目录
目 录 前言 Introduction to Wavelets 1.1 Wavelets and Wavelet Expansion Systems What is a Wavelet Expansion or a Wavelet Transform What is a Wavelet System More Specific Characteristics of Wavelet Systems Haar Scaling Functions and Wavelets What do Wavelets Look Like Why is Wavelet Analysis Effective 1.2 The Discrete Wavelet Transform 1.3 The Discrete-Time and Continuous Wavelet Transforms 1.4 Exercises and Experiments 1.5 This Chapter 2 A Multiresolution Formulation of Wavelet Systems 2.1 Signal Spaces 2.2 The Scaling Function Multiresolution Analysis 2.3 The Wavelet Functions 2.4 The Discrete Wavelet Transform 2.5 A Parseval''''s Theorem 2.6 Display of the Discrete Wavelet Transform and the Wavelet Expansion 2.7 Examples of Wavelet Expansions 2.8 An Example of the Haar Wavelet System Filter Banks and the Discrete Wavelet Transform 3.1 Analysis From Fine Scale to Coarse Scale Filtering and Down-Sampling or Decimating 3.2 Synthesis - From Coarse Scale to Fine Scale Filtering and Up-Sampling or Stretching 3.3 Input Coefficients 3.4 Lattices and Lifting 3.5 Different Points of View Multiresolution versus Time-Frequency Analysis Periodic versus Nonperiodic Discrete Wavelet Transforms The Discrete Wavelet Transform versus the Discrete-Time Wavelet Transform Numerical Complexity of the Discrete Wavelet Transform Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Tight Frames, and Un-conditional Bases 4.1 Bases, Orthogonal Bases, and Biorthogonal Bases Matrix Examples Fourier Series Example Sinc Expansion Example 4.2 Frames and Tight Frames Matrix Examples Sinc Expansion as a Tight Frame Example 4.3 Conditional and Unconditional Bases The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coeffi-cients 5.1 Tools and Definitions Signal Classes Fourier Transforms Refinement and Transition Matrices 5.2 Necessary Conditions 5.3 Frequency Domain Necessary Conditions 5.4 Sufficient Conditions Wavelet System Design 5.5 The Wavelet 5.6 Alternate Normalizations 5.7 Example Scaling Functions and Wavelets Haar Wavelets Sinc Wavelets Spline and Battle-Lemarie Wavelet Systems 5.8 Further Properties of the Scaling Function and Wavelet General Properties not Requiring Orthogonality Properties that Depend on Orthogonality 5.9 Parameterization of the Scaling Coefficients Length-2 Scaling Coefficient Vector Length-4 Scaling Coefficient Vector Length-6 Scaling Coefficient Vector 5.10 Calculating the Basic Scaling Function and Wavelet Successive Approximations or the Cascade Algorithm Iterating the Filter Bank Successive approximations in the frequency domain The Dyadic Expansion of the Scaling Function 6 Regularity, Moments, and Wavelet System Design 6.1 K-Regular Scaling Filters 6.2 Vanishing Wavelet Moments 6.3 Daubechies'''' Method for Zero Wavelet Moment Design 6.4 Non-Maximal Regularity Wavelet Design 6.5 Relation of Zero Wavelet Moments to Smoothness 6.6 Vanishing Scaling Function Moments 6.7 Approximation of Signals by Scaling Function Projection 6.8 Approximation of Scaling Coefficients by Samples of the Signal 6.9 Coifiets and Related Wavelet Systems Generalized Coifman Wavelet Systems 6.10 Minimization of Moments Rather than Zero Moments Generalizations of the Basic Uultiresolution Wavelet System 7.1 Tiling the Time-Frequency or Time-Scale Plane Nonstationary Signal Analysis Tiling with the Discrete-Time Short-Time Fourier Transform Tiling with the Discrete Two-Band Wavelet Transform General Tiling 7.2 Multiplicity-M M-Band Scaling Functions and Wavelets Properties of M-Band Wavelet Systems M-Band Scaling Function Design M-Band Wavelet Design and Cosine Modulated Methods 7.3 Wavelet Packets Full Wavelet Packet Decomposition Adaptive Wavelet Packet Systems 7.4 Biorthogonal Wavelet Systems Two-Channel Biorthogonal Filter Banks Biorthogonal Wavelets Comparisons of Orthogonal and Biorthogonal Wavelets Example Families of Biorthogonal Systems Cohen-Daubechies-Feauveau Family of Biorthogonal Spline Wavelets Cohen-Daubechies-Feauveau Family of Biorthogonal Wavelets with Less Dissimilar Filter Length Tian-Wells Family of Biorthogonal Coifiets Lifting Construction of Biorthogonal Systems 7.5 Multiwavelets Construction of Two-Band Multiwavelets Properties of Multiwavelets Approximation, Regularity and Smoothness Support Orthogonality Implementation of Multiwavelet Transform Examples Geronimo-Hardin-Massopust Multiwavelets Spline Multiwavelets Other Constructions Applications 7.6 Overcomptete Representations, Frames, Redundant Transforms, and Adaptive Bases Overcomplete Representations A Matrix Example Shift-Invariant Redundant Wavelet Transforms and Nondecimated Filter Banks Adaptive Construction of Frames and Bases 7.7 Local Trigonometric Bases Nonsmooth Local Trigonometric Bases Construction of Smooth Windows Folding and Unfolding Local Cosine and Sine Bases Signal Adaptive Local Trigonometric Bases 7.8 Discrete Multiresolution Analysis, the Discrete-Time Wavelet Transform, and the Continuous Wavelet Transform Discrete Multiresolution Analysis and the Discrete-Time Wavelet Transform Continuous Wavelet Transforms Analogies between Fourier Systems and Wavelet Systems 8 Filter Banks and Transmultiplexers 8.1 Introduction The Filter Bank Transmultiplexer Perfect Reconstruction--A Closer Look Direct Characterization of PR Matrix characterization of PR Polyphase Transform-Domain Characterization of PR 8.2 Unitary Filter Banks 8.3 Unitary Filter Banks--Some Illustrative Examples 8.4 M-band Wavelet Tight Frames 8.5 Modulated Filter Banks Unitary Modulated Filter Bank 8.6 Modulated Wavelet Tight Frames 8.7 Linear Phase Filter Banks Characterization of Unitary Hp z -- PS Symmetry Characterization of Unitary Hp z -- PCS Symmetry Characterization of Unitary Hp z -- Linear-Phase Symmetry Characterization of Unitary Hp z -- Linear Phase and PCS Symmetry Characterization of Unitary Hp z -- Linear Phase and PS Symmetry 8.8 Linear-Phase Wavelet Tight Frames 8.9 Linear-Phase Modulated Filter Banks DCT/DST I/II based 2M Channel Filter Bank 8.10 Linear Phase Modulated Wavelet Tight Frames 8.11 Time-Varying Filter Bank Trees Growing a Filter Bank Tree Pruning a Filter Bank Tree Wavelet Bases for the Interval Wavelet Bases for L2 [0, ∞] Wavelet Bases for L2 -∞, 0] Segmented Time-Varying Wavelet Packet Bases 8.12 Filter Banks and Wavelets--Summary 9 Calculation of the Discrete Wavelet Transform 9.1 Finite Wavelet Expansions and Transforms 9.2 Periodic or Cyclic Discrete Wavelet Transform 9.3 Filter Bank Structures for Calculation of the DWT and Complexity 9.4 The Periodic Case 9.5 Structure of the Periodic Discrete Wavelet Transform 9.6 More General Structures 10 Wavelet-Based Signal Processing and Applications 10.1 Wavelet-Based Signal Processing 10.2 Approximate FFT using the Discrete Wavelet Transform Introduction Review of the Discrete Fourier Transform and FFT Review of the Discrete Wavelet Transform The Algorithm Development Computational Complexity Fast Approximate Fourier Transform Computational Complexity Noise Reduction Capacity Summary 10.3 Nonlinear Filtering or Denoising with the DWT Denoising by Thresholding Shift-Invariant or Nondecimated Discrete Wavelet Transform Combining the Shensa-Beylkin-Mallat-a trous Algorithms and Wavelet Denoising Performance Analysis Examples of Denoising 10.4 Statistical Estimation 10.5 Signal and Image Compression Fundamentals of Data Compression Prototype Transform Coder Improved Wavelet Based Compression Algorithms 10.6 Why are Wavelets so Useful 10.7 Applications Numerical Solutions to Partial Differential Equations Seismic and Geophysical Signal Processing Medical and Biomedical Signal and Image Processing Application in Communications Fractals 10.8 Wavelet Software 11 Summary Overview 11.1 Properties of the Basic Multiresolution Scaling Function 11.2 Types of Wavelet Systems 12 References Bibliography Appendix A. Derivations for Chapter 5 on Scaling Functions Appendix B. Derivations for Section on Properties Appendix C. Matlab Programs Index
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