This book could be divided into two parts i.e. fundamental wavelet transform theory and method and some important applications of wavelet transform. In the first part, as preliminary knowledge, the Fourier analysis, inner product space, the characteristics of Haar functions, and concepts of multi-resolution analysis, are introduced followed by a description on how to construct wavelet functions both multi-band and multi wavelets, and finally introduces the design of integer wavelets via lifting schemes and its application to integer transform algorithm. In the second part, many applications are discussed in the field of image and signal processing by introducing other wavelet variants such as complex wavelets, ridgelets, and curvelets. Important application examples include image compression, image denoisingrestoration, image enhancement, digital watermarking, numerical solution of partial differential equations, and solving ill- conditioned Toeplitz system. The book is intended for senior undergraduate students and graduate students.
目錄：
Preface
Chapter 1 Overview of Fourier Analysis
1.1 Introduction
1.2 Fourier series preliminary
1.3 Continuous Fourier transform
1.3.1 Concept and basic properties of continuous Fourier transform
1.3.2 Fourier transform and linear filter
1.4 Sampling theorem and uncertainty principle
1.5 Discrete Fourier transform
Chapter 2 Mathematical Foundation
2.1 Euclidean algorithm and lifting scheme
2.2 Hilbert space
2.2.1 Orthogonality and orthogonal complement
2.2.2 Optimal approximation in closed convex set and orthogonal decomposition
2.3 0rthogonal family {ψ（x - k）｜k ∈ Z} in L2（R） space
2.4 Frames in Hilbert space
Chapter 3 Haar Wavelet Analysis
3.1 Short-time Fourier transform
3.2 Haar wavelet
3.3 Decomposition and reconstruction algorithms of signals based on Haar wavelet.
Chapter 4 Multiresolution Analysis and Wavelets Design
4.1 The multiresolution framework
4.2 The Mallat algorithm for signal decomposition and reconstruction
4.3 Implementation of Mallat algorithm
4.3.1 Initialization
4.3.2 Boundary extension
4.4 Wavelet packets
4.5 Computation of scaling function
4.5.1 Cascade algorithm
4.5.2 Matrix equation algorithm
4.6 Daubechies orthogonal compactly supported wavelets
4.7 Rationalized compactly supported orthogonal wavelets
4.8 Biorthogonal multiresolution analysis
4.9 Design of compactly supported biorthogonal wavelets
4.10 Design of perfect reconstruction filters and biorthogonal wavelets with rationalized coefficients
4.10.1 Decomposition
4.10.2 Reconstruction
4.10.3 Discussion about the parameters
Chapter 5 M-band Wavelets and Multiwavelets
5.1 Introduction
5.2 Fundamentals of multirate signal processing
5.3 The properties of the perfect reconstruction filter （PRF） banks
5.3.1 The two channel quardure mirror filter （QMF） banks
5.3.2 The two channel conjugated quardure filter （CQF） banks
5.3.3 The design of M-channel PR QMF banks
5.4 The block and lapped transform based on triangular basis functions
5.5 PR filter， banks and M-band wavelets
5.5.1 The construction of M-band orthogonal wavelets
5.5.2 The construction of M-band biorthogonal wavelets
5.5.3 Construct M-band wavelets based on cosine modulation
5.6 Multi-FB and multiwavelets
5.7 Multiwavelet MRA and discrete multiwavelet transform
5.8 The basic pricinple for the construction of multiwavelet
5.8.1 The conditions in time domain for orthogonal filters
5.8.2 The conditions in frequency domain for orthogonal filters
5.9 The construction of orthogonal multiwavelets
5.9.1 Construct multiwavelets based on approximation orders and regularity orders
5.9.2 Construct orthogonal multiwavelet based on OPTER
Chapter 6 The Wavelet Based on the Lifting Scheme and Integer Discrete Transform
6.1 Introduction
6.2 The design of the wavelet transform based on the lifting scheme
6.2.1 Perfect reconstruction filters and lifting decomposition
6.2.2 The lifting factorization of the symmetric biorthogonal wavelet
6.3 Integer DCTs and fast algorithms
6.3.1 Integer DCTs and algorithms
6.3.2 The design of scaled DCT-II
6.4 Integer implement of the lapped biorthogonal transform
Chapter 7 The Wavelet-based Image Compression
7.1 Introduction of image compression
7.1.1 Basic concept
7.1.2 Coding model
7.2 Progressive image coding
7.2.1 Basic conceptions in progressive image coding
7.2.2 Embedded zerotree wavelet （EZW） coding
7.2.3 Set partitioning in hierarchical trees （SPIHT）
7.2.4 Set partitioning embedded block （SPECK） coding
7.3 Line based image compression
7.3.1 Principle of low memory image compression
7.3.2 Line based entropy coding
7.3.3 Experiments and results
7.4 Embedded block coding with optimal truncation （EBCOT）
7.4.1 Code-block
7.4.2 Quality layers
7.4.3 Bit plan coding
7.5 Fast wavelet transform in image compression
7.5.1 Vanishing moment and high frequency coefficients
7.5.2 The lifting scheme in image compression
Chapter 8 Wavelet Based Image Denoising and Enhancement
8.1 The singularity detection of signal and wavelet transform modulus maxima ~
8.2 The thresholding methods for denoising
8.2.1 The selection of threshold function
8.2.2 The estimation of threshold
8.3 Scale factor shrinking method for denoising
8.4 Correlation based denoising methods
8.5 Image denoising based on DTCWs and multiwavelets
8.5.1 DTCWT based image denoising
8.5.2 Multiwavelet based denosing
8.6 Image enhancement based on the multiscale transform
8.6.1 The basic concepts of image enhancement
8.6.2 Image enhancement techniques based on multiscale method
Chapter 9 Ridgelets and Its Applications
9.1 Introduction
9.2 The ridgelet transform
9.2.1 Ridglet and continuous ridgelet transform
9.2.2 Discrete transform： ridgelet frames
9.2.3 Monoscale ridgelets
9.2.4 Curvelet
9.3 The application of ridgelet transform in signal processing
9.3.1 Image enhancement
9.3.2 Noise attenuation
9.3.3 Image reconstruction
Chapter 10 Application of Wavelet Transform in the Digital Watermarking.
10.1 Introduction
10.2 Digital watermarking method based on float wavelet transforms
10.3 Fragile digital watermarking method of integer wavelet transforms
10.3.1 The construction of Hash function through Rijndael encrypted algorithm
10.3.2 The insertion and test of watermark to pictures
10.3.3 Experiments and comparisons
10.4 The technology of the visible digital watermarking based on integral wavelet transform with parameters
10.4.1 Rijndael code with different type builds Hash function
10.4.2 Inserting and removing of visible digital watermark
10.4.3 Experiments and comparisons
10.5 The translucent digital watermarking technique based on integral wavelet transform with parameters
10.5.1 The visual weight analysis based on wavelet domain quantization
10.5.2 Translucent watermarking algorithm realization
10.5.3 Experimental results
10.6 Simultaneous embedding of various kinds of watermarks based on parametric integer wavelet transforms
Chapter 11 The Solution of PDE Based on Wavelets
11.1 Introduction
11.2 The representation of operator T by wavelet bases
11.2.1 The nonstandard form of representation
11.2.2 Standard form
11.2.3 The compression of operator by wavelet bases
11.3 Solving PDE based on wavelets
11.3.1 A classic example
11.3.2 The periodization of the original problem
11.3.3 Construct the inverse of periodized differential operator
11.4 Computing the inverse of ellipse differential operator based on multiscale method
Chapter 12 The Solution of Ill-conditioned Symmetric Toeplitz Systems via Two-grid and Wavelet Methods
12.1 Introduction
12.2 Multigrid method
12.3 The solution of ill-conditioned Toeplitz systems based on wavelets and MGM
12.3.1 Toeplitz systems
12.3.2 Solving Toeplitz system by TGM
12.3.3 Solving Toeplitz system by MGM
12.4 Numerical results
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