信号处理的小波导引（英文版·第2版）

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

.　　 Not all theorems of the book are proved in detail, but the important techniques are included. I hope that the reader will excuse the lack of mathematical rigor in the many instances where I have privileged ideas over details. Few proofs are long;they are concentrated to avoid diluting the mathematics into many intermediateresults, which would obscure the text.
　　Course Design Level numbers explained in Section 1.5.2 can help in designing an introductory or a more advanced course. Beginning with a review of the Fourier transform is often necessary. Although most applied mathematics students have already seen the Fourier transform, they have rarely had the time to understand it well. A non-technical review can stress applications, including the sampling theorem. Refreshing basic mathematical results is also needed for electrical engineering students. A mathematically oriented review of time-invariant signal
　　processing in Chapters 2 and 3 is the occasion to remind the student of elementary
　　properties of linear operators, projectors and vector spaces, which can be found
　　in Appendix A. For a course of a single semester, one can follow several paths,
　　oriented by different themes. Here are a few possibilities.
　　 One can teach a course that surveys the key ideas previously outlined. Chapter 4 is particularly important in introducing the concept of local time-frequency decompositions. Section 4.4 on instantaneous frequencies illustrates the limitations of time-frequency resolution. Chapter 6 gives a different perspective on the wavelet transform, by relating the local regularity of a signal to the decay of its wavelet coefficients across scales. It is useful to stress the importance of the wavelet vanishing moments. The course can continue with the presentation of wavelet bases in Chapter 7, and concentrate on Sections 7.1-7.3 on orthogonal bases, multireso-lution approximations and filter bank algorithms in one dimension. Linear and
　　non-linear approximations in wavelet bases are covered in Chapter 9. Depending
　　upon students' backgrounds and interests, the course can finish in Chapter 10 with
　　an application to signal estimation with wavelet thresholding, or in Chapter 11 by
　　presenting image transform codes in wavelet bases.
　　 A different course may study the construction of new orthogonal bases and their applications. Beginning with the wavelet basis, Chapter 7 also gives an introduction to filter banks. Continuing with Chapter 8 on wavelet packet and local cosine bases introduces different orthogonal tilings of the time-frequency plane.
　　It explains the main ideas of time-frequency decompositions. Chapter 9 on linear and non-linear approximation is then particularly important for understanding how to measure the efficiency of these bases, and for studying best bases search proce dures. To illustrate the differences between linear and non-linear approximation procedures, one can compare the linear and non-linear thresholding estimations studied in Chapter 10.
　　 The course can also concentrate on the construction of sparse representations with orthonormal bases, and study applications of non-linear diagonal operators in these bases. It may start in Chapter 10 with a comparison of linear and non-linear operators used to estimate piecewise regular signals contaminated by a white noise.
　　A quick excursion in Chapter 9 introduces linear and non-linear approximations to explain what is a sparse representation. Wavelet orthonormal bases are then presented in Chapter 7, with special emphasis on their non-linear approximation properties for piecewise regular signals. An application of non-linear diagonal operators to image compression or to thresholding estimation should then be studied in some detail, to motivate the use of modern mathematics for understanding these problems.
　　 A more advanced course can emphasize non-linear and adaptive signal processing. Chapter 5 on frames introduces flexible tools that are useful in analyzing the properties of non-linear representations such as irregularly sampled transforms.
　　The dyadic wavelet maxima representation illustrates the frame theory, with applications to multiscale edge detection. To study applications of adaptive repre sentations with orthonormal bases, one might start with non-linear and adaptive approximations, introduced in Chapter 9. Best bases, basis pursuit or matching pursuit algorithms are examples of adaptive transforms that construct sparse representations for complex signals. A central issue is to understand to what extent adaptivity improves applications such as noise removal or signal compression,
　　depending on the signal properties.
　　Responsibilities This book was a one-year project that ended up in a never will finish nightmare. Ruzena Bajcsy bears a major responsibility for not encouraging me to choose another profession, while guiding my first research steps. Herprofound scientific intuition opened my eyes to and well beyond computer vision.
　　Then of course, are all the collaborators who could have done a much better job of showing me that science is a selfish world where only competition counts. The wavelet story was initiated by remarkable scientists like Alex Grossmann, whose modesty created a warm atmosphere of collaboration, where strange new ideas and ingenuity were welcome as elements of creativity.
　　 I am also grateful to the few people who have been willing to work with me.
　　Some have less merit because they had to finish their degree but others did it on a voluntary basis. I am thinking of Amir Averbuch, Emmanuel Bacry, Francois Bergeaud, Geoff DaviS, Davi Geiger, Fr6d6ric Falzon, Wen Liang Hwang, Hamid Krim, George Papanicolaou, Jean-Jacques Slotine, Alan Willsky, Zifeng Zhang and Sifen Zhong. Their patience will certainly be rewarded in a future life.
　　 Although the reproduction of these 600 pages will probably not kill many trees, I do not want to bear the responsibility alone. After four years writing and rewriting each chapter, I first saw the end of the tunnel during a working retreat at the Fondation des Treilles, which offers an exceptional environment to think,write and eat in Provence. With WAvELAB, David Donoho saved me from spending the second half of my life programming wavelet algorithms. This opportunity was
　　beautifully implemented by Maureen Clerc and Jerome Kalifa, who made all the figures and found many more mistakes than I dare say. Dear reader, you should thank Barbara Burke Hubbard, who corrected my Franglais (remaining errors are mine), and forced me to modify many notations and explanations. I thank her for doing it with tact and humor. My editor, Chuck Glaser, had the patience to wait but I appreciate even more his wisdom to let me think that I would finish in a year.
　　 She will not read this book, yet my deepest gratitude goes to Branka with whom ,life has nothing to do with wavelets.
　　 Stephane Mallat