Ⅰ INTRODUCTION TO A TRANSIENT WORLD 2
1.1 Fourier Kingdom 2
1.2 Time-Frequency Wedding 2
1.2.1 Windowed Fourier Transform 3
1.2.2 Wavelet Transform 4
1.3 Bases of Time-Frequency Atoms 6
1.3.1 Wavelet Bases and Filter Banks 7
1.3.2 Tilings of Wavelet Packet and Local Cosine Bases 9
1.4 Bases for What? 11
1.4.1 Approximation 12
1.4.2 Estimation 14
1.4.3 Compression 16
1.5.1 Reproducible Computational Science 17
1.5 Travel Guide 17
1.5.2 Road Map 18
Ⅱ FOURIER KINGDOM 20
2.1 Linear Time-Invariant Filtering1 20
2.1.1 Impulse Response 21
2.1.2 Transfer Functions 22
2.2 Fourier Integrals1 22
2.2.1 Fourier Transform in L1(R) 23
2.2.2 Fourier Transform in L2(R) 25
2.2.3 Examples 27
2.3 Properties1 29
2.3.1 Regularity and Decay 29
2.3.2 Uncertainty Principle 30
2.3.3 Total Variation 33
2.4 Two-Dimensional Fourier Transform1 38
2.5 Problems 40
Ⅲ DISCRETE REVOLUTION 42
3.1 Sampling Analog Signals1 42
3.1.1 Whittaker Sampling Theorem 43
3.1.2 Aliasing 44
3.1.3 General Sampling Theorems 47
3.2 Discrete Time-Invariant Filters1 49
3.2.1 Impulse Response and Transfer Function 49
3.2.2 Fourier Series 51
3.3 Finite Signals1 54
3.3.1 Circular Convolutions 55
3.3.2 Discrete Fourier Transform 55
3.3.3 Fast Fourier Transform 57
3.3.4 Fast Convolutions 58
3.4 Discrete Image Processing1 59
3.4.1 Two-Dimensional Sampling Theorem 60
3.4.2 Discrete Image Filtering 61
3.4.3 Circular Convolutions and Fourier Basis 62
3.5 Problems 64
Ⅳ TIME MEETS FREQUENCY 67
4.1 Time-Frequency Atoms1 67
4.2 Windowed Fourier Transform1 69
4.2.1 Completeness and Stability 72
4.2.2 Choice of Window2 75
4.2.3 Discrete Windowed Fourier Transform2 77
4.3 Wavelet Transforms1 79
4.3.1 Real Wavelets 80
4.3.2 Analytic Wavelets 84
4.3.3 Discrete Wavelets2 89
4.4 Instantaneous Frequency2 91
4.4.1 Windowed Fourier Ridges 94
4.4.2 Wavelet Ridges 102
4.5 Quadratic Time-Frequency Energy1 107
4.5.1 Wigner-Ville Distribution 107
4.5.2 Interferences and Positivity 112
4.5.3 Cohen s Class2 116
4.5.4 Discrete Wigner-Ville Computations2 120
4.6 Problems 121
Ⅴ FRAMES 125
5.1 Frame Theory2 125
5.1.1 Frame Definition and Sampling 125
5.1.2 Pseudo Inverse 127
5.1.3 Inverse Frame Computations 132
5.1.4 Frame Projector and Noise Reduction 135
5.2 Windowed Fourier Frames2 138
5.3 Wavelet Frames2 143
5.4 Translation Invariance1 146
5.5 Dyadic Wavelet Transform2 148
5.5.1 Wavelet Design 150
5.5.2 Algorithme ? Trous 153
5.5.3 Oriented Wavelets for a Vision3 156
5.6 Problems 160
Ⅵ WAVELET ZOOM 163
6.1 Lipschitz Regularity1 163
6.1.1 Lipschitz Definition and Fourier Analysis 164
6.1.2 Wavelet Vanishing Moments 166
6.1.3 Regularity Measurements with Wavelets 169
6.2.1 Detection of Singularities 176
6.2 Wavelet Transform Modulus Maxima2 176
6.2.2 Reconstruction From Dyadic Maxima3 183
6.3 Multiscale Edge Detection2 189
6.3.1 Wavelet Maxima for Images2 189
6.3.2 Fast Multiscale Edge Computations3 197
6.4 Multifractals2 200
6.4.1 Fractal Sets and Self-Similar Functions 200
6.4.2 Singularity Spectrum3 205
6.4.3 Fractal Noises3 211
6.5 Problems 216
Ⅶ WAVELET BASES 220
7.1 Orthogonal Wavelet Bases1 220
7.1.1 Multiresolution Approximations 221
7.1.2 Scaling Function 224
7.1.3 Conjugate Mirror Filters 228
7.1.4 In Which Orthogonal Wavelets Finally Arrive 235
7.2 Classes of Wavelet Bases1 241
7.2.1 Choosing a Wavelet 241
7.2.2 Shannon, Meyer and Battle-Lemarié Wavelets 246
7.2.3 Daubechies Compactly Supported Wavelets 249
7.3 Wavelets and Filter Banks1 255
7.3.1 Fast Orthogonal Wavelet Transform 255
7.3.2 Perfect Reconstruction Filter Banks 259
7.3.3 Biorthogonal Bases of 12(Z)2 263
7.4 Biorthogonal Wavelet Bases2 265
7.4.1 Construction of Biorthogonal Wavelet Bases 265
7.4.2 Biorthogonal Wavelet Design2 268
7.4.3 Compactly Supported Biorthogonal Wavelets2 270
7.4.4 Lifting Wavelets3 273
7.5 Wavelet Bases on an Interval2 281
7.5.1 Periodic Wavelets 282
7.5.2 Folded Wavelets 284
7.5.3 Boundary Wavelets3 286
7.6 Multiscale Interpolations2 293
7.6.1 Interpolation and Sampling Theorems 293
7.6.2 Interpolation Wavelet Basis3 299
7.7 Separable Wavelet Bases1 303
7.7.1 Separable Multiresolutions 304
7.7.2 Two-Dimensional Wavelet Bases 306
7.7.3 Fast Two-Dimensional Wavelet Transform 310
7.7.4 Wavelet Bases in Higher Dimensions2 313
7.8 Problems 314
8.1.1 Wavelet Packet Tree 322
8.1 Wavelet Packets2 322
Ⅷ WAVELET PACKET AND LOCAL COSINE BASES 322
8.1.2 Time-Frequency Localization 327
8.1.3 Particular Wavelet Packet Bases 333
8.1.4 Wavelet Packet Filter Banks 336
8.2 Image Wavelet Packets2 339
8.2.1 Wavelet Packet Quad-Tree 339
8.2.2 Separable Filter Banks 341
8.3 Block Transforms1 343
8.3.1 Block Bases 344
8.3.2 Cosine Bases 346
8.3.3 Discrete Cosine Bases 349
8.3.4 Fast Discrete Cosine Transforms2 350
8.4 Lapped Orthogonal Transforms2 353
8.4.1 Lapped Projectors 353
8.4.2 Lapped Orthogonal Bases 359
8.4.3 Local Cosine Bases 361
8.4.4 Discrete Lapped Transforms 364
8.5 Local Cosine Trees2 368
8.5.1 Binary Tree of Cosine Bases 369
8.5.2 Tree of Discrete Bases 371
8.5.3 Image Cosine Quad-Tree 372
8.6 Problems 374
Ⅸ AN APPROXIMATION TOUR 377
9.1 Linear Approximations1 377
9.1.1 Linear Approximation Error 377
9.1.2 Linear Fourier Approximations 378
9.1.3 Linear Multiresolution Approximations 382
9.1.4 Karhunen-Loève Approximations2 385
9.2.1 Non-Linear Approximation Error 389
9.2 Non-Linear Approximations1 389
9.2.2 Wavelet Adaptive Grids 391
9.2.3 Besov Spaces3 394
9.3 Image Approximations with Wavelets1 398
9.4 Adaptive Basis Selection2 405
9.4.1 Best Basis and Schur Concavity 406
9.4.2 Fast Best Basis Search in Trees 411
9.4.3 Wavelet Packet and Local Cosine Best Bases 413
9.5 Approximations with Pursuits3 417
9.5.1 Basis Pursuit 418
9.5.2 Matching Pursuit 421
9.5.3 Orthogonal Matching Pursuit 428
9.6 Problems 430
10.1.1 Bayes Estimation 435
10.1 Bayes Versus Minimax2 435
Ⅹ ESTIMATIONS ARE APPROXIMATIONS 435
10.1.2 Minimax Estimation 442
10.2 Diagonal Estimation in a Basis2 446
10.2.1 Diagonal Estimation with Oracles 446
10.2.2 Thresholding Estimation 450
10.2.3 Thresholding Refinements3 455
10.2.4 Wavelet Thresholding 458
10.2.5 Best Basis Thresholding3 466
10.3 Minimax Optimality3 469
10.3.1 Linear Diagonal Minimax Estimation 469
10.3.2 Orthosymmetric Sets 474
10.3.3 Nearly Minimax with Wavelets 479
10.4.1 Estimation in Arbitrary Gaussian Noise 486
10.4 Restoration3 486
10.4.2 Inverse Problems and Deconvolution 491
10.5 Coherent Estimation3 501
10.5.1 Coherent Basis Thresholding 502
10.5.2 Coherent Matching Pursuit 505
10.6 Spectrum Estimation2 507
10.6.1 Power Spectrum 508
10.6.2 Approximate Karhunen-Loève Search3 512
10.6.3 Locally Stationary Processes3 516
10.7 Problems 520
Ⅺ TRANSFORM CODING 526
11.1 Signal Compression2 526
11.1.1 State of the Art 526
11.1.2 Compression in Orthonormal Bases 527
11.2 Distortion Rate of Quantization2 528
11.2.1 Entropy Coding 529
11.2.2 Scalar Quantization 537
11.3 High Bit Rate Compression2 540
11.3.1 Bit Allocation 540
11.3.2 Optimal Basis and Karhunen-Loève 542
11.3.3 Transparent Audio Code 544
11.4 Image Compression2 548
11.4.1 Deterministic Distortion Rate 548
11.4.2 Wavelet Image Coding 557
11.4.3 Block Cosine Image Coding 561
11.4.4 Embedded Transform Coding 566
11.4.5 Minimax Distortion Rate3 571
11.5 Video Signals2 577
11.5.1 Optical Flow 577
11.5.2 MPEG Video Compression 585
11.6 Problems 587
Appendix A MATHEMATICAL COMPLEMENTS 591
A.1 Functions and Integration 591
A.2 Banach and Hilbert Spaces 593
A.3 Bases of Hilbert Spaces 595
A.4 Linear Operators 596
A.5 Separable Spaces and Bases 598
A.6 Random Vectors and Covariance Operators 599
A.7 Diracs 601
Appendix B SOFTWARE TOOLBOXES 603
B.1 WAVELAB 603
8.2 LASTWAVE 609
B.3 Freeware Wavelet Toolboxes 610
BIBLIOGRAPHY 612
INDEX 629