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A Wavelet Tour of Signal Processing - 3rd Edition - ISBN: 9780123743701, 9780080922027

A Wavelet Tour of Signal Processing

3rd Edition

The Sparse Way

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Author: Stephane Mallat
Hardcover ISBN: 9780123743701
eBook ISBN: 9780080922027
Imprint: Academic Press
Published Date: 11th December 2008
Page Count: 832
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Description

Mallat's book is the undisputed reference in this field - it is the only one that covers the essential material in such breadth and depth. - Laurent Demanet, Stanford University

The new edition of this classic book gives all the major concepts, techniques and applications of sparse representation, reflecting the key role the subject plays in today's signal processing. The book clearly presents the standard representations with Fourier, wavelet and time-frequency transforms, and the construction of orthogonal bases with fast algorithms. The central concept of sparsity is explained and applied to signal compression, noise reduction, and inverse problems, while coverage is given to sparse representations in redundant dictionaries, super-resolution and compressive sensing applications.

Features:

* Balances presentation of the mathematics with applications to signal processing
* Algorithms and numerical examples are implemented in WaveLab, a MATLAB toolbox

New in this edition

* Sparse signal representations in dictionaries
* Compressive sensing, super-resolution and source separation
* Geometric image processing with curvelets and bandlets
* Wavelets for computer graphics with lifting on surfaces
* Time-frequency audio processing and denoising
* Image compression with JPEG-2000
* New and updated exercises

A Wavelet Tour of Signal Processing: The Sparse Way, Third Edition, is an invaluable resource for researchers and R&D engineers wishing to apply the theory in fields such as image processing, video processing and compression, bio-sensing, medical imaging, machine vision and communications engineering.

Stephane Mallat is Professor in Applied Mathematics at École Polytechnique, Paris, France. From 1986 to 1996 he was a Professor at the Courant Institute of Mathematical Sciences at New York University, and between 2001 and 2007, he co-founded and became CEO of an image processing semiconductor company.

Key Features

  • Includes all the latest developments since the book was published in 1999, including its application to JPEG 2000 and MPEG-4
  • Algorithms and numerical examples are implemented in Wavelab, a MATLAB toolbox
  • Balances presentation of the mathematics with applications to signal processing
  • Readership

    R&D engineers and university researchers in image and signal processing; Signal processing and applied mathematics graduates

    Table of Contents

    Preface to the Sparse Edition

    Notations

    1. Sparse Representations

    2. 1 Computational Harmonic Analysis

    3. 1.1 Fourier Kingdom

    4. 1.2 Wavelet Bases

    5. 2 Approximation and Processing in Bases

    6. 2.1 Sampling with Linear Approximations

    7. 2.2 Sparse Non-linear Approximations

    8. 2.3 Compression

    9. 2.4 Denoising

    10. 3 Time-Frequency Dictionaries

    11. 3.1 Heisenberg Uncertainty

    12. 3.2 Windowed Fourier Transform

    13. 3.3 Continuous Wavelet Transform

    14. 3.4 Time-Frequency Orthonormal Bases

    15. 4 Sparsity in Redundant Dictionaries

    16. 4.1 Frame Analysis and Synthesis

    17. 4.2 Ideal Dictionary Approximations

    18. 4.3 Pursuit in Dictionaries

    19. 5 Inverse Problems

    20. 5.1 Diagonal Inverse Estimation

    21. 5.2 Super-Resolution and Compressive Sensing

    22. 6 Travel Guide

    23. Fourier Kingdom

    24. 1 Linear Time-Invariant Filtering

    25. 1.1 Impulse Response

    26. 1.2 Transfer Functions

    27. 2 Fourier Integrals

    28. 2.1 Fourier Transform in L1(R)

    29. 2.2 Fourier Transform in L2(R)

    30. 2.3 Examples

    31. 3 Properties

    32. 3.1 Regularity and Decay

    33. 3.2 Uncertainty Principle

    34. 3.3 Total Variation

    35. 4 Two-Dimensional Fourier Transform

    36. 5 Exercises

    37. Discrete Revolution

    38. 1 Sampling Analog Signals

    39. 1.1 Shannon-Whittaker Sampling Theorem

    40. 1.2 Aliasing

    41. 1.3 General Sampling and Linear Analog Conversions

    42. 2 Discrete Time-Invariant Filters

    43. 2.1 Impulse Response and Transfer Function

    44. 2.2 Fourier Series

    45. 3 Finite Signals

    46. 3.1 Circular Convolutions

    47. 3.2 Discrete Fourier Transform

    48. 3.3 Fast Fourier Transform

    49. 3.4 Fast Convolutions

    50. 4 Discrete Image Processing

    51. 4.1 Two-Dimensional Sampling Theorems

    52. 4.2 Discrete Image Filtering

    53. 4.3 Circular Convolutions and Fourier Basis

    54. 5 Exercises

    4 Time Meets Frequency 4.1 Time-Frequency Atoms 4.2 Windowed Fourier Transform 4.2.1 Completeness and Stability 4.2.2 Choice of Window 4.2.3 Discrete Windowed Fourier Transform 4.3 Wavelet Transforms 4.3.1 Real Wavelets 4.3.2 Analytic Wavelets 4.3.3 Discrete Wavelets 4.4 Time-Frequency Geometry of Instantaneous Frequencies 4.4.1 Windowed Fourier Ridges 4.4.2 Wavelet Ridges 4.5 Quadratic Time-Frequency Energy 4.5.1 Wigner-Ville Distribution 4.5.2 Interferences and Positivity 4.5.3 Cohen¡¯s Class 4.5.4 Discrete Wigner-Ville Computations 4.6 Exercises

    1. Frames

    2. 1 Frames and Riesz Bases

    3. 1.1 Stable Analysis and Synthesis Operators

    4. 1.2 Dual Frame and Pseudo Inverse

    5. 1.3 Dual Frame Analysis and Synthesis Computations

    6. 1.4 Frame Projector and Reproducing Kernel

    7. 1.5 Translation Invariant Frames

    8. 2 Translation Invariant Dyadic Wavelet Transform

    9. 2.1 Dyadic Wavelet Design

    10. 2.2 ¡°Algorithme `a Trous¡±

    11. 3 Subsampled Wavelet Frames

    12. 4 Windowed Fourier Frames

    13. 5 Multiscale Directional Frames for Images

    14. 5.1 Directional Wavelet Frames

    15. 5.2 Curvelet Frames

    16. 6 Exercises

    17. Wavelet Zoom

    18. 1 Lipschitz Regularity

    19. 1.1 Lipschitz Definition and Fourier Analysis

    20. 1.2 Wavelet Vanishing Moments

    21. 1.3 Regularity Measurements with Wavelets

    22. 2 Wavelet Transform Modulus Maxima

    23. 2.1 Detection of Singularities

    24. 2.2 Dyadic Maxima Representation

    25. 3 Multiscale Edge Detection

    26. 3.1 Wavelet Maxima for Images

    27. 3.2 Fast Multiscale Edge Computations

    28. 4 Multifractals

    29. 4.1 Fractal Sets and Self-Similar Functions

    30. 4.2 Singularity Spectrum

    31. 4.3 Fractal Noises

    32. 5 Exercises

    33. Wavelet Bases

    34. 1 Orthogonal Wavelet Bases

    35. 1.1 Multiresolution Approximations

    36. 1.2 Scaling Function

    37. 1.3 Conjugate Mirror Filters

    38. 1.4 In Which Orthogonal Wavelets Finally Arrive

    39. 2 Classes of Wavelet Bases

    40. 2.1 Choosing a Wavelet

    41. 2.2 Shannon, Meyer and Battle-Lemari¢¥e Wavelets

    42. 2.3 Daubechies Compactly Supported Wavelets

    43. 3 Wavelets and Filter Banks

    44. 3.1 Fast Orthogonal Wavelet Transform

    45. 3.2 Perfect Reconstruction Filter Banks

    46. 3.3 Biorthogonal Bases of §¤2(Z)

    47. 4 Biorthogonal Wavelet Bases

    48. 4.1 Construction of Biorthogonal Wavelet Bases

    49. 4.2 Biorthogonal Wavelet Design

    50. 4.3 Compactly Supported Biorthogonal Wavelets

    51. 5 Wavelet Bases on an Interval

    52. 5.1 Periodic Wavelets

    53. 5.2 Folded Wavelets

    54. 5.3 Boundary Wavelets

    55. 6 Multiscale Interpolations

    56. 6.1 Interpolation and Sampling Theorems

    57. 6.2 Interpolation Wavelet Basis

    58. 7 Separable Wavelet Bases

    59. 7.1 Separable Multiresolutions

    60. 7.2 Two-Dimensional Wavelet Bases

    61. 7.3 Fast Two-Dimensional Wavelet Transform

    62. 7.4 Wavelet Bases in Higher Dimensions

    63. 8 Lifting Wavelets

    64. 8.1 Biorthogonal Bases over Non-stationary Grids

    65. 8.2 The Lifting Scheme

    66. 8.3 Quincunx Wavelet Bases

    67. 8.4 Wavelets on Bounded Domains and Surfaces

    68. 8.5 Faster Wavelet Transform with Lifting

    69. 9 Exercises

    70. Wavelet Packet and Local Cosine Bases

    71. 1 Wavelet Packets

    72. 1.1 Wavelet Packet Tree

    73. 1.2 Time-Frequency Localization

    74. 1.3 Particular Wavelet Packet Bases

    75. 1.4 Wavelet Packet Filter Banks

    76. 2 Image Wavelet Packets

    77. 2.1 Wavelet Packet Quad-Tree

    78. 2.2 Separable Filter Banks

    79. 3 Block Transforms

    80. 3.1 Block Bases

    81. 3.2 Cosine Bases

    82. 3.3 Discrete Cosine Bases

    83. 3.4 Fast Discrete Cosine Transforms

    84. 4 Lapped Orthogonal Transforms

    85. 4.1 Lapped Projectors

    86. 4.2 Lapped Orthogonal Bases

    87. 4.3 Local Cosine Bases

    88. 4.4 Discrete Lapped Transforms

    89. 5 Local Cosine Trees

    90. 5.1 Binary Tree of Cosine Bases

    91. 5.2 Tree of Discrete Bases

    92. 5.3 Image Cosine Quad-Tree

    93. 6 Exercises

    94. Approximations in Bases

    95. 1 Linear Approximations

    96. 1.1 Sampling and Approximation Error

    97. 1.2 Linear Fourier Approximations .

    98. 1.3 Multiresolution Approximation Errors with Wavelets

    99. 1.4 Karhunen-Lo`eve Approximations

    100. 2 Non-Linear Approximations

    101. 2.1 Non-Linear Approximation Error

    102. 2.2 Wavelet Adaptive Grids

    103. 2.3 Approximations in Besov and Bounded Variation Spaces

    104. 3 Sparse Image Representations

    105. 3.1 Wavelet Image Approximations

    106. 3.2 Geometric Image Models and Adaptive Triangulations

    107. 3.3 Curvelet Approximations

    108. 4 Exercises

    109. Compression

    110. 1 Transform Coding

    111. 1.1 Compression State of the Art

    112. 1.2 Compression in Orthonormal Bases

    113. 2 Distortion Rate of Quantization

    114. 2.1 Entropy Coding

    115. 2.2 Scalar Quantization

    116. 3 High Bit Rate Compression

    117. 3.1 Bit Allocation

    118. 3.2 Optimal Basis and Karhunen-Lo`eve

    119. 3.3 Transparent Audio Code

    120. 4 Sparse Signal Compression

    121. 4.1 Distortion Rate and Wavelet Image Coding

    122. 4.2 Embedded Transform Coding

    123. 5 Image Compression Standards

    124. 5.1 JPEG Block Cosine Coding

    125. 5.2 JPEG-2000 Wavelet Coding

    126. 6 Exercises

    127. Denoising

    128. 1 Estimation with Additive Noise

    129. 1.1 Bayes Estimation

    130. 1.2 Minimax Estimation

    131. 2 Diagonal Estimation in a Basis

    132. 2.1 Diagonal Estimation with Oracles

    133. 2.2 Thresholding Estimation

    134. 2.3 Thresholding Refinements

    135. 2.4 Wavelet Thresholding

    136. 2.5 Wavelet and Curvelet Image Denoising

    137. 2.6 Audio Denoising by Time-Frequency Thresholding

    138. 3 Non-Diagonal Block Thresholding

    139. 3.1 Block Thresholding in Bases and Frames

    140. 3.2 Wavelet Block Thresholding

    141. 3.3 Time-Frequency Audio Block Thresholding

    142. 4 Denoising Minimax Optimality

    143. 4.1 Linear Diagonal Minimax Estimation

    144. 4.2 Orthosymmetric Sets

    145. 4.3 Nearly Minimax with Wavelet Thresholding

    146. 5 Exercises

    147. Sparse in Redundant Dictionaries

    148. 1 Ideal Sparse Processing in Dictionaries

    149. 1.1 Best Approximation

    150. 1.2 Compression by Support Coding in a Dictionary

    151. 1.3 Denoising in a Dictionary

    152. 2 Dictionaries of Orthonormal Bases

    153. 2.1 Approximation, Compression and Denoising in a Best Basis

    154. 2.2 Fast Best Basis Search in Tree Dictionaries

    155. 2.3 Wavelet Packet and Local Cosine Best Bases

    156. 2.4 Bandlet Dictionaries for Geometric Processing

    157. 3 Greedy Pursuits

    158. 3.1 Matching Pursuit

    159. 3.2 Orthogonal Matching Pursuit .

    160. 3.3 Gabor Dictionaries

    161. 3.4 Learning Dictionaries

    162. 3.5 Coherent Matching Pursuit Denoising

    163. 4 l1 Pursuits

    164. 4.1 Basis Pursuit

    165. 4.2 l1 Lagrangian Pursuit

    166. 5 Approximation Performance of Pursuits

    167. 5.1 Support Identification and Stability

    168. 5.2 Support Dependent Success of Pursuits

    169. 5.3 Sparsity Dependent Criterions and Mutual-Coherence

    170. 6 Inverse Problems

    171. 6.1 Linear Estimation and Singular Value Decompositions

    172. 6.2 Thresholding Inverse Problem Estimators

    173. 6.3 Super-Resolution

    174. 6.4 Compressive Sensing

    175. 6.5 Source Separation

    176. 7 Exercises

    A. Mathematical Complements A.1 Functions and Integration A.2 Banach and Hilbert Spaces A.3 Bases of Hilbert Spaces A.4 Linear Operators A.5 Separable Spaces and Bases A.6 Random Vectors and Covariance Operators A.7 Diracs

    Details

    No. of pages:
    832
    Language:
    English
    Copyright:
    © Academic Press 2009
    Published:
    11th December 2008
    Imprint:
    Academic Press
    Hardcover ISBN:
    9780123743701
    eBook ISBN:
    9780080922027

    About the Author

    Stephane Mallat

    Stéphane Mallat is a 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 ccole Polytechnique, Paris, France. He has been a visiting professor in the ElectricalEngineering Department at Massachusetts Institute of Technology and in the Applied Mathematics Department at the University of Tel Aviv. Dr. Mallat received the 1990 IEEE Signal Processing Society's paper award, the 1993 Alfred Sloan fellowship in Mathematics, the 1997Outstanding Achievement Award from the SPIE Optical Engineering Society, and the 1997 Blaise Pascal Prize in applied mathematics, from theFrench Academy of Sciences.

    Affiliations and Expertise

    École Polytechique, Centre de Mathématiques Appliquées, Paris, France

    Reviews

    "There is no question that this revision should be published. Mallat’s book is the undisputed reference in this field – it is the only one that covers the essential material in such breadth and depth." - Laurent Demanet, Stanford University

    Ratings and Reviews