小波分析導論 (英文版)

preface
Ⅰ preliminaries
1 functions and convergence
1.1 functions
1.1.1 bounded (l∞) functions
1.1.2 integrable (l1) functions
1.1.3 square integrable (l2) functions
1.1.4 differentiable (cn) functions
1.2 convergence of sequences of functions
1.2.1 numerical convergence
1.2.2 pointwise convergence
1.2.3 uniform (l∞) convergence
1.2.4 mean (ll) convergence
1.2.5 mean-square (l2) convergence
1.2.6 interchange of limits and integrals
2 fourier series
2.1 trigonometric series
2.1.1 periodic functions
2.1.2 the trigonometric system
2.1.3 the fourier coefficients
2.1.4 convergence of fourier series
2.2 approximate identities
2.2.1 motivation from fourier series
2.2.2 definition and examples
2.2.3 convergence theorems
2.3 generalized fourier series
2.3.1 orthogonality
2.3.2 generalized fourier series
2.3.3 completeness
3 the fourier transform
3.1 motivation and definition
3.2 basic properties of the fourier transform
3.3 fourier inversion
3.4 convolution
3.5 plancherel’’s formula
3.6 the fourier transform for l2 functions
3.7 smoothness versus decay
3.8 dilation, translation, and modulation
3.9 bandlimited functions and the sampling formula
4 signals and systems
4.1 signals
4.2 systems
4.2.1 causality and stability
4.3 periodic signals and the discrete fourier transform
4.3.1 the discrete fourier transform
4.4 the fast fourier transform
4.5 l2 fourier series
Ⅱ the haar system
5 the haar system
5.1 dyadic step functions
5.1.1 the dyadic intervals
5.1.2 the scale j dyadic step functions
5.2 the haar system
5.2.1 the haar scaling functions and the haar functions.
5.2.2 orthogonality of the haar system
5.2.3 the splitting lemma
5.3 haar bases on [0, 1]
5.4 comparison of haar series with fourier series
5.4.1 representation of functions with small support
5.4.2 behavior of haar coefficients near jump discontinuities
5.4.3 haar coefficients and global smoothness
5.5 haar bases on r
5.5.1 the approximation and detail operators
5.5.2 the scale j haar system on r
5.5.3 the hair system on r
6 the discrete haar transform
6.1 motivation
6.1.1 the discrete haar transform (dht)
6.2 the dht in two dimensions
6.2.1 the row-wise and column-wise approximations and details
6.2.2 the dht for matrices
6.3 image analysis with the dht
6.3.1 approximation and blurring
6.3.2 horizontal, vertical, and diagonal edges
6.3.3 ”naive” image compression
Ⅲ orthonormal wavelet bases
7 multiresolution analysis
7.1 orthonormal systems of translates
7.2 definition of multiresolution analysis
7.2.1 some basic properties of mras
7.3 examples of multiresolution analysis
7.3.1 the haar mra
7.3.2 the piecewise linear mra
7.3.3 the bandlimited mra
7.3.4 the meyer mra
7.4 construction and examples of orthonormal wavelet bases
7.4.1 examples of wavelet bases
7.4.2 wavelets in two dimensions
7.4.3 localization of wavelet bases
7.5 proof of theorem 7.35
7.5.1 sufficient conditions for a wavelet basis
7.5.2 proof of theorem 7.35
7.6 necessary properties of the scaling function
7.7 general spline wavelets
7.7.1 basic properties of spline functions
7.7.2 spline multiresolution analyses
8 the discrete wavelet transform
8.1 motivation: from mra to a discrete transform
8.2 the quadrature mirror filter conditions
8.2.1 motivation from mra
8.2.2 the approximation and detail operators and their adjoints
8.2.3 the quadrature mirror filter (qmf) conditions
8.3 the discrete wavelet transform (dwt)
8.3.1 the dwt for signals
8.3.2 the dwt for finite signals
8.3.3 the dwt as an orthogonal transformation
8.4 scaling functions from scaling sequences
8.4.1 the infinite product formula
8.4.2 the cascade algorithm
8.4.3 the support of the scaling function
9 smooth, compactly supported wavelets
9.1 vanishing moments
9.1.1 vanishing moments and smoothness
9.1.2 vanishing moments and approximation
9.1.3 vanishing moments and the reproduction of polynomials
9.1.4 equivalent conditions for vanishing moments
9.2 the daubechies wavelets
9.2.1 the daubechies polynomials
9.2.2 spectral factorization
9.3 image analysis with smooth wavelets
9.3.1 approximation and blurring
9.3.2 ”naive” image compression with smooth wavelets
Ⅳ other wavelet constructions
10 biorthogonal wavelets
10.1 linear independence and biorthogonality
10.2 riesz bases and the frame condition
10.3 riesz bases of translates
10.4 generalized multiresolution analysis (gmra)
10.4.1 basic properties of gmra
10.4.2 dual gmra and riesz bases of wavelets
10.5 riesz bases orthogonal across scales
10.5.1 example: the piecewise linear gmra
10.6 a discrete transform for biorthogonal wavelets
10.6.1 motivation from gmra
10.6.2 the qmf conditions
10.7 compactly supported biorthogonal wavelets
10.7.1 compactly supported spline wavelets
10.7.2 symmetric biorthogonal wavelets
10.7.3 using symmetry in the dwt
11 wavelet packets
11.1 motivation: completing the wavelet tree
11.2 localization of wavelet packets
11.2.1 time/spatial localization
11.2.2 frequency localization
11.3 0rthogonality and completeness properties of wavelet packets
11.3.1 wavelet packet bases with a fixed scale
11.3.2 wavelet packets with mixed scales
11.4 the discrete wavelet packet transform (dwpt)
11.4.1 the dwpt for signals
11.4.2 the dwpt for finite signals
11.5 the best-basis algorithm
11.5.1 the discrete wavelet packet library
11.5.2 the idea of the best basis
11.5.3 description of the algorithm
Ⅴ applications
12 image compression
12.1 the transform step
12.1.1 wavelets or wavelet packets?
12.1.2 choosing a filter
12.2 the quantization step
12.3 the coding step
12.3.1 sources and codes
12.3.2 entropy and information
12.3.3 coding and compression
12.4 the binary huffman code
12.5 a model wavelet transform image coder
12.5.1 examples
13 integral operators
13.1 examples of integral operators
13.1.1 sturm-liouville boundary value problems
13.1.2 the hilbert transform
13.1.3 the radon transform
13.2 the bcr algorithm
13.2.1 the scale j approximation to t
13.2.2 description of the algorithm
Ⅵ appendixes
a review of advanced calculus and linear algebra
a.1 glossary of basic terms from advanced calculus and linear algebra
a.2 basic theorems from advanced calculus
b excursions in wavelet theory
b.1 other wavelet constructions
b.1.1 m-band wavelets
b.1.2 wavelets with rational noninteger dilation factors
b.1.3 local cosine bases
b.1.4 the continuous wavelet transform
b.1.5 non~mra wavelets
b.1.6 multiwavelets
b.2 wavelets in other domains
b.2.1 wavelets on intervals
b.2.2’’ wavelets in higher dimensions
b.2.3 the lifting scheme
b.3 applications of wavelets
b.3.1 wavelet denoising
b.3.2 multiscale edge detection
b.3.3 the fbi fingerprint compression standard
c references cited in the text
index