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 I)ictionary 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 TotalVariation
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 a 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. l 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-Lemarie 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 e2 (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
……
CHAPTER 9 Approximations in Bases
CHAPTER 10 Compression
CHAPTER 11 Denoising
CHAPTER 12 Sparsity in Redundant Dictionaries
CHAPTER 13 Inverse Problems
APPENDIX Mathematical Complements
Bibliography
Index
