Fundamentals of Wavelets Theory, Algorithms, and Applications

Cover Page
WILEY SERIES IN MICROWAVE AND OPTICAL ENGINEERING
Title Page
ISBN 9780470484135
Contents
Preface to the Second Edition xv
Preface to the First Edition xvii
1 What Is This Book All About? 1
2 Mathematical Preliminary 6
3 Fourier Analysis 34
4 Time-Frequency Analysis 61
5 Multiresolution Analysis 94
6 Construction of Wavelets 114
7 DWT and Filter Bank Algorithms 147
8 Special Topics in Wavelets and Algorithms 197
9 Digital Signal Processing Applications 239
10 Wavelets in Boundary Value Problems 308
Index 353
Preface to the Second Edition
Preface to the First Edition
1 What Is This Book All About?
2 Mathematical Preliminary
2.1 LINEAR SPACES
2.2 VECTORS AND VECTOR SPACES
2.3 BASIS FUNCTIONS, ORTHOGONALITY, AND BIORTHOGONALITY
2.3.1 Example
2.3.2 Orthogonality and Biorthogonality
2.4 LOCAL BASIS AND RIESZ BASIS
2.4.1 Haar Basis
2.4.2 Shannon Basis
2.5 DISCRETE LINEAR NORMED SPACE
2.5.1 Example 1
2.5.2 Example 2
2.6 APPROXIMATION BY ORTHOGONAL PROJECTION
2.7 MATRIX ALGEBRA AND LINEAR TRANSFORMATION
2.7.1 Elements of Matrix Algebra
2.7.2 Eigenmatrix
2.7.3 Linear Transformation
2.7.4 Change of Basis
2.7.5 Hermitian Matrix, Unitary Matrix, and Orthogonal Transformation
2.8 DIGITAL SIGNALS
2.8.1 Sampling of Signal
2.8.2 Linear Shift Invariant Systems
2.8.3 Convolution
2.8.4 z - Transform
2.8.5 Region of Convergence
2.8.6 Inverse z - Transform
2.9 EXERCISES
2.10 REFERENCES
3 Fourier Analysis
3.1 FOURIER SERIES
3.2 EXAMPLES
3.2.1 Rectified Sine Wave
3.2.2 Comb Function and the Fourier Series Kernel KN(t)
3.3 FOURIER TRANSFORM
3.4 PROPERTIES OF FOURIER TRANSFORM
3.4.1 Linearity
3.4.2 Time Shifting and Time Scaling
3.4.3 Frequency Shifting and Frequency Scaling
3.4.4 Moments
3.4.5 Convolution
3.4.6 Parseval's Theorem
3.5 EXAMPLES OF FOURIER TRANSFORM
3.5.1 The Rectangular Pulse
3.5.2 The Triangular Pulse
3.5.3 The Gaussian Function
3.6 POISSON'S SUM AND PARTITION OF UNITY
3.6.1 Partition of Unity
3.7 SAMPLING THEOREM
3.8 PARTIAL SUM AND GIBB'S PHENOMENON
3.9 FOURIER ANALYSIS OF DISCRETE TIME SIGNALS
3.9.1 Discrete Fourier Basis and Discrete Fourier Series
3.9.2 Discrete Time Fourier Transform ( DTFT )
3.10 DISCRETE FOURIER TRANSFORM ( DFT )
3.11 EXERCISES
3.12 REFERENCES
4 Time Frequency Analysis
4.1 WINDOW FUNCTION
4.2 SHORT TIME FOURIER TRANSFORM
4.2.1 Inversion Formula
4.2.2 Gabor Transform
4.2.3 Time Frequency Window
4.2.4 Properties of STFT
4.3 DISCRETE SHORT TIME FOURIER TRANSFORM
4.4 DISCRETE GABOR REPRESENTATION
4.5 CONTINUOUS WAVELET TRANSFORM
4.5.1 Inverse Wavelet Transform
4.5.2 Time Frequency Window
4.6 DISCRETE WAVELET TRANSFORM
4.7 WAVELET SERIES
4.8 INTERPRETATIONS OF THE TIME FREQUENCY PLOT
4.9 WIGNER VILLE DISTRIBUTION
4.9.1 Gaussian Modulated Chirp
4.9.2 Sinusoidal Modulated Chirp
4.9.3 Sinusoidal Signal
4.10 PROPERTIES OF WIGNER VILLE DISTRIBUTION
4.10.1 A Real Quantity
4.10.2 Marginal Properties
4.10.3 Correlation Function
4.11 QUADRATIC SUPERPOSITION PRINCIPLE
4.12 AMBIGUITY FUNCTION
4.13 EXERCISES
4.14 COMPUTER PROGRAMS
4.14.1 Short Time Fourier Transform
4.14.2 Wigner Ville Distribution
4.15 REFERENCES
5 Multiresolution Analysis
5.1 MULTIRESOLUTION SPACES
5.2 ORTHOGONAL, BIORTHOGONAL, AND SEMIORTHOGONAL DECOMPOSITION
5.3 TWO SCALE RELATIONS
5.4 DECOMPOSITION RELATION
5.5 SPLINE FUNCTIONS AND PROPERTIES
5.5.1 Properties of Splines
5.6 MAPPING A FUNCTION INTO MRA SPACE
5.6.1 Linear Splines (m=2)
5.6.2 Cubic Splines (m=4)
5.7 EXERCISES
5.8 COMPUTER PROGRAMS 5.8.1 B Splines
5.9 REFERENCES
6 Construction of Wavelets
6.1 NECESSARY INGREDIENTS FOR WAVELET CONSTRUCTION
6.1.1 Relationship between the Two Scale Sequences
6.1.2 Relationship between Reconstruction and Decomposition Sequences
6.2 CONSTRUCTION OF SEMIORTHOGONAL SPLINE WAVELETS
6.2.1 Expression for {g0[k]}
6.2.2 Remarks
6.3 CONSTRUCTION OF ORTHONORMAL WAVELETS
6.4 ORTHONORMAL SCALING FUNCTIONS
6.4.1 Shannon Scaling Function
6.4.2 Meyer Scaling Function
6.4.3 Battle Lemarié Scaling Function
6.4.4 Daubechies Scaling Function
6.5 CONSTRUCTION OF BIORTHOGONAL WAVELETS
6.6 GRAPHICAL DISPLAY OF WAVELET
6.6.1 Iteration Method
6.6.2 Spectral Method
6.6.3 Eigenvalue Method
6.7 EXERCISES
6.8 COMPUTER PROGRAMS
6.8.1 Daubechies Wavelet
6.8.2 Iteration Method
6.9 REFERENCES
7 DWT and Filter Bank Algorithms
7.1 DECIMATION AND INTERPOLATION
7.1.1 Decimation
7.1.2 Interpolation
7.1.3 Convolution Followed by Decimation
7.1.4 Interpolation Followed by Convolution
7.2 SIGNAL REPRESENTATION IN THE APPROXIMATION SUBSPACE
7.3 WAVELET DECOMPOSITION ALGORITHM
7.4 RECONSTRUCTION ALGORITHM
7.5 CHANGE OF BASES
7.6 SIGNAL RECONSTRUCTION IN SEMIORTHOGONAL SUBSPACES
7.6.1 Change of Basis for Spline Functions
7.6.2 Change of Basis for Spline Wavelets
7.7 EXAMPLES
7.8 TWO CHANNEL PERFECT RECONSTRUCTION FILTER BANK
7.8.1 Spectral Domain Analysis of a Two Channel PR Filter Bank
7.8.2 Time Domain Analysis
7.9 POLYPHASE REPRESENTATION FOR FILTER BANKS
7.9.1 Signal Representation in Polyphase Domain
7.9.2 Filter Bank in the Polyphase Domain
7.10 COMMENTS ON DWT AND PR FILTER BANKS
7.11 EXERCISES
7.12 COMPUTER PROGRAM
7.12.1 Decomposition and Reconstruction Algorithm
7.13 REFERENCES
8 Special Topics in Wavelets and Algorithms
8.1 FAST INTEGRAL WAVELET TRANSFORM
8.1.1 Finer Time Resolution
8.1.2 Finer Scale Resolution
8.1.3 Function Mapping into the Interoctave Approximation Subspaces
8.1.4 Examples
8.2 RIDGELET TRANSFORM
8.3 CURVELET TRANSFORM
8.4 COMPLEX WAVELETS
8.4.1 Linear Phase Biorthogonal Approach
8.4.2 Quarter Shift Approach
8.4.3 Common Factor Approach
8.5 LIFTING WAVELET TRANSFORM
8.5.1 Linear Spline Wavelet
8.5.2 Construction of Scaling Function and Wavelet from Lifting Scheme
8.5.3 Linear Interpolative Subdivision
8.6 REFERENCES
9 Digital Signal Processing Applications
9.1 WAVELET PACKET
9.2 WAVELET PACKET ALGORITHMS
9.3 THRESHOLDING
9.3.1 Hard Thresholding
9.3.2 Soft Thresholding
9.3.3 Percentage Thresholding
9.3.4 Implementation
9.4 INTERFERENCE SUPPRESSION
9.4.1 Best Basis Selection
9.5 FAULTY BEARING SIGNATURE IDENTIFICATION
9.5.1 Pattern Recognition of Acoustic Signals
9.5.2 Wavelets, Wavelet Packets, and FFT Features
9.6 TWO DIMENSIONAL WAVELETS AND WAVELET PACKETS
9.6.1 Two Dimensional Wavelets
9.6.2 Two Dimensional Wavelet Packets
9.6.3 Two Dimensional Wavelet Algorithm
9.6.4 Wavelet Packet Algorithm
9.7 EDGE DETECTION
9.7.1 Sobel Edge Detector
9.7.2 Laplacian of Gaussian Edge Detector
9.7.3 Canny Edge Detector
9.7.4 Wavelet Edge Detector
9.8 IMAGE COMPRESSION
9.8.1 Basics of Data Compression
9.8.2 Wavelet Tree Coder
9.8.3 EZW Code
9.8.4 EZW Example
9.8.5 Spatial Oriented Tree ( SOT )
9.8.6 Generalized Self Similarity Tree ( GST )
9.9 MICROCALCIFICATION CLUSTER DETECTION
9.9.1 CAD Algorithm Structure
9.9.2 Partitioning of Image and Nonlinear Contrast Enhancement
9.9.3 Wavelet Decomposition of the Subimages
9.9.4 Wavelet Coefficient Domain Processing
9.9.5 Histogram Thresholding and Dark Pixel Removal
9.9.6 Parametric ART 2 Clustering
9.9.7 Results
9.10 MULTICARRIER COMMUNICATION SYSTEMS ( MCCS )
9.10.1 OFDM Multicarrier Communication Systems
9.10.2 Wavelet Packet – Based MCCS
9.11 THREE DIMENSIONAL MEDICAL IMAGE VISUALIZATION
9.11.1 Three Dimensional Wavelets and Algorithms
9.11.2 Rendering Techniques
9.11.3 Region of Interest
9.11.4 Summary
9.12 GEOPHYSICAL APPLICATIONS
9.12.1 Boundary Value Problems and Inversion
9.12.2 Well Log Analysis
9.12.3 Reservoir Data Analysis
9.12.4 Downhole Pressure Gauge Data Analysis
9.13 COMPUTER PROGRAMS
9.13.1 Two Dimensional Wavelet Algorithms
9.13.2 Wavelet Packet Algorithms
9.14 REFERENCES
10 Wavelets in Boundary Value Problems
10.1 INTEGRAL EQUATIONS
10.2 METHOD OF MOMENTS
10.3 WAVELET TECHNIQUES
10.3.1 Use of Fast Wavelet Algorithm
10.3.2 Direct Application of Wavelets
10.3.3 Wavelets in Spectral Domain
10.3.4 Wavelet Packets
10.4 WAVELETS ON THE BOUNDED INTERVAL
10.5 SPARSITY AND ERROR CONSIDERATIONS
10.6 NUMERICAL EXAMPLES
10.7 SEMIORTHOGONAL VERSUS ORTHOGONAL WAVELETS
10.8 DIFFERENTIAL EQUATIONS
10.8.1 Multigrid Method
10.8.2 Multiresolution Time Domain ( MRTD ) Method
10.8.3 Haar MRTD Derivation
10.8.4 Subcell Modeling in MRTD
10.8.5 Examples
10.9 EXPRESSIONS FOR SPLINES AND WAVELETS
10.10 REFERENCES
Index
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