A Wavelet Tour of Signal Processing
3rd Edition
The Sparse Way
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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
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
Sparse Representations
1 Computational Harmonic Analysis
1.1 Fourier Kingdom
1.2 Wavelet Bases
2 Approximation and Processing in Bases
2.1 Sampling with Linear Approximations
2.2 Sparse Non-linear Approximations
2.3 Compression
2.4 Denoising
3 Time-Frequency Dictionaries
3.1 Heisenberg Uncertainty
3.2 Windowed Fourier Transform
3.3 Continuous Wavelet Transform
3.4 Time-Frequency Orthonormal Bases
4 Sparsity in Redundant Dictionaries
4.1 Frame Analysis and Synthesis
4.2 Ideal Dictionary Approximations
4.3 Pursuit in Dictionaries
5 Inverse Problems
5.1 Diagonal Inverse Estimation
5.2 Super-Resolution and Compressive Sensing
6 Travel Guide
Fourier Kingdom
1 Linear Time-Invariant Filtering
1.1 Impulse Response
1.2 Transfer Functions
2 Fourier Integrals
2.1 Fourier Transform in L1(R)
2.2 Fourier Transform in L2(R)
2.3 Examples
3 Properties
3.1 Regularity and Decay
3.2 Uncertainty Principle
3.3 Total Variation
4 Two-Dimensional Fourier Transform
5 Exercises
Discrete Revolution
1 Sampling Analog Signals
1.1 Shannon-Whittaker Sampling Theorem
1.2 Aliasing
1.3 General Sampling and Linear Analog Conversions
2 Discrete Time-Invariant Filters
2.1 Impulse Response and Transfer Function
2.2 Fourier Series
3 Finite Signals
3.1 Circular Convolutions
3.2 Discrete Fourier Transform
3.3 Fast Fourier Transform
3.4 Fast Convolutions
4 Discrete Image Processing
4.1 Two-Dimensional Sampling Theorems
4.2 Discrete Image Filtering
4.3 Circular Convolutions and Fourier Basis
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
Frames
1 Frames and Riesz Bases
1.1 Stable Analysis and Synthesis Operators
1.2 Dual Frame and Pseudo Inverse
1.3 Dual Frame Analysis and Synthesis Computations
1.4 Frame Projector and Reproducing Kernel
1.5 Translation Invariant Frames
2 Translation Invariant Dyadic Wavelet Transform
2.1 Dyadic Wavelet Design
2.2 ¡°Algorithme `a Trous¡±
3 Subsampled Wavelet Frames
4 Windowed Fourier Frames
5 Multiscale Directional Frames for Images
5.1 Directional Wavelet Frames
5.2 Curvelet Frames
6 Exercises
Wavelet Zoom
1 Lipschitz Regularity
1.1 Lipschitz Definition and Fourier Analysis
1.2 Wavelet Vanishing Moments
1.3 Regularity Measurements with Wavelets
2 Wavelet Transform Modulus Maxima
2.1 Detection of Singularities
2.2 Dyadic Maxima Representation
3 Multiscale Edge Detection
3.1 Wavelet Maxima for Images
3.2 Fast Multiscale Edge Computations
4 Multifractals
4.1 Fractal Sets and Self-Similar Functions
4.2 Singularity Spectrum
4.3 Fractal Noises
5 Exercises
Wavelet Bases
1 Orthogonal Wavelet Bases
1.1 Multiresolution Approximations
1.2 Scaling Function
1.3 Conjugate Mirror Filters
1.4 In Which Orthogonal Wavelets Finally Arrive
2 Classes of Wavelet Bases
2.1 Choosing a Wavelet
2.2 Shannon, Meyer and Battle-Lemari¢¥e Wavelets
2.3 Daubechies Compactly Supported Wavelets
3 Wavelets and Filter Banks
3.1 Fast Orthogonal Wavelet Transform
3.2 Perfect Reconstruction Filter Banks
3.3 Biorthogonal Bases of §¤2(Z)
4 Biorthogonal Wavelet Bases
4.1 Construction of Biorthogonal Wavelet Bases
4.2 Biorthogonal Wavelet Design
4.3 Compactly Supported Biorthogonal Wavelets
5 Wavelet Bases on an Interval
5.1 Periodic Wavelets
5.2 Folded Wavelets
5.3 Boundary Wavelets
6 Multiscale Interpolations
6.1 Interpolation and Sampling Theorems
6.2 Interpolation Wavelet Basis
7 Separable Wavelet Bases
7.1 Separable Multiresolutions
7.2 Two-Dimensional Wavelet Bases
7.3 Fast Two-Dimensional Wavelet Transform
7.4 Wavelet Bases in Higher Dimensions
8 Lifting Wavelets
8.1 Biorthogonal Bases over Non-stationary Grids
8.2 The Lifting Scheme
8.3 Quincunx Wavelet Bases
8.4 Wavelets on Bounded Domains and Surfaces
8.5 Faster Wavelet Transform with Lifting
9 Exercises
Wavelet Packet and Local Cosine Bases
1 Wavelet Packets
1.1 Wavelet Packet Tree
1.2 Time-Frequency Localization
1.3 Particular Wavelet Packet Bases
1.4 Wavelet Packet Filter Banks
2 Image Wavelet Packets
2.1 Wavelet Packet Quad-Tree
2.2 Separable Filter Banks
3 Block Transforms
3.1 Block Bases
3.2 Cosine Bases
3.3 Discrete Cosine Bases
3.4 Fast Discrete Cosine Transforms
4 Lapped Orthogonal Transforms
4.1 Lapped Projectors
4.2 Lapped Orthogonal Bases
4.3 Local Cosine Bases
4.4 Discrete Lapped Transforms
5 Local Cosine Trees
5.1 Binary Tree of Cosine Bases
5.2 Tree of Discrete Bases
5.3 Image Cosine Quad-Tree
6 Exercises
Approximations in Bases
1 Linear Approximations
1.1 Sampling and Approximation Error
1.2 Linear Fourier Approximations .
1.3 Multiresolution Approximation Errors with Wavelets
1.4 Karhunen-Lo`eve Approximations
2 Non-Linear Approximations
2.1 Non-Linear Approximation Error
2.2 Wavelet Adaptive Grids
2.3 Approximations in Besov and Bounded Variation Spaces
3 Sparse Image Representations
3.1 Wavelet Image Approximations
3.2 Geometric Image Models and Adaptive Triangulations
3.3 Curvelet Approximations
4 Exercises
Compression
1 Transform Coding
1.1 Compression State of the Art
1.2 Compression in Orthonormal Bases
2 Distortion Rate of Quantization
2.1 Entropy Coding
2.2 Scalar Quantization
3 High Bit Rate Compression
3.1 Bit Allocation
3.2 Optimal Basis and Karhunen-Lo`eve
3.3 Transparent Audio Code
4 Sparse Signal Compression
4.1 Distortion Rate and Wavelet Image Coding
4.2 Embedded Transform Coding
5 Image Compression Standards
5.1 JPEG Block Cosine Coding
5.2 JPEG-2000 Wavelet Coding
6 Exercises
Denoising
1 Estimation with Additive Noise
1.1 Bayes Estimation
1.2 Minimax Estimation
2 Diagonal Estimation in a Basis
2.1 Diagonal Estimation with Oracles
2.2 Thresholding Estimation
2.3 Thresholding Refinements
2.4 Wavelet Thresholding
2.5 Wavelet and Curvelet Image Denoising
2.6 Audio Denoising by Time-Frequency Thresholding
3 Non-Diagonal Block Thresholding
3.1 Block Thresholding in Bases and Frames
3.2 Wavelet Block Thresholding
3.3 Time-Frequency Audio Block Thresholding
4 Denoising Minimax Optimality
4.1 Linear Diagonal Minimax Estimation
4.2 Orthosymmetric Sets
4.3 Nearly Minimax with Wavelet Thresholding
5 Exercises
Sparse in Redundant Dictionaries
1 Ideal Sparse Processing in Dictionaries
1.1 Best Approximation
1.2 Compression by Support Coding in a Dictionary
1.3 Denoising in a Dictionary
2 Dictionaries of Orthonormal Bases
2.1 Approximation, Compression and Denoising in a Best Basis
2.2 Fast Best Basis Search in Tree Dictionaries
2.3 Wavelet Packet and Local Cosine Best Bases
2.4 Bandlet Dictionaries for Geometric Processing
3 Greedy Pursuits
3.1 Matching Pursuit
3.2 Orthogonal Matching Pursuit .
3.3 Gabor Dictionaries
3.4 Learning Dictionaries
3.5 Coherent Matching Pursuit Denoising
4 l1 Pursuits
4.1 Basis Pursuit
4.2 l1 Lagrangian Pursuit
5 Approximation Performance of Pursuits
5.1 Support Identification and Stability
5.2 Support Dependent Success of Pursuits
5.3 Sparsity Dependent Criterions and Mutual-Coherence
6 Inverse Problems
6.1 Linear Estimation and Singular Value Decompositions
6.2 Thresholding Inverse Problem Estimators
6.3 Super-Resolution
6.4 Compressive Sensing
6.5 Source Separation
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
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