A Wavelet Tour of Signal Processing The Sparse Way

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1 A Wavelet Tour of Signal Processing The Sparse Way Stephane Mallat with contributions from Gabriel Peyre AMSTERDAM BOSTON HEIDELBERG LONDON NEWYORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY»TOKYO Ж Academic Press is an imprint of Elsevier

2 Preface to the Sparse Edition Notations xv xix CHAPTER 1 Sparse Representations Computational Harmonic Analysis The Fourier Kingdom Wavelet Bases Approximation and Processing in Bases Sampling with Linear Approximations Sparse Nonlinear Approximations Compression Denoising Time-Frequency Dictionaries Heisenberg Uncertainty Windowed Fourier Transform Continuous Wavelet Transform Time-Frequency Orthonormal Bases Sparsity in Redundant Dictionaries Frame Analysis and Synthesis Ideal Dictionary Approximations Pursuit in Dictionaries Inverse Problems Diagonal Inverse Estimation Super-resolution and Compressive Sensing Travel Guide Reproducible Computational Science Book Road Map 30 CHAPTER 2 The Fourier Kingdom Linear Time-Invariant Filtering Impulse Response Transfer Functions Fourier Integrals Fourier Transform in L J (R) Fourier Transform in L 2 (K) Examples Properties Regularity and Decay Uncertainty Principle 43 vi

3 viii 2.3.З Total Variation Two-Dimensional Fourier Transform Exercises 55 CHAPTER 3 Discrete Revolution Sampling Analog Signals Shannon-Whittaker Sampling Theorem Aliasing General Sampling and Linear Analog Conversions Discrete Time-Invariant Filters Impulse Response and Transfer Function Fourier Series Finite Signals Circular Convolutions Discrete Fourier Transform Fast Fourier Transform Fast Convolutions Discrete Image Processing Two-Dimensional Sampling Theorems Discrete Image Filtering Circular Convolutions and Fourier Basis Exercises 85 CHAPTER 4 Time Meets Frequency Time-Frequency Atoms Windowed Fourier Transform Completeness and Stability Choice of Window Discrete Windowed Fourier Transform Wavelet Transforms Real Wavelets Analytic Wavelets Discrete Wavelets 112 А Л Time-Frequency Geometry of Instantaneous Frequencies Analytic Instantaneous Frequency Windowed Fourier Ridges Wavelet Ridges Quadratic Time-Frequency Energy Wigner-Ville Distribution Interferences and Positivity Cohen's Class Discrete Wigner-Ville Computations Exercises 151

4 ix CHAPTER 5 Frames Frames and Riesz Bases Stable Analysis and Synthesis Operators Dual Frame and Pseudo Inverse Dual-Frame Analysis and Synthesis Computations Frame Projector and Reproducing Kernel Translation-Invariant Frames Translation-Invariant Dyadic Wavelet Transform Dyadic Wavelet Design Algorithme ätrous Subsampled Wavelet Frames Windowed Fourier Frames Tight Frames General Frames Multiscale Directional Frames for Images Directional Wavelet Frames Curvelet Frames Exercises 201 CHAPTER 6 Wavelet Zoom Lipschitz Regularity Lipschitz Definition and Fourier Analysis Wavelet Vanishing Moments Regularity Measurements with Wavelets Wavelet Transform Modulus Maxima Detection of Singularities Dyadic Maxima Representation Multiscale Edge Detection Wavelet Maxima for Images Fast Multiscale Edge Computations Multifractals Fractal Sets and Self-Similar Functions Singularity Spectrum Fractal Noises Exercises 259 CHAPTER 7 Wavelet Bases Orthogonal Wavelet Bases Multiresolution Approximations Scaling Function Conjugate Mirror Filters In Which Orthogonal Wavelets Finally Arrive Classes ofwavelet Bases Choosing a Wavelet 284

5 x Shannon, Meyer, Haar, and Battle-Lemarie Wavelets Daubechies Compactly Supported Wavelets Wavelets and Filter Banks Fast Orthogonal Wavelet Transform Perfect Reconstruction Filter Banks Biorthogonal Bases of l 2 (Z) Biorthogonal Wavelet Bases Construction of Biorthogonal Wavelet Bases Biorthogonal Wavelet Design Compactly Supported Biorthogonal Wavelets Wavelet Bases on an Interval Periodic Wavelets Folded Wavelets Boundary Wavelets Multiscale Interpolations Interpolation and Sampling Theorems Interpolation Wavelet Basis Separable Wavelet Bases Separable Multiresolutions Two-Dimensional Wavelet Bases Fast Two-Dimensional Wavelet Transform Wavelet Bases in Higher Dimensions Lifting Wavelets Biorthogonal Bases over Nonstationary Grids Lifting Scheme Quincunx Wavelet Bases Wavelets on Bounded Domains and Surfaces Faster Wavelet Transform with Lifting Exercises 370 CHAPTER 8 Wavelet Packet and Local Cosine Bases Wavelet Packets Wavelet Packet Tree Time-Frequency Localization Particular Wavelet Packet Bases Wavelet Packet Filter Banks Image Wavelet Packets Wavelet Packet Quad-Tree Separable Filter Banks Block Transforms Block Bases Cosine Bases Discrete Cosine Bases Fast Discrete Cosine Transforms 407

6 8.4 Lapped Orthogonal Transforms Lapped Projectors Lapped Orthogonal Bases Local Cosine Bases Discrete Lapped Transforms Local Cosine Trees Binary Tree of Cosine Bases Tree of Discrete Bases Image Cosine Quad-Tree Exercises 432 CHAPTER 9 Approximations in Bases Linear Approximations Sampling and Approximation Error Linear Fourier Approximations Multiresolution Approximation Errors with Wavelets Karhunen-LoeveApproximations Nonlinear Approximations Nonlinear Approximation Error Wavelet Adaptive Grids Approximations in Besov and Bounded Variation Spaces Sparse Image Representations Wavelet Image Approximations Geometric Image Models and Adaptive Triangulations Curvelet Approximations Exercises 478 CHAPTER 10 Compression Transform Coding Compression State of theart Compression in Orthonormal Bases Distortion Rate of Quantization Entropy Coding Scalar Quantization High Bit Rate Compression Bit Allocation Optimal Basis and Karhunen-Loeve Transparent Audio Code Sparse Signal Compression Distortion Rate and Wavelet Image Coding Embedded Transform Coding 516

7 xii 10.5 Image-Compression Standards JPEG Block Cosine Coding JPEG-2000 Wavelet Coding Exercises 531 CHAPTER 11 Denoising Estimation with Additive Noise Bayes Estimation Minimax Estimation Diagonal Estimation in a Basis Diagonal Estimation with Oracles Thresholding Estimation Thresholding Improvements Thresholding Sparse Representations Wavelet Thresholding Wavelet and Curvelet Image Denoising Audio Denoising by Time-Frequency Thresholding Nondiagonal Block Thresholding Block Thresholding in Bases and Frames Wavelet Block Thresholding Time-Frequency Audio Block Thresholding Denoising Minimax Optimality Linear Diagonal Minimax Estimation Thresholding Optimality over Orthosymmetric Sets Nearly Minimax with Wavelet Estimation Exercises 606 CHAPTER 12 Sparsity in Redundant Dictionaries Ideal Sparse Processing in Dictionaries Best M-TermApproximations Compression by Support Coding Denoising by Support Selection in a Dictionary Dictionaries of Orthonormal Bases Approximation, Compression, and Denoising in a Best Basis Fast Best-Basis Search in Tree Dictionaries Wavelet Packet and Local Cosine Best Bases Bandlets for Geometric Image Regularity Greedy Matching Pursuits Matching Pursuit Orthogonal Matching Pursuit Gabor Dictionaries Coherent Matching Pursuit Denoising 655

8 xiii 12.4 V Pursuits Basis Pursuit l 1 Lagrangian Pursuit Computations of l 1 Minimizations Sparse Synthesis versus Analysis andtotal Variation Regularization Pursuit Recovery Stability and Incoherence Support Recovery with Matching Pursuit Support Recovery with l 1 Pursuits Multichannel Signals Approximation and Denoising by Thresholding in Bases Multichannel Pursuits Learning Dictionaries Exercises 696 CHAPTER 13 Inverse Problems 699 I3.I Linear Inverse Estimation Quadratic and Tikhonov Regularizations Singular Value Decompositions Thresholding Estimators for Inverse Problems Thresholding in Bases of Almost Singular Vectors Thresholding Deconvolutions Super-resolution Sparse Super-resolution Estimation Sparse Spike Deconvolution Recovery of Missing Data Compressive Sensing Incoherence with Random Measurements Approximations with Compressive Sensing Compressive Sensing Applications Blind Source Separation Blind Mixing Matrix Estimation Source Separation Exercises 752 APPENDIX Mathematical Complements 753 Bibliography 765 Index 795

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