Wavelets. Tools for Science & Technology

Title page
Preface to Revised Edition
Preface from the First Edition
Chapter 1. Signals and Wavelets
	1.1 What is a signal?
	1.2 The language and goals of signal and image proccssing
	1.3 Stationary signals, transient signals, and adaptive coding
	1.4 Grossmann Morlet time-scale wavelets
	1.5 Time-frequcncy wavelets from Gabor to Malvar and Wilson
	1.6 Optimal algorithms in signal processing
	1.7 Optimal representation according to Marr
	1.8 Terminology
	1.9 Reader's guide
Chapter 2. Wavelets from a Historical Perspective
	2.1 Introduction
	2.2 From Fourier (1807) to Haar (1909), frequency analysis becomcs scale analysis
	2.3 New directions of the 1930s: Paul Lévy and Brownian motion
	2.4 New directions of the 1930s: Littlewood and Paley
	2.5 New directions of the 1930s: The Franklin system
	2.6 New directions of the 1930s: The wavelets of Lusin
	2.7 Atomic decompositions from 1960 to 1980
	2.8 Strömberg's wavelets
	2.9 A first synthesis: Wavelet analysis
	2.10 The advent of signal processing
	2.11 Conclusions
Chapter 3. Quadrature Mirror Filters
	3.1 Introduction
	3.2 Subband coding: The case of ideal filters
	3.3 Quadrature mirror filters
	3.4 Trend and fluctuation
	3.5 The time-scale algorithm of Mallat and the time-frequency algorithm of Galand
	3.6 Trends and fluctuations with orthonormal wavelet bases
	3.7 Convergence to wavelets
	3.8 The wavelets of Daubechies
	3.9 Conclusions
Chapter 4. Pyramid Algorithms for Numerical Image Processing
	4.1 Introduction
	4.2 The pyramid algorithms of Burt and Adelson
	4.3 Examples of pyramid algorithms
	4.4 Pyramid algorithms and image compression
	4.5 Pyramid algorithms and multiresolution analysis
	4.6 The orthogonal pyramids and wavelets
	4.7 Biorthogonal wavelets
Chapter 5. Time-Frequency Analysis for Signal Processing
	5.1 Introduction
	5.2 The collections Ω of time-frequency atoms
	5.3 Mallat's matching pursuit algorithm
	5.4 Best-basis search
	5.5 The Wigner-Ville transforrn
	5.6 Properties of the Wigner-Ville transform
	5.7 The Wigner-Ville transform and pseudodifferential calculus
	5.8 Return to the definition of time-frequency atoms
	5.9 The Wigner Ville transform and instantaneous frequency
	5.10 The Wigner Ville transforrn of asyrnptotic signals
	5.11 Instantaneous frequency and the matching pursuit algorithm
	5.12 Matching pursuit and the Wigner-Ville transform
	5.13 Several spectral lines
	5.14 Conclusions
	5.15 Historical remarks
Chapter 6. Time-Frequency Algorithms Using Malvar-Wilson Wavelets
	6.1 Introduction
	6.2 Malvar-Wilson wavelets: A historical perspective
	6.3 Windows with variable lengths
	6.4 Malvar-Wilson wavelets and time-scale wavelets
	6.5 Adaptive segmentation and the split-and-merge algorithm
	6.6 The entropy of a vector with respect to an orthonormal basis
	6.7 The algorithm for finding the optimal Malvar-Wilson basis
	6.8 An example where this algorithm works
	6.9 The discrete case
	6.10 Modulated Malvar-Wilson bases
	6.11 Examples
	6.12 Conclusions
Chapter 7. Time-Frequency Analysis and Wavelet Packets
	7.1 Heuristic considerations
	7.2 The definition of basic wavelet packets
	7.3 General wavelet packets
	7.4 Splitting algorithms
	7.5 Conclusions
Chapter 8. Computer Vision and Human Vision
	8.1 Marr's program
	8.2 The theory of zero-crossings
	8.3 A counterexample to Marr's conjecture
	8.4 Mallat's conjecture
	8.5 The two-dimensional version of Mallat's algorithm
	8.6 Conclusions
Chapter 9. Wavelets and Turbulence
	9.1 Introduction
	9.2 The statistical theory of turbulence and Fourier analysis
	9.3 Multifractal probability measures and turbulent flows
	9.4 Multifractal modeling of the velocity field
	9.5 Coherent structures
	9.6 Couder's experiments
	9.7 Marie Farge's numerical experiments
	9.8 Modeling and detecting chirps in turbulent flows
	9.9 Wavelets, paraproducts, and Navier-Stokes equations
	9.10 Hausdorff measure and dimension
Chapter 10. Wavelets and Multifractal Functions
	10.1 Introduction
	10.2 The Weierstrass function
	10.3 Regular points in an irregular background
	10.4 The Riemann function
	10.4.1 Hölder regularity at irrationals
	10.4.2 Riemann's function near x₀ = 1
	10.5 Conclusions and comments
Chapter 11. Data Compression and Restoration of Noisy Images
	11.1 Introduction
	11.2 Nonlinear approximation and sparse wavelet expansions
	11.3 Denoising
	11.4 Modeling images
	Il.5 Ridgelets
	11.6 Conclusions
Chapter 12. Wavelets and Astronomy
	12.1 The Hubble Space Telescope and deconvolving its images
	12.1.1 The model
	12.1.2 Discovering and fixing the problem
	12.1.3 IDEA
	12.2 Data compression
	12.2.1 ht_compress
	12.2.2 Smooth restoration
	12.2.3 Comments
	12.3 The hierarchical organization of the universe
	12.3.1 A fractal universe
	12.4 Conclusions
Appendix A. Filter Fundamentals
	A.l The l²(Z) theory and definitions
	A.2 The general two-channel filter bank
Appendix B. Wavelet Transforms
	B.l The L² theory
	B.2 Inversion formulas
	B.2.l L² inversion
	B.2.2 Inversion with the Lusin wavelet
	B.3 Generalizations
Appendix C. A Counterexample
	C.l Introduction
	C.2 The function θ
	C.3 Representations of f₀*θ_ρ and its derivatives
	C.4 Hunting the zeros of (f₀*θ_ρ)"
	C.5 The functions R, R*θ_ρ, (R*θ_ρ)', and (R*θ_ρ)"
	C.6 (R*θ_ρ)" and (R*θ_ρ)' vanish at the zeros of (f\0*θ_ρ)"
	C.7 The behavior of (R*θ_ρ)" /(f₀*θ_ρ)"
	C.8 Remarks
	C.9 A case of perfect reconstruction
Appendix D. Hölder Spaces and Besov Spaces
	D.1 Hölder spaces
	D.2 Besov spaces
	D.3 Examples
Bibliography
Author Index
Subject Index