A Wavelet Tour of Signal Processing,
Résumé
This book is intended to serve as an invaluable reference for anyone concerned with the application of wavelets to signal processing. It has evolved from material used to teach "wavelet signal processing" courses in electrical engineering departments at Massachusetts Institute of Technology and Tel Aviv University, as well as applied mathematics departments at the Courant Institute of New York Universite and Ecole Polytechnique in Paris. Features: Provides a broad perspective on the principles and applications of transient signal processing with wavelets Emphasizes intuitive understanding, while providing the mathematical foundations and description of fast algorithms Numerous examples of real applications to noise removal, deconvolution, audio and image compression, singularity and edge detection, multifractal analysis, and time-varying frequency measurements Algorithms and numerical examples are implemented in Wavelab, which is a Matlab toolbox freely available over the Internet Content is accessible on several level of complexity, depending on the individual reader's needs New to the Second Edition: Optical flow calculation and video compression algorithms Image models with bounded variation functions Bayes and Minimax theories for signal estimation 200 pages rewritten and most illustrations redrawn More problems and topics for a graduate course in wavelet signal processing, in engineering and applied mathematics
Preface to the Second Edition xx
Notation xxii
Introduction to a Transient World
Fourier Kingdom 2
Time-Frequency Wedding 2
Windowed Fourier Transform 3
Wavelet Transform 4
Bases of Time-Frequency Atoms 6
Wavelet Bases and Filter Banks 7
Tilings of Wavelet Packet and Local 9
Cosine Bases
Bases for What? 11
Approximation 12
Estimation 14
Compression 16
Travel Guide 17
Reproducible Computational Science 17
Road Map 18
Fourier Kingdom
Linear Time-Invariant Filtering1 20
Impulse Response 21
Transfer Functions 22
Fourier Integrals1 22
Fourier Transform in L1(R) 23
Fourier Transform in L2(R) 25
Examples 27
Properties1 29
Regularity and Decay 29
Uncertainty Principle 30
Total Variation 33
Two-Dimensional Fourier Transform1 38
Problems 40
Discrete Revolution
Sampling Analog Signals1 42
Whittaker Sampling Theorem 43
Aliasing 44
General Sampling Theorems 47
Discrete Time-Invariant Filters1 49
Impulse Response and Transfer Function 49
Fourier Series 51
Finite Signals1 54
Circular Convolutions 55
Discrete Fourier Transform 55
Fast Fourier Transform 57
Fast Convolutions 58
Discrete Image Processing1 59
Two-Dimensional Sampling Theorem 60
Discrete Image Filtering 61
Circular Convolutions and Fourier Basis 62
Problems 64
Time Meets Frequency
Time-Frequency Atoms1 67
Windowed Fourier Transform1 69
Completeness and Stability 72
Choice of Window2 75
Discrete Windowed Fourier Transform2 77
Wavelet Transforms1 79
Real Wavelets 80
Analytic Wavelets 84
Discrete Wavelets2 89
Instantaneous Frequency2 91
Windowed Fourier Ridges 94
Wavelet Ridges 102
Quadratic Time-Frequency Energy1 107
Wigner-Ville Distribution 107
Interferences and Positivity 112
Cohen's Class2 116
Discrete Wigner-Ville Computations2 120
Problems 121
Frames
Frame Theory2 125
Frame Definition and Sampling 125
Pseudo Inverse 127
Inverse Frame Computations 132
Frame Projector and Noise Reduction 135
Windowed Fourier Frames2 138
Wavelet Frames2 143
Translation Invariance1 146
Dyadic Wavelet Transform2 148
Wavelet Design 150
``Algorithme a Trous' 153
Oriented Wavelets for a Vision3 156
Problems 160
Wavelet Zoom
Lipschitz Regularity1 163
Lipschitz Definition and Fourier Analysis 164
Wavelet Vanishing Moments 166
Regularity Measurements with Wavelets 169
Wavelet Transform Modulus Maxima2 176
Detection of Singularities 176
Reconstruction From Dyadic Maxima3 183
Multiscale Edge Detection2 189
Wavelet Maxima for Images2 189
Fast Multiscale Edge Computations3 197
Multifractals2 200
Fractal Sets and Self-Similar Functions 200
Singularity Spectrum3 205
Fractal Noises3 211
Problems 216
Wavelet Bases
Orthogonal Wavelet Bases1 220
Multiresolution Approximations 221
Scaling Function 224
Conjugate Mirror Filters 228
In Which Orthogonal Wavelets Finally 235
Arrive
Classes of Wavelet Bases1 241
Choosing a Wavelet 241
Shannon, Meyer and Battle-Lemarie Wavelets 246
Daubechies Compactly Supported Wavelets 249
Wavelets and Filter Banks1 255
Fast Orthogonal Wavelet Transform 255
Perfect Reconstruction Filter Banks 259
Biorthogonal Bases of l2(Z)2 263
Biorthogonal Wavelet Bases2 265
Construction of Biorthogonal Wavelet Bases 265
Biorthogonal Wavelet Design2 268
Compactly Supported Biorthogonal Wavelets2 270
Lifting Wavelets3 273
Wavelet Bases on an Interval2 281
Periodic Wavelets 282
Folded Wavelets 284
Boundary Wavelets3 286
Multiscale Interpolations2 293
Interpolation and Sampling Theorems 293
Interpolation Wavelet Basis3 299
Separable Wavelet Bases1 303
Separable Multiresolutions 304
Two-Dimensional Wavelet Bases 306
Fast Two-Dimensional Wavelet Transform 310
Wavelet Bases in Higher Dimensions2 313
Problems 314
Wavelet Packet and Local Cosine Bases
Wavelet Packets2 322
Wavelet Packet Tree 322
Time-Frequency Localization 327
Particular Wavelet Packet Bases 333
Wavelet Packet Filter Banks 336
Image Wavelet Packets2 339
Wavelet Packet Quad-Tree 339
Separable Filter Banks 341
Block Transforms1 343
Block Bases 344
Cosine Bases 346
Discrete Cosine Bases 349
Fast Discrete Cosine Transforms2 350
Lapped Orthogonal Transforms2 353
Lapped Projectors 353
Lapped Orthogonal Bases 359
Local Cosine Bases 361
Discrete Lapped Transforms 364
Local Cosine Trees2 368
Binary Tree of Cosine Bases 369
Tree of Discrete Bases 371
Image Cosine Quad-Tree 372
Problems 374
An Approximation Tour
Linear Approximations1 377
Linear Approximation Error 377
Linear Fourier Approximations 378
Linear Multiresolution Approximations 382
Karhunen-Loeve Approximations2 385
Non-Linear Approximations1 389
Non-Linear Approximation Error 389
Wavelet Adaptive Grids 391
Besov Spaces3 394
Image Approximations with Wavelets1 398
Adaptive Basis Selection2 405
Best Basis and Schur Concavity 406
Fast Best Basis Search in Trees 411
Wavelet Packet and Local Cosine Best Bases 413
Approximations with Pursuits3 417
Basis Pursuit 418
Matching Pursuit 421
Orthogonal Matching Pursuit 428
Problems 430
Estimations are Approximations
Bayes Versus Minimax2 435
Bayes Estimation 435
Minimax Estimation 442
Diagonal Estimation in a Basis2 446
Diagonal Estimation with Oracles 446
Thresholding Estimation 450
Thresholding Refinements3 455
Wavelet Thresholding 458
Best Basis Thresholding3 466
Minimax Optimality3 469
Linear Diagonal Minimax Estimation 469
Orthosymmetric Sets 474
Nearly Minimax with Wavelets 479
Restoration3 486
Estimation in Arbitrary Gaussian Noise 486
Inverse Problems and Deconvolution 491
Coherent Estimation3 501
Coherent Basis Thresholding 502
Coherent Matching Pursuit 505
Spectrum Estimation2 507
Power Spectrum 508
Approximate Karhunen-Loeve Search3 512
Locally Stationary Processes3 516
Problems 520
Transform Coding
Signal Compression2 526
State of the Art 526
Compression in Orthonormal Bases 527
Distortion Rate of Quantization2 528
Entropy Coding 529
Scalar Quantization 537
High Bit Rate Compression2 540
Bit Allocation 540
Optimal Basis and Karhunen-Loeve 542
Transparent Audio Code 544
Image Compression2 548
Deterministic Distortion Rate 548
Wavelet Image Coding 557
Block Cosine Image Coding 561
Embedded Transform Coding 566
Minimax Distortion Rate3 571
Video Signals2 577
Optical Flow 577
MPEG Video Compression 585
Problems 587
APPENDIX A MATHEMATICAL COMPLEMENTS
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
B.1 WaveLab 603
B.2 LastWave 609
B.3 Freeware Wavelet Toolboxes 610
Bibliography 612
Index 629
L'auteur - Stephane Mallat
is an Associate Professor in the Computer Science
Department of the Courant institute of Mathematical
Sciences at New York University, and a Professor in the
Applied Mathematics Department at Ecole Polytechnique,
Paris, France. He has been a visiting Professor in the
Electrical Engineering Department at Massachusetts
Institute of Technology and in the Applied Mathematics
Department at the University of Tel Aviv. His research
interests include computer vision, signal processing and
diverse applications of wavelet transforms. Dr. Mallat
received the 1990 IEEE Signal Processing Society's paper
award, the 1993 Alfred Sloan fellowship in Mathematics, the
1997 Outstanding Achievement Award from the SPIE Optical
Engineering Society, and the 1997 Blaise Pascal Prize in
applied mathematics, from the French Academy of
Sciences.
Autres livres de Stephane Mallat
Caractéristiques techniques
| PAPIER | |
| Éditeur(s) | Apress |
| Auteur(s) | Stephane Mallat |
| Parution | 01/09/1999 |
| Édition | 2eme édition |
| Nb. de pages | 637 |
| Intérieur | Noir et Blanc |
| EAN13 | 9780124666061 |
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