Front Cover -- A Wavelet Tour of Signal Processing -- Copyright Page -- Dedication Page -- Table of Contents -- Preface to the Sparse Edition -- Notations -- Chapter 1. Sparse Representations -- 1.1 Computational Harmonic Analysis -- 1.1.1 The 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 Nonlinear 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 -- 1.6.1 Reproducible Computational Science -- 1.6.2 Book Road Map -- Chapter 2. The 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 -- Chapter 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 -- Chapter 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 Analytic Instantaneous Frequency -- 4.4.2 Windowed Fourier Ridges -- 4.4.3 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 -- Chapter 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 à Trous -- 5.3 Subsampled Wavelet Frames -- 5.4 Windowed Fourier Frames -- 5.4.1 Tight Frames -- 5.4.2 General Frames -- 5.5 Multiscale Directional Frames For Images -- 5.5.1 Directional Wavelet Frames -- 5.5.2 Curvelet Frames -- 5.6 Exercises -- Chapter 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 -- Chapter 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, Haar, and Battle-Lemarié 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 l2(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 Nonstationary Grids -- 7.8.2 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 -- Chapter 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 -- Chapter 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-Loève Approximations -- 9.2 Nonlinear Approximations -- 9.2.1 Nonlinear 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 -- Chapter 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-Loève -- 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 -- Chapter 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 Improvements.

11.3 Thresholding Sparse Representations -- 11.3.1 Wavelet Thresholding -- 11.3.2 Wavelet and Curvelet Image Denoising -- 11.3.3 Audio Denoising by Time-Frequency Thresholding -- 11.4 Nondiagonal Block Thresholding -- 11.4.1 Block Thresholding in Bases and Frames -- 11.4.2 Wavelet Block Thresholding -- 11.4.3 Time-Frequency Audio Block Thresholding -- 11.5 Denoising Minimax Optimality -- 11.5.1 Linear Diagonal Minimax Estimation -- 11.5.2 Thresholding Optimality over Orthosymmetric Sets -- 11.5.3 Nearly Minimax with Wavelet Estimation -- 11.6 Exercises -- Chapter 12. Sparsity in Redundant Dictionaries -- 12.1 Ideal Sparse Processing in Dictionaries -- 12.1.1 Best M-Term Approximations -- 12.1.2 Compression by Support Coding -- 12.1.3 Denoising by Support Selection 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 Bandlets for Geometric Image Regularity -- 12.3 Greedy Matching Pursuits -- 12.3.1 Matching Pursuit -- 12.3.2 Orthogonal Matching Pursuit -- 12.3.3 Gabor Dictionaries -- 12.3.4 Coherent Matching Pursuit Denoising -- 12.4 l1 Pursuits -- 12.4.1 Basis Pursuit -- 12.4.2 l1 Lagrangian Pursuit -- 12.4.3 Computations of l1 Minimizations -- 12.4.4 Sparse Synthesis versus Analysis and Total Variation Regularization -- 12.5 Pursuit Recovery -- 12.5.1 Stability and Incoherence -- 12.5.2 Support Recovery with Matching Pursuit -- 12.5.3 Support Recovery with l1 Pursuits -- 12.6 Multichannel Signals -- 12.6.1 Approximation and Denoising by Thresholding in Bases -- 12.6.2 Multichannel Pursuits -- 12.7 Learning Dictionaries -- 12.8 Exercises -- Chapter 13. Inverse Problems -- 13.1 Linear Inverse Estimation -- 13.1.1 Quadratic and Tikhonov Regularizations.

13.1.2 Singular Value Decompositions.