PREFACE xiii
PREFACE TO THE SECOND EDITION xviii
NOTATION xx
I
INTRODUCTION TO A TRANSIENT WORLD
1.1 Fourier Kingdom
1.2 Time-Frequency Wedding
1.2.1 Windowed Fourier Transform
1.2.2 Wavelet Transform
1.3 Bases of Time-Frequency Atoms
1.3.1 Wavelet Bases and Filter Banks
1.3.2 Tilings of Wavelet Packet and Local Cosine Bases
1.4 Bases for What?
1.4.1 Approximation
1.4.2 Estimation
1.4.3 Compression
1.5 Travel Guide
1.5.1 Reproducible Computational Science
1.5.2 Road Map
II
FOURIER KINGDOM
2.1 Linear Time-Invariant Filtering 1
2.1.1 Impulse Response
2.1.2 Transfer Functions
2.2 Fourier Integrals l
2.2.1 Fourier Transform in L1(R)
2.2.2 Fourier Transform in L2(R)
2.2.3 Examples
2.3 Properties 1
2.3.1 Regularity and Decay
2.3.2 Uncertainty Principle
2.3.3 Total Variation
2.4 Two-Dimensional Fourier Transform 1
2.5 Problems
III
DISCRETE REVOLUTION
3.1 Sampling Analog Signals 1
3.1.1 Whittaker Sampling Theorem
3.1.2 Aliasing
3.1.3 General Sampling Theorems
3.2 Discrete Time-Invariant Filters 1
3.2.1 Impulse Response and Transfer Function
3.2.2 Fourier Series
3.3 Finite Signals 1
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 1
3.4.1 Two-Dimensional Sampling Theorem
3.4.2 Discrete Image Filtering
3.4.3 Circular Convolutions and Fourier Basis
3.5 Problems
IV
TIME MEETS FREQUENCY
4.1 Time-Frequency Atoms 1
4.2 Windowed Fourier Transform 1
4.2.1 Completeness and Stability
4.2.2 Choice of Window 2
4.2.3 Discrete Windowed Fourier Transform 2
4.3 Wavelet Transforms 1
4.3.1 Real Wavelets
4.3.2 Analytic Wavelets
4.3.3 Discrete Wavelets 2
4.4 Instantaneous Frequency 2
4.4.1 Windowed Fourier Ridges
4.4.2 Wavelet Ridges
4.5 Quadratic Tune-Frequency Energy 1
4.5.1 Wigner-Ville Distribution
4.5.2 Interferences and Positivity
4.5.3 Cohen's Class 2
4.5.4 Discrete Wigner-Ville Computations 2
4.6 Problems
V
FRAMES
5.1 Frame Theory 2
5.1.1 Frame Definition and SampLing
5.1.2 Pseudo Inverse
5.1.3 Inverse Frame Computations
5.1.4 Frame Projector and Noise Reduction
5.2 Windowed Fourier Frames 2
5.3 Wavelet Frames 2
5.4 Translation Invariance 1
5.5 Dyadic Wavelet Transform 2
5.5.1 Wavelet Design
5.5.2 "Algorithme a Trous"
5.5.3 Oriented Wavelets for a Vision 3
5.6 Problems
VI
WAVELET ZOOM
6.1 Lipschitz Regularity 1
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 2
6.2.1 Detection of Singularities
6.2.2 Reconstruction From Dyadic Maxima 3
6.3 Multiscale Edge Detection 2
6.3.1 Wavelet Maxima for Images 2
6.3.2 Fast Multiscale Edge Computations 3
6.4 Multifractals 2
6.4.1 Fractal Sets and Self-Similar Functions
6.4.2 Singularity Spectrum 3
6.4.3 Fractal Noises 3
6.5 Problems
VII
WAVELET BASES
7.1 Orthogonal Wavelet Bases 1
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 1
7.2.1 Choosing a Wavelet
7.2.2 Shannon, Meyer and Battle-Lemarie Wavelets
7.2.3 Daubechies Compactly Supported Wavelets
7.3 Wavelets and Filter Banks 1
7.3.1 Fast Orthogonal Wavelet Transform
7.3.2 Perfect Reconstruction Filter Banks
7.3.3 Biorthogonal Bases of I2(z) z
7.4 Biorthogonal Wavelet Bases 2
7.4.1 Construction of Biorthogonal Wavelet Bases
7.4.2 Biorthogonal Wavelet Design 2
7.4.3 Compactly Supported Biorthogonal Wavelets 2
7.4.4 Lifting Wavelets 3
7.5 Wavelet Bases on an Interval 2
7.5.1 Periodic Wavelets
IX
AN APPROXIMATION TOUR
9.1 Linear Approximations 1
9.1.1 Linear Approximation Error
9.1.2 Linear Fourier Approximations
9.1.3 Linear Multiresolution Approximations
9.1.4 Karhunen-Loeve Approximations 2
9.2 Non-Linear Approximations 1
9.2.1 Non-Linear Approximation Error
9.2.2 Wavelet Adaptive Grids
9.2.3 Besov Spaces 3
9.3 Image Approximations with Wavelets 1
9.4 Adaptive Basis Selection 2
9.4.1 Best Basis and Schur Concavity
9.4.2 Fast Best Basis Search in Trees
9.4.3 Wavelet Packet and Local Cosine Best Bases
9.5 Approximations with Pursuits 3
9.5.1 Basis Pursuit
9.5.2 Matching Pursuit
9.5.3 Orthogonal Matching Pursuit
9.6 Problems
X
ESTIMATIONS ARE APPROXIMATIONS
10.1 Bayes Versus Minimax 2
10.1.1 Bayes Estimation
10.1.2 Minimax Estimation
10.2 Diagonal Estimation in a Basis 2
10.2.1 Diagonal Estimation with Oracles
10.2.2 Thresholding Estimation
10.2.3 Thresholding Refinements 3
10.2.4 Wavelet Thresholding
10.2.5 Best Basis Thresholding 3
10.3 Minimax Optimality 3
10.3.1 Linear Diagonal Minimax Estimation
10.3.2 0rthosymmetric Sets
10.3.3 Nearly Minimax with Wavelets
10.4 Restoration3
10.4.1 Estimation in Arbitrary Gaussian Noise
10.4.2 Inverse Problems and Deconvolution
10.5 Coherent Estimation 3
1O.5. I Coherent Basis Thresholding
10.5.2 Coherent Matching Pursuit
10.6 Spectrum Estimation 2
10.6.1 Power Spectrum
10.6.2 Approximate Karhunen-Lotve Search 3
10.6.3 Locally Stationary Processes 3
10.7 Problems
Xl
TRANSFORM CODING
11.1 Signal Compression z
11.1.1 State of the Art
11.1.2 Compression in Orthonormal Bases
11.2 Distortion Rate of Quantization 2
11.2.1 Entropy Coding
11.2.2 Scalar Quantization
11.3 High Bit Rate Compression 2
11.3.1 Bit Allocation
11.3.2 Optimal Basis and Karhunen-Loeve
11.3.3 Transparent Audio Code
11.4 Image Compression 2
11.4.1 Deterministic Distortion Rate
11.4.2 Wavelet Image Coding
11.4.3 Block Cosine Image Coding
11.4.4 Embedded Transform Coding
11.4.5 Minimax Distortion Rate 3
11.5 Video Signals 2
11.5.1 Optical Flow
11.5.2 MPEG Video Compression
11.6 Problems
Appendix 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
Appendix B
SOFTWARE TOOLBOXES
B.1 WAVELAB
B.2 LASTWAVE
B.3 Freeware Wavelet Toolboxes
BIBLIOGRAPHY
INDEX