Front Cover	1
	A Wavelet Tour of Signal Processing	4
	Copyright Page	5
	Dedication Page	6
	Table of Contents	8
	Preface to the Sparse Edition	16
	Notations	20
	Chapter 1. Sparse Representations	24
	1.1 Computational Harmonic Analysis	24
	1.1.1 The Fourier Kingdom	25
	1.1.2 Wavelet Bases	25
	1.2 Approximation and Processing in Bases	28
	1.2.1 Sampling with Linear Approximations	30
	1.2.2 Sparse Nonlinear Approximations	31
	1.2.3 Compression	34
	1.2.4 Denoising	34
	1.3 Time-Frequency Dictionaries	37
	1.3.1 Heisenberg Uncertainty	38
	1.3.2 Windowed Fourier Transform	39
	1.3.3 Continuous Wavelet Transform	40
	1.3.4 Time-Frequency Orthonormal Bases	42
	1.4 Sparsity in Redundant Dictionaries	44
	1.4.1 Frame Analysis and Synthesis	44
	1.4.2 Ideal Dictionary Approximations	46
	1.4.3 Pursuit in Dictionaries	47
	1.5 Inverse Problems	49
	1.5.1 Diagonal Inverse Estimation	50
	1.5.2 Super-resolution and Compressive Sensing	51
	1.6 Travel Guide	53
	1.6.1 Reproducible Computational Science	53
	1.6.2 Book Road Map	53
	Chapter 2. The Fourier Kingdom	56
	2.1 Linear Time-Invariant Filtering	56
	2.1.1 Impulse Response	56
	2.1.2 Transfer Functions	58
	2.2 Fourier Integrals	58
	2.2.1 Fourier Transform in L1(R)	58
	2.2.2 Fourier Transform in L2(R)	61
	2.2.3 Examples	63
	2.3 Properties	65
	2.3.1 Regularity and Decay	65
	2.3.2 Uncertainty Principle	66
	2.3.3 Total Variation	69
	2.4 Two-Dimensional Fourier Transform	74
	2.5 Exercises	78
	Chapter 3. Discrete Revolution	82
	3.1 Sampling Analog Signals	82
	3.1.1 Shannon-Whittaker Sampling Theorem	82
	3.1.2 Aliasing	84
	3.1.3 General Sampling and Linear Analog Conversions	88
	3.2 Discrete Time-Invariant Filters	93
	3.2.1 Impulse Response and Transfer Function	93
	3.2.2 Fourier Series	95
	3.3 Finite Signals	98
	3.3.1 Circular Convolutions	99
	3.3.2 Discrete Fourier Transform	99
	3.3.3 Fast Fourier Transform	101
	3.3.4 Fast Convolutions	102
	3.4 Discrete Image Processing	103
	3.4.1 Two-Dimensional Sampling Theorems	103
	3.4.2 Discrete Image Filtering	105
	3.4.3 Circular Convolutions and Fourier Basis	106
	3.5 Exercises	108
	Chapter 4. Time Meets Frequency	112
	4.1 Time-Frequency Atoms	112
	4.2 Windowed Fourier Transform	115
	4.2.1 Completeness and Stability	117
	4.2.2 Choice of Window	121
	4.2.3 Discrete Windowed Fourier Transform	124
	4.3 Wavelet Transforms	125
	4.3.1 Real Wavelets	126
	4.3.2 Analytic Wavelets	130
	4.3.3 Discrete Wavelets	135
	4.4 Time-Frequency Geometry of Instantaneous Frequencies	138
	4.4.1 Analytic Instantaneous Frequency	138
	4.4.2 Windowed Fourier Ridges	141
	4.4.3 Wavelet Ridges	152
	4.5 Quadratic Time-Frequency Energy	157
	4.5.1 Wigner-Ville Distribution	159
	4.5.2 Interferences and Positivity	163
	4.5.3 Cohen’s Class	168
	4.5.4 Discrete Wigner-Ville Computations	172
	4.6 Exercises	174
	Chapter 5. Frames	178
	5.1 Frames and Riesz Bases	178
	5.1.1 Stable Analysis and Synthesis Operators	178
	5.1.2 Dual Frame and Pseudo Inverse	182
	5.1.3 Dual-Frame Analysis and Synthesis Computations	184
	5.1.4 Frame Projector and Reproducing Kernel	189
	5.1.5 Translation-Invariant Frames	191
	5.2 Translation-Invariant Dyadic Wavelet Transform	193
	5.2.1 Dyadic Wavelet Design	195
	5.2.2 Algorithme à Trous	198
	5.3 Subsampled Wavelet Frames	201
	5.4 Windowed Fourier Frames	204
	5.4.1 Tight Frames	206
	5.4.2 General Frames	207
	5.5 Multiscale Directional Frames For Images	211
	5.5.1 Directional Wavelet Frames	212
	5.5.2 Curvelet Frames	217
	5.6 Exercises	224
	Chapter 6. Wavelet Zoom	228
	6.1 Lipschitz Regularity	228
	6.1.1 Lipschitz Definition and Fourier Analysis	228
	6.1.2 Wavelet Vanishing Moments	231
	6.1.3 Regularity Measurements with Wavelets	234
	6.2 Wavelet Transform Modulus Maxima	241
	6.2.1 Detection of Singularities	241
	6.2.2 Dyadic Maxima Representation	247
	6.3 Multiscale Edge Detection	253
	6.3.1 Wavelet Maxima for Images	253
	6.3.2 Fast Multiscale Edge Computations	262
	6.4 Multifractals	265
	6.4.1 Fractal Sets and Self-Similar Functions	265
	6.4.2 Singularity Spectrum	269
	6.4.3 Fractal Noises	277
	6.5 Exercises	282
	Chapter 7. Wavelet Bases	286
	7.1 Orthogonal Wavelet Bases	286
	7.1.1 Multiresolution Approximations	287
	7.1.2 Scaling Function	290
	7.1.3 Conjugate Mirror Filters	293
	7.1.4 In Which Orthogonal Wavelets Finally Arrive	301
	7.2 Classes of Wavelet Bases	307
	7.2.1 Choosing a Wavelet	307
	7.2.2 Shannon, Meyer, Haar, and Battle-Lemarié Wavelets	312
	7.2.3 Daubechies Compactly Supported Wavelets	315
	7.3 Wavelets and Filter Banks	321
	7.3.1 Fast Orthogonal Wavelet Transform	321
	7.3.2 Perfect Reconstruction Filter Banks	325
	7.3.3 Biorthogonal Bases of l2(Z)	329
	7.4 Biorthogonal Wavelet Bases	331
	7.4.1 Construction of Biorthogonal Wavelet Bases	331
	7.4.2 Biorthogonal Wavelet Design	334
	7.4.3 Compactly Supported Biorthogonal Wavelets	336
	7.5 Wavelet Bases on an Interval	340
	7.5.1 Periodic Wavelets	341
	7.5.2 Folded Wavelets	343
	7.5.3 Boundary Wavelets	345
	7.6 Multiscale Interpolations	351
	7.6.1 Interpolation and Sampling Theorems	351
	7.6.2 Interpolation Wavelet Basis	356
	7.7 Separable Wavelet Bases	361
	7.7.1 Separable Multiresolutions	361
	7.7.2 Two-Dimensional Wavelet Bases	363
	7.7.3 Fast Two-Dimensional Wavelet Transform	369
	7.7.4 Wavelet Bases in Higher Dimensions	371
	7.8 Lifting Wavelets	373
	7.8.1 Biorthogonal Bases over Nonstationary Grids	373
	7.8.2 Lifting Scheme	375
	7.8.3 Quincunx Wavelet Bases	382
	7.8.4 Wavelets on Bounded Domains and Surfaces	384
	7.8.5 Faster Wavelet Transform with Lifting	390
	7.9 Exercises	393
	Chapter 8. Wavelet Packet and Local Cosine Bases	400
	8.1 Wavelet Packets	400
	8.1.1 Wavelet Packet Tree	400
	8.1.2 Time-Frequency Localization	406
	8.1.3 Particular Wavelet Packet Bases	411
	8.1.4 Wavelet Packet Filter Banks	416
	8.2 Image Wavelet Packets	418
	8.2.1 Wavelet Packet Quad-Tree	418
	8.2.2 Separable Filter Banks	422
	8.3 Block Transforms	423
	8.3.1 Block Bases	424
	8.3.2 Cosine Bases	426
	8.3.3 Discrete Cosine Bases	429
	8.3.4 Fast Discrete Cosine Transforms	430
	8.4 Lapped Orthogonal Transforms	433
	8.4.1 Lapped Projectors	433
	8.4.2 Lapped Orthogonal Bases	439
	8.4.3 Local Cosine Bases	442
	8.4.4 Discrete Lapped Transforms	445
	8.5 Local Cosine Trees	449
	8.5.1 Binary Tree of Cosine Bases	449
	8.5.2 Tree of Discrete Bases	452
	8.5.3 Image Cosine Quad-Tree	452
	8.6 Exercises	455
	Chapter 9. Approximations in Bases	458
	9.1 Linear Approximations	458
	9.1.1 Sampling and Approximation Error	458
	9.1.2 Linear Fourier Approximations	461
	9.1.3 Multiresolution Approximation Errors with Wavelets	465
	9.1.4 Karhunen-Loève Approximations	469
	9.2 Nonlinear Approximations	473
	9.2.1 Nonlinear Approximation Error	474
	9.2.2 Wavelet Adaptive Grids	478
	9.2.3 Approximations in Besov and Bounded Variation Spaces	482
	9.3 Sparse Image Representations	486
	9.3.1 Wavelet Image Approximations	487
	9.3.2 Geometric Image Models and Adaptive Triangulations	494
	9.3.3 Curvelet Approximations	499
	9.4 Exercises	501
	Chapter 10. Compression	504
	10.1 Transform Coding	504
	10.1.1 Compression State of the Art	505
	10.1.2 Compression in Orthonormal Bases	506
	10.2 Distortion Rate of Quantization	508
	10.2.1 Entropy Coding	508
	10.2.2 Scalar Quantization	516
	10.3 High Bit Rate Compression	519
	10.3.1 Bit Allocation	519
	10.3.2 Optimal Basis and Karhunen-Loève	521
	10.3.3 Transparent Audio Code	524
	10.4 Sparse Signal Compression	529
	10.4.1 Distortion Rate and Wavelet Image Coding	529
	10.4.2 Embedded Transform Coding	539
	10.5 Image-Compression Standards	542
	10.5.1 JPEG Block Cosine Coding	542
	10.5.2 JPEG-2000 Wavelet Coding	546
	10.6 Exercises	554
	Chapter 11. Denoising	558
	11.1 Estimation with Additive Noise	558
	11.1.1 Bayes Estimation	559
	11.1.2 Minimax Estimation	567
	11.2 Diagonal Estimation in a Basis	571
	11.2.1 Diagonal Estimation with Oracles	571
	11.2.2 Thresholding Estimation	575
	11.2.3 Thresholding Improvements	581
	11.3 Thresholding Sparse Representations	585
	11.3.1 Wavelet Thresholding	586
	11.3.2 Wavelet and Curvelet Image Denoising	591
	11.3.3 Audio Denoising by Time-Frequency Thresholding	594
	11.4 Nondiagonal Block Thresholding	598
	11.4.1 Block Thresholding in Bases and Frames	598
	11.4.2 Wavelet Block Thresholding	604
	11.4.3 Time-Frequency Audio Block Thresholding	605
	11.5 Denoising Minimax Optimality	608
	11.5.1 Linear Diagonal Minimax Estimation	610
	11.5.2 Thresholding Optimality over Orthosymmetric Sets	613
	11.5.3 Nearly Minimax with Wavelet Estimation	618
	11.6 Exercises	629
	Chapter 12. Sparsity in Redundant Dictionaries	634
	12.1 Ideal Sparse Processing in Dictionaries	634
	12.1.1 Best M-Term Approximations	635
	12.1.2 Compression by Support Coding	637
	12.1.3 Denoising by Support Selection in a Dictionary	639
	12.2 Dictionaries of Orthonormal Bases	644
	12.2.1 Approximation, Compression, and Denoising in a Best Basis	645
	12.2.2 Fast Best-Basis Search in Tree Dictionaries	646
	12.2.3 Wavelet Packet and Local Cosine Best Bases	649
	12.2.4 Bandlets for Geometric Image Regularity	654
	12.3 Greedy Matching Pursuits	665
	12.3.1 Matching Pursuit	665
	12.3.2 Orthogonal Matching Pursuit	671
	12.3.3 Gabor Dictionaries	673
	12.3.4 Coherent Matching Pursuit Denoising	678
	12.4 l1 Pursuits	682
	12.4.1 Basis Pursuit	682
	12.4.2 l1 Lagrangian Pursuit	687
	12.4.3 Computations of l1 Minimizations	691
	12.4.4 Sparse Synthesis versus Analysis and Total Variation Regularization	696
	12.5 Pursuit Recovery	700
	12.5.1 Stability and Incoherence	700
	12.5.2 Support Recovery with Matching Pursuit	702
	12.5.3 Support Recovery with l1 Pursuits	707
	12.6 Multichannel Signals	711
	12.6.1 Approximation and Denoising by Thresholding in Bases	712
	12.6.2 Multichannel Pursuits	713
	12.7 Learning Dictionaries	716
	12.8 Exercises	719
	Chapter 13. Inverse Problems	722
	13.1 Linear Inverse Estimation	723
	13.1.1 Quadratic and Tikhonov Regularizations	723
	13.1.2 Singular Value Decompositions	725
	13.2 Thresholding Estimators for Inverse Problems	726
	13.2.1 Thresholding in Bases of Almost Singular Vectors	726
	13.2.2 Thresholding Deconvolutions	732
	13.3 Super-Resolution	736
	13.3.1 Sparse Super-resolution Estimation	736
	13.3.2 Sparse Spike Deconvolution	742
	13.3.3 Recovery of Missing Data	745
	13.4 Compressive Sensing	751
	13.4.1 Incoherence with Random Measurements	752
	13.4.2 Approximations with Compressive Sensing	758
	13.4.3 Compressive Sensing Applications	765
	13.5 Blind Source Separation	767
	13.5.1 Blind Mixing Matrix Estimation	768
	13.5.2 Source Separation	774
	13.6 Exercises	775
	Appendix Mathematical Complements	776
	Bibliography	788
	Index	818