the of Theorem 4.1 (Exercise transform, exactly 4.1). proof The energy conservation (4.29) proves that this windowed Fourier transform defines in Chapter 5. The reconstruction formula a tight frame, as explained is (4.28) This

discrete

theorem Fourier

rewritten f[n]

=

11

N-\\

^-^ 1^ ^n \317\207 m=0

N-l

~ m] ^-^ 1^ 1=0

s^m^/]

// exp

-o 7 \\ \316\2712\317\200\316\231\316\267 \\ -

(~^r~) ^

'

102

4

CHAPTER

Meets

Time

Frequency

The second sum computes, for each 0 ^ m < N, the inverse with respect to /. This is calculated with TV FFT procedures, log2 N) operations. A discrete windowed

DFT

of

Sf[m,

a total

requiring

I] of

0(N2

redundant is characterized equivalent

To

signal

analyze

atoms

frequency

signals

a zero

a

by

of (4.20)

4.1).

(Exercise

TRANSFORMS

WAVELET

4.3

TV2 m] that is very image Sf[l, a of size N. The signal entirely specified by redundancy / discrete reproducing kernel equation, which is the discrete

is an

transform

Fourier

it is

because

sizes, it is necessary to

structures of very different with different time supports.

over dilated

transform

A wavelet is

wavelets.

translated

and

The wavelet

a function

use

time-

decomposes e L2(R) \317\210

with

average:

+ 00 = \316\277. \317\210(\316\220)\316\261\316\220

/ normalized

It is

=

is obtained

atoms

time-frequency

in the

and centered

\\\\\317\206\\\\1

by scaling

1

v-atoms remain

These time

\320\270 and

scale

5 and

\317\210 by

s

\\

/+00

Linear

of

wavelet transform

of

at

/eL2(R)

5 is / .

-,

The

0. A dictionary translating it by u:

ueR,seR+

= l.The

||^M,S||

t =

neighborhood of

(t-u

^s

normalized:

(4.30)

\\

\342\200\224

f(t)

il,*{\342\200\224^-\\dt.

(4.31)

Filtering

wavelet

transform

can

as a

be rewritten foo

Wf(u,

1

fit)

convolution product:

,*(t-u dt=f*4>s(.u),

(4.32)

\302\253>-/: \320\2550

with Mt)

The

Fourier

transform

of

ijjs

(t)

=

Vs

\\

s

)

is &()

= s/si]j*(s(u).

(4.33)

4.3

Since

=

\317\210(0)

= 0, it is the transfer appears that \317\210 the wavelet transform (4.32) computes

\\fj{t) dt

f_O0

convolution

filter.The

103

Transforms

Wavelet

function of a band-pass dilated

with

band-pass

filters.

Versus Real Wavelets Like a windowed Fourier a wavelet transform can measure the time transform, evolution of frequency transients. This requires using a complex wavelet, analytic which can separate amplitude and phase The properties of this analytic components. in Section wavelet transform are described 4.3.2, and its application to the In contrast, measurementof instantaneous is real 4.4.3. frequencies explained in Section wavelets are often used to detect sharp signal transitions. Section introduces 4.3.1 in Chapter 6. which are developed elementary properties of real wavelets, Analytic

Real Wavelets

4.3.1

is a \317\210

that

Suppose

real wavelet.

Since it has a zero

the

wavelet

integral

\342\200\224 /\317\206 \342\200\224\317\210*\316\257dt

Wf(u,s)=l the variation

average,

J

completeness

a neighborhood of \320\270 to 5. Section 6.1.3 proportional to zero, the decay of the wavelet coefficient regularity of/ in the neighborhood of u. This has important applications transients and analyzing fractals. This section concentrates on the and redundancy transforms. properties of real wavelet

EXAMPLE

4.6

measures

proves that when characterizes for

the

detecting

/ in

of

5 goes

scale

Wavelets equal to the second derivative of a Gaussian are called in computer vision to detect multiscale edges [487]. The first used wavelet is

Mexican normalized

hats.

They were hat

Mexican

(434) ^^(^MiS)\302\267 For

\317\203= 1,

Figure

4.6

plots

and -\317\210

its Fourier

.

-VSa^ir1/4 \321\207 = \316\250(\317\211)

Figure 4.7 everywhere,

transform: 2 \317\2112 exp

\316\257-\317\2032\317\2112\\ \\ . \316\257 \342\200\224^\342\200\224

r/

(4.35)

\320\273

regular signal on the left and, almost smaller than 1 because the support of / is normalized to [0,1]. The minimum scale is limited of the by the sampling interval discretized signal used in numerical calculations. When the scale decreases, the wavelet transform has a rapid decay to zero in the regions where the signal is regular. The isolated on the left create cones of large-amplitude wavelet coefficients that converge to singularities the locations of the singularities. This is further explained in Chapter 6. shows

the

singular

wavelet

on

the

transform

right. The

of

a piecewise

maximum scale

is

104

4

CHAPTER

Time

Meets

Frequency

(\317\216) -\317\206

-\320\244\320\241\320\236

FIGURE 4.6

Mexican-hat

wavelet

(4.34)

for

1 and \317\203=

its Fourier transform.

\320\224\320\236

log200

FIGURE

0.8

0.6

0.2 4.7

Real wavelet transform Wf(u,s) computed axis represents log2s. Black, gray, and white and negative wavelet coefficients.

with

a Mexican-hat

points

correspond,

wavelet

(4.34).

respectively, to

The vertical positive,

zero,

4.3

105

Transforms

Wavelet

maintains an energy conservation condition, specified by long admissibility in 1964 by the mathematician Calderon Theorem 4A. This theorem was first proved from a different point of view. Wavelets did not appear as such, but Calderon [132] defines a wavelet transform as a convolution operator that decomposes the identity. Grossmann and Morlet [288] were not aware of Calderon's work when they proved the same formula for signal processing. A real

wavelet

4.4:

Theorem

is

transform

wavelet

as the

as

and

Grossmann

Calderon,

and

complete

a weak

satisfies

Morlet

real function

Let

be a \321\204\320\265\320\2542(\320\250)

such

that

+ 00

\316\231,?,/,.\316\233|2

J*^

\320\241\321\204=

*\302\273)=f(a>) Since

= (\317\211) 0 \317\210

at negative

and

frequencies,

=

/5(\317\211)

which

is the

Fourier

transform

Vsf-i*\302\273).

fa(cu)

=

(\317\211) Vs \321\202\321\203, then 0 |\317\211| \317\210(\317\211)

for

4.10.

\\kv)\\ A Gabor

as a

a scalogram

of

du \316\254\316\276. \316\276)

Pwfiu,

J-t Jo \320\241\321\204

expO'r/f)-

window:

(4.62)

112

The

Meets

Time

4

CHAPTER

g((o) ~0 for approximately analytic.

1 then \320\243>> \317\2032\316\2672

be

EXAMPLE

4.9

The wavelet

transform

s) =

Wf(u,

Observe that the

of

Gabor

such

Thus, \\\317\211\\>\316\267.

=

f(t)

is g((o) =

window

this

of

transform

Fourier

Frequency

=

=

\\/sg(s)\\

of e(u,

\316\276)

\\\317\211\\^5\\\316\270'(\316\267)\\

if

is negligible

\316\224\317\211 2?\342\200\224. \316\270'(\320\270)

Points

Ridge

suppose that ait)

small variations over intervals of size 5 and \316\230'(t)have in (4.77) the corrective term \316\265(\316\274,\316\276) can be neglected. = Since is maximum at \317\211 shows that for each \320\270 the 0, (4.77) |\302\243(\317\211)| = = is maximum at The \316\276(\317\215) \316\270'(\317\215). spectrogram \\Sf(u,\302\243)\\2 corresponding \\{f,gs,U\302\243)\\2

Let us

that

so 0'(\316\257)^\316\224\317\211/$

and

that

points (u,

time-frequency

5/(\302\253, f)

are called \316\276(\317\215))

=

^

a(u)

ridges. -

exp(/[0(*/)

At ridge

\316\276\317\215\\)

(g(0)

points, (4.77) +

\316\265(\316\274, \316\276)).

becomes

(4.9D

4.6 proves that the \316\265(\316\274,\316\276) is smaller at a ridge point because the first-order in (4.81). This is shown by verifying term \316\265a, that \\ becomes | g'(2s0'(u)) | negligible is negligible when s0'(u)^Aw. At ridge the second-order terms \316\265\316\261^ and points, in \316\265{\316\267, \316\265 are \316\276). \316\262,2 predominant

Theorem

The

ridge

amplitude

frequency

gives the

instantaneous frequency

=

and \316\276(\316\267) \316\270'(\316\267)

the

is calculated by

a(u) =

'

^V

;bV

n\\

(4.92)

V^\\g(o)\\

Let

be \316\276)

Ss(u,

then (4.91)

proves

the

complex phase of ridges are also

that

If we neglect the corrective term, Sf(u, \316\276). stationary phase points:

du

Testing

the

stationarity

of the phase locates

the ridges

more

precisely.

4.4

of Instantaneous

Geometry

Time-Frequency

Frequencies

123

Multiple Frequencies the signal contains several spectral lines having frequencies sufficiently the windowed Fourier transform each of these components and the separates detect the evolution in time of each spectral component. Let us consider

When

f(t) = ai (f) a^it) and

where

0i (f) +

cos \320\2572(t)

02(f),

have small variations over intervals Fourier transform is linear, we apply the corrective terms: neglect

and

component

ai (u)g(s[% Sf(u^) = \342\200\224

-

0i'(\320\270)])

(4.77)

-

(u)

exp(/[0i

sOk(t) ^ \316\224\317\211. to each spectral

5 and

size

of

6k(t)

windowed

the

Since

cos

apart, ridges

\316\276\317\215\\)

(4.93)

+ :ye2(\302\253)g(5[f-02,(\302\253)])exp(\302\25302(\302\253)-f\302\253]).

two

The

are

components

spectral

all

if for

discriminated

\320\270

(4.94)

\302\243(5|0i'(\302\253)-02'(\302\253)l)\302\253l,

means

which

that the

is larger

difference

frequency

than the

of g(s \317\211):

bandwidth

(4.95)

|0i'(\302\253)-02'(\302\253)l^\342\200\224\302\267

In this

term

the first

a second distributed

too

for

close,

of (4.93) can

term

be

and

neglected

a ridge

generates

=

two

=

and \316\276 This result lines, \316\276 \316\271' (\320\270) \316\270 {\320\270) (\320\270) 02' (\320\270). time-frequency of time-varying spectral components, as long as the distance any number If two spectral lines are satisfies (4.95). any two instantaneous frequencies

along

between

=

the second \316\276\316\270\316\271'(\317\215),

(u) and a\\ (u) point from which we may recover \316\230\316\212 = if \316\276 term can be neglected and we have 02' (u), the first (4.92). Similarly, that characterizes The 02' (u) and