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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
h
1
!!
A
of
Tour
Wavelet
Processing
Signal
The Sparse
Way
Mallat
Stephane
with
contributions
from
Gabriel Peyre
AMSTERDAM
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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