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 University and ?‰colePolytechnique in Paris. Key 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 needsNew 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
Author(s): Stephane Mallat
Edition: 2
Year: 1999
Language: English
Pages: 620
Front Cover......Page 1
A Wavelet Tour of Signal Processing......Page 4
Copyright Page......Page 5
Contents......Page 8
Preface......Page 16
Preface to the Second Edition......Page 21
Notation......Page 23
CHAPTER I. INTRODUCTION TO A TRANSIENT WORLD......Page 26
1.2 Time-Frequency Wedding......Page 27
1.3 Bases of Time-Frequency Atoms......Page 31
1.4 Bases for What?......Page 36
1.5 Travel Guide......Page 42
2.1 Linear Time-Invariant Filtering......Page 45
2.2 Fourier Integrals......Page 47
2.3 Properties......Page 54
2.4 Two-Dimensional Fourier Transform......Page 63
2.5 Problems......Page 65
3.1 Sampling Analog Signals......Page 67
3.2 Discrete Time-Invariant Filters......Page 74
3.3 Finite Signals......Page 79
3.4 Discrete Image Processing......Page 84
3.5 Problems......Page 89
4.1 Time-Frequency Atoms......Page 92
4.2 Windowed Fourier Transform......Page 94
4.3 Wavelet Transforms......Page 104
4.4 Instantaneous Frequency......Page 116
4.5 Quadratic Time-Frequency Energy......Page 132
4.6 Problems......Page 146
5.1 Frame Theory......Page 150
5.2 Windowed Fourier Frames......Page 163
5.3 Wavelet Frames......Page 168
5.4 Translation Invariance......Page 171
5.5 Dyadic Wavelet Transform......Page 173
5.6 Problems......Page 185
6.1 Lipschitz Regularity......Page 188
6.2 Wavelet Transform Modulus Maxima......Page 201
6.3 Multiscale Edge Detection......Page 214
6.4 Multifractals......Page 225
6.5 Problems......Page 241
7.1 Orthogonal Wavelet Bases......Page 245
7.2 Classes of Wavelet Bases......Page 266
7.3 Wavelets and Filter Banks......Page 280
7.4 Biorthogonal Wavelet Bases......Page 290
7.5 Wavelet Bases on an Interval......Page 306
7.6 Multiscale Interpolations......Page 318
7.7 Separable Wavelet Bases......Page 328
7.8 Problems......Page 339
CHAPTER VIII. WAVELET PACKET AND LOCAL COSINE BASES......Page 346
8.1 Wavelet Packets......Page 347
8.2 Image Wavelet Packets......Page 364
8.3 Block Transforms......Page 368
8.4 Lapped Orthogonal Transforms......Page 378
8.5 Local Cosine Trees......Page 393
8.6 Problems......Page 399
CHAPTER IX. AN APPROXIMATION TOUR......Page 401
9.1 Linear Approximations......Page 402
9.2 Non-Linear Approximations......Page 414
9.3 Image Approximations with Wavelets......Page 423
9.4 Adaptive Basis Selection......Page 430
9.5 Approximations with Pursuits......Page 442
9.6 Problems......Page 455
CHAPTER X. ESTIMATIONS ARE APPROXIMATIONS......Page 459
10.1 Bayes Versus Minimax......Page 460
10.2 Diagonal Estimation in a Basis......Page 471
10.3 Minimax Optimality......Page 494
10.4 Restoration......Page 511
10.5 Coherent Estimation......Page 526
10.6 Spectrum Estimation......Page 532
10.7 Problems......Page 545
CHAPTER XI. TRANSFORM CODING......Page 550
11.1 Signal Compression......Page 551
11.2 Distortion Rate of Quantization......Page 553
11.3 High Bit Rate Compression......Page 565
11.4 Image Compression......Page 573
11.5 Video Signals......Page 602
11.6 Problems......Page 612
A.1 Functions and Integration......Page 616
A.2 Banach and Hilbert Spaces......Page 618
A.3 Bases of Hilbert Spaces......Page 620
A.4 Linear Operators......Page 621
A.5 Separable Spaces and Bases......Page 623
A.6 Random Vectors and Covariance Operators......Page 624
A.7 Diracs......Page 626
B.1 Wavelas......Page 628
B.2 Lastwave......Page 634
B.3 Freeware Wavelet Toolboxes......Page 635
Bibliography......Page 637
Index......Page 654
