A Wavelet Tour of Signal Processing

A Wavelet Tour of Signal Processing

The Sparse Way

3rd Edition - December 11, 2008
This is the Latest Edition
  • Author: Stephane Mallat
  • eBook ISBN: 9780080922027
  • Hardcover ISBN: 9780123743701

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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 UniversityThe 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 toolboxNew 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 exercisesA 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
      1.1 Computational Harmonic Analysis
      1.1.1 Fourier Kingdom
      1.1.2 Wavelet Bases
      1.2 Approximation and Processing in Bases
      1.2.1 Sampling with Linear Approximations
      1.2.2 Sparse Non-linear Approximations
      1.2.3 Compression
      1.2.4 Denoising
      1.3 Time-Frequency Dictionaries
      1.3.1 Heisenberg Uncertainty
      1.3.2 Windowed Fourier Transform
      1.3.3 Continuous Wavelet Transform
      1.3.4 Time-Frequency Orthonormal Bases
      1.4 Sparsity in Redundant Dictionaries
      1.4.1 Frame Analysis and Synthesis
      1.4.2 Ideal Dictionary Approximations
      1.4.3 Pursuit in Dictionaries
      1.5 Inverse Problems
      1.5.1 Diagonal Inverse Estimation
      1.5.2 Super-Resolution and Compressive Sensing
      1.6 Travel Guide

      2. Fourier Kingdom
      2.1 Linear Time-Invariant Filtering
      2.1.1 Impulse Response
      2.1.2 Transfer Functions
      2.2 Fourier Integrals
      2.2.1 Fourier Transform in L1(R)
      2.2.2 Fourier Transform in L2(R)
      2.2.3 Examples
      2.3 Properties
      2.3.1 Regularity and Decay
      2.3.2 Uncertainty Principle
      2.3.3 Total Variation
      2.4 Two-Dimensional Fourier Transform
      2.5 Exercises

      3. Discrete Revolution
      3.1 Sampling Analog Signals
      3.1.1 Shannon-Whittaker Sampling Theorem
      3.1.2 Aliasing
      3.1.3 General Sampling and Linear Analog Conversions
      3.2 Discrete Time-Invariant Filters
      3.2.1 Impulse Response and Transfer Function
      3.2.2 Fourier Series
      3.3 Finite Signals
      3.3.1 Circular Convolutions
      3.3.2 Discrete Fourier Transform
      3.3.3 Fast Fourier Transform
      3.3.4 Fast Convolutions
      3.4 Discrete Image Processing
      3.4.1 Two-Dimensional Sampling Theorems
      3.4.2 Discrete Image Filtering
      3.4.3 Circular Convolutions and Fourier Basis
      3.5 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

      5. Frames
      5.1 Frames and Riesz Bases
      5.1.1 Stable Analysis and Synthesis Operators
      5.1.2 Dual Frame and Pseudo Inverse
      5.1.3 Dual Frame Analysis and Synthesis Computations
      5.1.4 Frame Projector and Reproducing Kernel
      5.1.5 Translation Invariant Frames
      5.2 Translation Invariant Dyadic Wavelet Transform
      5.2.1 Dyadic Wavelet Design
      5.2.2 ¡°Algorithme `a Trous¡±
      5.3 Subsampled Wavelet Frames
      5.4 Windowed Fourier Frames
      5.5 Multiscale Directional Frames for Images
      5.5.1 Directional Wavelet Frames
      5.5.2 Curvelet Frames
      5.6 Exercises

      6. Wavelet Zoom
      6.1 Lipschitz Regularity
      6.1.1 Lipschitz Definition and Fourier Analysis
      6.1.2 Wavelet Vanishing Moments
      6.1.3 Regularity Measurements with Wavelets
      6.2 Wavelet Transform Modulus Maxima
      6.2.1 Detection of Singularities
      6.2.2 Dyadic Maxima Representation
      6.3 Multiscale Edge Detection
      6.3.1 Wavelet Maxima for Images
      6.3.2 Fast Multiscale Edge Computations
      6.4 Multifractals
      6.4.1 Fractal Sets and Self-Similar Functions
      6.4.2 Singularity Spectrum
      6.4.3 Fractal Noises
      6.5 Exercises

      7. Wavelet Bases
      7.1 Orthogonal Wavelet Bases
      7.1.1 Multiresolution Approximations
      7.1.2 Scaling Function
      7.1.3 Conjugate Mirror Filters
      7.1.4 In Which Orthogonal Wavelets Finally Arrive
      7.2 Classes of Wavelet Bases
      7.2.1 Choosing a Wavelet
      7.2.2 Shannon, Meyer and Battle-Lemari¢¥e Wavelets
      7.2.3 Daubechies Compactly Supported Wavelets
      7.3 Wavelets and Filter Banks
      7.3.1 Fast Orthogonal Wavelet Transform
      7.3.2 Perfect Reconstruction Filter Banks
      7.3.3 Biorthogonal Bases of §¤2(Z)
      7.4 Biorthogonal Wavelet Bases
      7.4.1 Construction of Biorthogonal Wavelet Bases
      7.4.2 Biorthogonal Wavelet Design
      7.4.3 Compactly Supported Biorthogonal Wavelets
      7.5 Wavelet Bases on an Interval
      7.5.1 Periodic Wavelets
      7.5.2 Folded Wavelets
      7.5.3 Boundary Wavelets
      7.6 Multiscale Interpolations
      7.6.1 Interpolation and Sampling Theorems
      7.6.2 Interpolation Wavelet Basis
      7.7 Separable Wavelet Bases
      7.7.1 Separable Multiresolutions
      7.7.2 Two-Dimensional Wavelet Bases
      7.7.3 Fast Two-Dimensional Wavelet Transform
      7.7.4 Wavelet Bases in Higher Dimensions
      7.8 Lifting Wavelets
      7.8.1 Biorthogonal Bases over Non-stationary Grids
      7.8.2 The Lifting Scheme
      7.8.3 Quincunx Wavelet Bases
      7.8.4 Wavelets on Bounded Domains and Surfaces
      7.8.5 Faster Wavelet Transform with Lifting
      7.9 Exercises

      8. Wavelet Packet and Local Cosine Bases
      8.1 Wavelet Packets
      8.1.1 Wavelet Packet Tree
      8.1.2 Time-Frequency Localization
      8.1.3 Particular Wavelet Packet Bases
      8.1.4 Wavelet Packet Filter Banks
      8.2 Image Wavelet Packets
      8.2.1 Wavelet Packet Quad-Tree
      8.2.2 Separable Filter Banks
      8.3 Block Transforms
      8.3.1 Block Bases
      8.3.2 Cosine Bases
      8.3.3 Discrete Cosine Bases
      8.3.4 Fast Discrete Cosine Transforms
      8.4 Lapped Orthogonal Transforms
      8.4.1 Lapped Projectors
      8.4.2 Lapped Orthogonal Bases
      8.4.3 Local Cosine Bases
      8.4.4 Discrete Lapped Transforms
      8.5 Local Cosine Trees
      8.5.1 Binary Tree of Cosine Bases
      8.5.2 Tree of Discrete Bases
      8.5.3 Image Cosine Quad-Tree
      8.6 Exercises

      9. Approximations in Bases
      9.1 Linear Approximations
      9.1.1 Sampling and Approximation Error
      9.1.2 Linear Fourier Approximations .
      9.1.3 Multiresolution Approximation Errors with Wavelets
      9.1.4 Karhunen-Lo`eve Approximations
      9.2 Non-Linear Approximations
      9.2.1 Non-Linear Approximation Error
      9.2.2 Wavelet Adaptive Grids
      9.2.3 Approximations in Besov and Bounded Variation Spaces
      9.3 Sparse Image Representations
      9.3.1 Wavelet Image Approximations
      9.3.2 Geometric Image Models and Adaptive Triangulations
      9.3.3 Curvelet Approximations
      9.4 Exercises

      10. Compression
      10.1 Transform Coding
      10.1.1 Compression State of the Art
      10.1.2 Compression in Orthonormal Bases
      10.2 Distortion Rate of Quantization
      10.2.1 Entropy Coding
      10.2.2 Scalar Quantization
      10.3 High Bit Rate Compression
      10.3.1 Bit Allocation
      10.3.2 Optimal Basis and Karhunen-Lo`eve
      10.3.3 Transparent Audio Code
      10.4 Sparse Signal Compression
      10.4.1 Distortion Rate and Wavelet Image Coding
      10.4.2 Embedded Transform Coding
      10.5 Image Compression Standards
      10.5.1 JPEG Block Cosine Coding
      10.5.2 JPEG-2000 Wavelet Coding
      10.6 Exercises

      11. Denoising
      11.1 Estimation with Additive Noise
      11.1.1 Bayes Estimation
      11.1.2 Minimax Estimation
      11.2 Diagonal Estimation in a Basis
      11.2.1 Diagonal Estimation with Oracles
      11.2.2 Thresholding Estimation
      11.2.3 Thresholding Refinements
      11.2.4 Wavelet Thresholding
      11.2.5 Wavelet and Curvelet Image Denoising
      11.2.6 Audio Denoising by Time-Frequency Thresholding
      11.3 Non-Diagonal Block Thresholding
      11.3.1 Block Thresholding in Bases and Frames
      11.3.2 Wavelet Block Thresholding
      11.3.3 Time-Frequency Audio Block Thresholding
      11.4 Denoising Minimax Optimality
      11.4.1 Linear Diagonal Minimax Estimation
      11.4.2 Orthosymmetric Sets
      11.4.3 Nearly Minimax with Wavelet Thresholding
      11.5 Exercises

      12. Sparse in Redundant Dictionaries
      12.1 Ideal Sparse Processing in Dictionaries
      12.1.1 Best Approximation
      12.1.2 Compression by Support Coding in a Dictionary
      12.1.3 Denoising in a Dictionary
      12.2 Dictionaries of Orthonormal Bases
      12.2.1 Approximation, Compression and Denoising in a Best Basis
      12.2.2 Fast Best Basis Search in Tree Dictionaries
      12.2.3 Wavelet Packet and Local Cosine Best Bases
      12.2.4 Bandlet Dictionaries for Geometric Processing
      12.3 Greedy Pursuits
      12.3.1 Matching Pursuit
      12.3.2 Orthogonal Matching Pursuit .
      12.3.3 Gabor Dictionaries
      12.3.4 Learning Dictionaries
      12.3.5 Coherent Matching Pursuit Denoising
      12.4 l1 Pursuits
      12.4.1 Basis Pursuit
      12.4.2 l1 Lagrangian Pursuit
      12.5 Approximation Performance of Pursuits
      12.5.1 Support Identification and Stability
      12.5.2 Support Dependent Success of Pursuits
      12.5.3 Sparsity Dependent Criterions and Mutual-Coherence
      12.6 Inverse Problems
      12.6.1 Linear Estimation and Singular Value Decompositions
      12.6.2 Thresholding Inverse Problem Estimators
      12.6.3 Super-Resolution
      12.6.4 Compressive Sensing
      12.6.5 Source Separation
      12.7 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

    Product details

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