A wavelet tour of signal processing the Sparse way

Cover Page......Page 1
Copyright Page......Page 2
Dedication\r......Page 3
Preface to the Sparse Edition......Page 4
Acknowledgments......Page 7
Acknowledgments......Page 8
Notations......Page 9
Computational Harmonic Analysis......Page 11
Wavelet Bases......Page 12
Approximation and Processing in Bases......Page 15
Sampling with Linear Approximations......Page 17
Sparse Nonlinear Approximations......Page 18
Denoising......Page 21
Time-Frequency Dictionaries......Page 24
Heisenberg Uncertainty......Page 25
Windowed Fourier Transform......Page 26
Continuous Wavelet Transform......Page 27
Time-Frequency Orthonormal Bases......Page 29
Frame Analysis and Synthesis......Page 31
Ideal Dictionary Approximations......Page 33
Pursuit in Dictionaries......Page 34
Inverse Problems......Page 36
Diagonal Inverse Estimation......Page 37
Super-resolution and Compressive Sensing......Page 38
Book Road Map......Page 40
Impulse Response......Page 42
Fourier Transform in L1(R)......Page 44
Fourier Transform in L2(R)......Page 47
Examples......Page 49
Regularity and Decay......Page 51
Uncertainty Principle......Page 52
Total Variation......Page 55
Two-Dimensional Fourier Transform......Page 60
Exercises......Page 64
Shannon-Whittaker Sampling Theorem......Page 67
Aliasing......Page 69
General Sampling and Linear Analog Conversions......Page 73
Impulse Response and Transfer Function......Page 78
Fourier Series......Page 80
Finite Signals......Page 83
Discrete Fourier Transform......Page 84
Fast Fourier Transform......Page 86
Fast Convolutions......Page 87
Two-Dimensional Sampling Theorems......Page 88
Discrete Image Filtering......Page 90
Circular Convolutions and Fourier Basis......Page 91
Exercises......Page 93
Time-Frequency Atoms......Page 97
Windowed Fourier Transform......Page 100
Completeness and Stability......Page 102
Choice of Window......Page 106
Discrete Windowed Fourier Transform......Page 109
Wavelet Transforms......Page 110
Real Wavelets......Page 111
Analytic Wavelets......Page 115
Discrete Wavelets......Page 120
Analytic Instantaneous Frequency......Page 123
Windowed Fourier Ridges......Page 126
Wavelet Ridges......Page 137
Quadratic Time-Frequency Energy......Page 142
Wigner-Ville Distribution......Page 144
Interferences and Positivity......Page 148
Cohen\'s Class......Page 153
Discrete Wigner-Ville Computations......Page 157
Exercises......Page 159
Stable Analysis and Synthesis Operators......Page 162
Dual Frame and Pseudo Inverse......Page 166
Dual-Frame Analysis and Synthesis Computations......Page 168
Frame Projector and Reproducing Kernel......Page 173
Translation-Invariant Frames......Page 175
Translation-Invariant Dyadic Wavelet Transform......Page 177
Dyadic Wavelet Design......Page 179
Algorithme à Trous......Page 182
Subsampled Wavelet Frames......Page 185
Windowed Fourier Frames......Page 188
Tight Frames......Page 190
General Frames......Page 191
Multiscale Directional Frames for Images......Page 195
Directional Wavelet Frames......Page 196
Curvelet Frames......Page 201
Exercises......Page 208
Lipschitz Definition and Fourier Analysis......Page 212
Wavelet Vanishing Moments......Page 215
Regularity Measurements with Wavelets......Page 218
Detection of Singularities......Page 225
Dyadic Maxima Representation......Page 231
Wavelet Maxima for Images......Page 237
Fast Multiscale Edge Computations......Page 246
Fractal Sets and Self-Similar Functions......Page 249
Singularity Spectrum......Page 253
Fractal Noises......Page 261
Exercises......Page 266
Orthogonal Wavelet Bases......Page 269
Multiresolution Approximations......Page 270
Scaling Function......Page 273
Conjugate Mirror Filters......Page 276
In Which Orthogonal Wavelets Finally Arrive......Page 284
Choosing a Wavelet......Page 290
Shannon, Meyer, Haar, and Battle-Lemarié Wavelets......Page 295
Daubechies Compactly Supported Wavelets......Page 298
Fast Orthogonal Wavelet Transform......Page 304
Perfect Reconstruction Filter Banks......Page 308
Biorthogonal Bases of l2(Z)\r......Page 312
Construction of Biorthogonal Wavelet Bases......Page 314
Biorthogonal Wavelet Design......Page 317
Compactly Supported Biorthogonal Wavelets......Page 319
Wavelet Bases on an Interval......Page 323
Periodic Wavelets......Page 324
Folded Wavelets......Page 326
Boundary Wavelets......Page 328
Interpolation and Sampling Theorems......Page 334
Interpolation Wavelet Basis......Page 339
Separable Multiresolutions......Page 344
Two-Dimensional Wavelet Bases......Page 346
Fast Two-Dimensional Wavelet Transform......Page 352
Wavelet Bases in Higher Dimensions......Page 354
Biorthogonal Bases over Nonstationary Grids......Page 356
Lifting Scheme......Page 358
Quincunx Wavelet Bases......Page 365
Wavelets on Bounded Domains and Surfaces......Page 367
Faster Wavelet Transform with Lifting......Page 373
Exercises......Page 376
Wavelet Packet Tree......Page 383
Time-Frequency Localization......Page 389
Particular Wavelet Packet Bases......Page 394
Wavelet Packet Filter Banks......Page 399
Wavelet Packet Quad-Tree......Page 401
Separable Filter Banks......Page 405
Block Transforms......Page 406
Block Bases......Page 407
Cosine Bases......Page 409
Discrete Cosine Bases......Page 412
Fast Discrete Cosine Transforms......Page 413
Lapped Projectors......Page 416
Lapped Orthogonal Bases......Page 422
Local Cosine Bases......Page 425
Discrete Lapped Transforms......Page 428
Binary Tree of Cosine Bases......Page 432
Image Cosine Quad-Tree......Page 435
Exercises......Page 438
Sampling and Approximation Error......Page 441
Linear Fourier Approximations......Page 444
Multiresolution Approximation Errors with Wavelets......Page 448
Karhunen-Loève Approximations......Page 452
NonLinear Approximations......Page 456
Nonlinear Approximation Error......Page 457
Wavelet Adaptive Grids......Page 461
Approximations in Besov and Bounded Variation Spaces......Page 465
Sparse Image Representations......Page 469
Wavelet Image Approximations......Page 470
Geometric Image Models and Adaptive Triangulations......Page 477
Curvelet Approximations......Page 482
Exercises......Page 484
Transform Coding......Page 487
Compression State of the Art......Page 488
Compression in Orthonormal Bases......Page 489
Entropy Coding......Page 491
Scalar Quantization......Page 499
Bit Allocation......Page 502
Optimal Basis and Karhunen-Loève......Page 504
Transparent Audio Code......Page 507
Distortion Rate and Wavelet Image Coding......Page 512
Embedded Transform Coding......Page 522
JPEG Block Cosine Coding......Page 525
JPEG-2000 Wavelet Coding......Page 529
Exercises......Page 537
Estimation with Additive Noise......Page 540
Bayes Estimation......Page 541
Minimax Estimation......Page 549
Diagonal Estimation with Oracles......Page 553
Thresholding Estimation......Page 557
Thresholding Improvements......Page 563
Thresholding Sparse Representations......Page 567
Wavelet Thresholding......Page 568
Wavelet and Curvelet Image Denoising......Page 573
Audio Denoising by Time-Frequency Thresholding......Page 576
Block Thresholding in Bases and Frames......Page 580
Wavelet Block Thresholding......Page 586
Time-Frequency Audio Block Thresholding......Page 587
Denoising Minimax Optimality......Page 590
Linear Diagonal Minimax Estimation......Page 592
Thresholding Optimality over Orthosymmetric Sets......Page 595
Nearly Minimax with Wavelet Estimation......Page 600
Exercises......Page 611
Ideal Sparse Processing in Dictionaries......Page 615
Best M-Term Approximations......Page 616
Compression by Support Coding......Page 618
Denoising by Support Selection in a Dictionary......Page 620
Dictionaries of Orthonormal Bases......Page 625
Approximation, Compression, and Denoising in a Best Basis......Page 626
Fast Best-Basis Search in Tree Dictionaries......Page 627
Wavelet Packet and Local Cosine Best Bases......Page 630
Bandlets for Geometric Image Regularity......Page 635
Matching Pursuit......Page 646
Orthogonal Matching Pursuit......Page 652
Gabor Dictionaries......Page 654
Coherent Matching Pursuit Denoising......Page 659
Basis Pursuit......Page 663
l1 Lagrangian Pursuit......Page 668
Computations of l1 Minimizations......Page 672
Sparse Synthesis versus Analysis and TotalVariation Regularization......Page 677
Stability and Incoherence......Page 681
Support Recovery with Matching Pursuit......Page 683
Support Recovery with l1 Pursuits......Page 688
Multichannel Signals......Page 692
Approximation and Denoising by Thresholding in Bases......Page 693
Multichannel Pursuits......Page 694
Learning Dictionaries......Page 697
Exercises......Page 700
13 Inverse Problems......Page 703
Quadratic and Tikhonov Regularizations......Page 704
Singular Value Decompositions......Page 706
Thresholding in Bases of Almost Singular Vectors......Page 707
Thresholding Deconvolutions......Page 713
Sparse Super-resolution Estimation......Page 717
Sparse Spike Deconvolution......Page 723
Recovery of Missing Data......Page 726
Compressive Sensing......Page 732
Incoherence with Random Measurements......Page 733
Approximations with Compressive Sensing......Page 739
Compressive Sensing Applications......Page 746
Blind Source Separation......Page 748
Blind Mixing Matrix Estimation......Page 749
Source Separation......Page 755
Exercises......Page 756
Functions and Integration......Page 757
Banach and Hilbert Spaces......Page 758
Bases of Hilbert Spaces......Page 761
Linear Operators......Page 762
Separable Spaces and Bases......Page 764
Random Vectors and Covariance Operators......Page 765
Diracs......Page 767
Books......Page 769
Articles......Page 772
B......Page 798
C......Page 799
D......Page 800
F......Page 801
I......Page 802
M......Page 803
P......Page 804
S......Page 805
T......Page 806
W......Page 807
Z......Page 808