A Wavelet Tour of Signal Processing

3rd Edition

The Sparse Way

Authors: Stephane Mallat
Print ISBN: 9780123743701
eBook ISBN: 9780080922027
Imprint: Academic Press
Published Date: 11th December 2008
Page Count: 832
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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.


  • Balances presentation of the mathematics with applications to signal processing
  • Algorithms and numerical examples are implemented in WaveLab, a MATLAB toolbox
  • Companion website for instructors and selected solutions and code available for students

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 engi

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


    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


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    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


    "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