Fundamentals of Wavelets: Theory, Algorithms, and Applications,9780471197485

Fundamentals of Wavelets: Theory, Algorithms, and Applications

by ;
ISBN13: 9780471197485
ISBN10: 0471197483
Format: Hardcover
Pub. Date: 1999-02-01
Publisher(s): Wiley-Interscience
List Price: $148.02

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Summary

Wavelet theory originated from research activities in many areas of science and engineering. As a result, it finds applications in a wide range of practical problems. Wavelet techniques are specifically suited for nonstationary signals for which classic Fourier methods are ineffective. Based on courses taught by the authors at Texas A&M University as well as related conferences, Fundamentals of Wavelets is a textbook offering an up-to-date engineering approach to wavelet theory. It balances a discussion of wavelet theory and algorithms with its far-ranging practical applications in signal processing, image processing, electromagnetic wave scattering, and boundary value problems. In a clear, progressive format, the book describes: * Basic concepts of linear algebra, Fourier analysis, and discrete signal analysis * Theoretical aspects of time-frequency analysis and multiresolution analysis * Construction of various wavelets * Algorithms for computing wavelet transformations. Concluding chapters present interesting applications of wavelets to signal processing and boundary value problems. Fundamentals of Wavelets is an essential introduction to wavelet theory for students and professionals alike in a practical, real-world engineering context.

Author Biography

JAIDEVA C. GOSWAMI is an engineer at Schlumberger Well Services in Sugar- land, Texas. He has taught several short courses on wavelets and contributed to the Encyclopedia of Electrical and Electronics Engineering. Dr. Goswami received his PhD in electrical engineering from Texas A&M University.<br> <br> K. CHAN is on the faculty of Texas A&M University and the coauthor of Wavelets in a Box and Wavelet Toolware. Dr. Chan received his PhD in electrical engineering from the University of Washington.

Table of Contents

Preface xv
1 What This Book Is About
1(4)
2 Mathematical Preliminaries
5(26)
2.1 Linear Spaces
5(2)
2.2 Vectors and Vector Spaces
7(2)
2.3 Basis Functions
9(4)
2.3.1 Orthogonality and Biorthogonality
10(3)
2.4 Local Basis and Riesz Basis
13(2)
2.5 Discrete Linear Normed Space
15(1)
2.6 Approximation by Orthogonal Projection
16(2)
2.7 Matrix Algebra and Linear Transformation
18(5)
2.7.1 Elements of Matrix Algebra
18(1)
2.7.2 Eigenmatrix
19(1)
2.7.3 Linear Transformation
20(1)
2.7.4 Change of Basis
21(1)
2.7.5 Hermitian Matrix, Unitary Matrix, and Orthogonal Transformation
22(1)
2.8 Digital Signals
23(6)
2.8.1 Sampling of Signal
23(1)
2.8.2 Linear Shift-Invariant Systems
24(1)
2.8.3 Convolution
24(1)
2.8.4 z-Transform
25(1)
2.8.5 Region of Convergence
26(2)
2.8.6 Inverse z-Transform
28(1)
2.9 Exercises
29(1)
References
30(1)
3 Fourier Analysis
31(26)
3.1 Fourier Series
31(1)
3.2 Examples
32(3)
3.2.1 Rectified Sine Wave
32(1)
3.2.2 Comb Function and the Fourier Series Kernel K(N)(t)
33(2)
3.3 Fourier Transform
35(2)
3.4 Properties of the Fourier Transform
37(3)
3.4.1 Linearity
37(1)
3.4.2 Time Shifting and Time Scaling
38(1)
3.4.3 Frequency Shifting and Frequency Scaling
38(1)
3.4.4 Moments
38(1)
3.4.5 Convolution
39(1)
3.4.6 Parseval's Theorem
40(1)
3.5 Examples of the Fourier Transform
40(3)
3.5.1 Rectangular Pulse
41(1)
3.5.2 Triangular Pulse
41(1)
3.5.3 Gaussian Function
42(1)
3.6 Poisson's Sum
43(3)
3.6.1 Partition of Unity
45(1)
3.7 Sampling Theorem
46(3)
3.8 Partial Sum and the Gibbs Phenomenon
49(1)
3.9 Fourier Analysis of Discrete-Time Signals
50(4)
3.9.1 Discrete Fourier Basis and Discrete Fourier Series
50(2)
3.9.2 Discrete-Time Fourier Transform
52(2)
3.10 Discrete Fourier Transform
54(1)
3.11 Exercises
55(1)
References
56(1)
4 Time-Frequency Analysis
57(32)
4.1 Window Function
58(2)
4.2 Short-Time Fourier Transform
60(4)
4.2.1 Inversion Formula
61(1)
4.2.2 Gabor Transform
61(1)
4.2.3 Time-Frequency Window
62(1)
4.2.4 Properties of STFT
62(2)
4.3 Discrete Short-Time Fourier Transform
64(1)
4.3.1 Examples of STFT
64(1)
4.4 Discrete Gabor Representation
65(2)
4.5 Continuous Wavelet Transform
67(5)
4.5.1 Inverse Wavelet Transform
69(1)
4.5.2 Time-Frequency Window
70(2)
4.6 Discrete Wavelet Transform
72(1)
4.7 Wavelet Series
73(1)
4.8 Interpretations of the Time-Frequency Plot
74(2)
4.9 Wigner-Ville Distribution
76(4)
4.10 Properties of the Wigner-Ville Distribution
80(1)
4.10.1 A Real Quantity
80(1)
4.10.2 Marginal Properties
80(1)
4.10.3 Correlation Function
81(1)
4.11 Quadratic Superposition Principle
81(2)
4.12 Ambiguity Function
83(1)
4.13 Exercises
84(1)
4.14 Computer Programs
85(3)
4.14.1 Short-Time Fourier Transform
85(1)
4.14.2 Wigner-Ville Distribution
86(2)
References
88(1)
5 Multiresolution Analysis
89(19)
5.1 Multiresolution Spaces
89(3)
5.2 Orthogonal, Biorthogonal, and Semiorthogonal Decomposition
92(4)
5.3 Two-Scale Relations
96(1)
5.4 Decomposition Relation
97(1)
5.5 Spline Functions
98(5)
5.5.1 Properties of Splines
102(1)
5.6 Mapping a Function into MRA Space
103(1)
5.7 Exercises
104(2)
5.8 Computer Programs
106(1)
5.8.1 B-Splines
106(1)
References
107(1)
6 Construction of Wavelets
108(33)
6.1 Necessary Ingredients for Wavelet Construction
109(3)
6.1.1 Relationship Between Two-Scale Sequences
109(1)
6.1.2 Relationship Between Reconstruction and Decomposition Sequences
110(2)
6.2 Construction of Semiorthogonal Spline Wavelets
112(2)
6.2.1 Expression for {go[k]}
113(1)
6.3 Construction of Orthonormal Wavelets
114(4)
6.4 Orthonormal Scaling Functions
118(11)
6.4.1 Shannon Scaling Function
118(1)
6.4.2 Meyer Scaling Function
119(4)
6.4.3 Battle-Lemarie Scaling Function
123(2)
6.4.4 Daubechies Scaling Function
125(4)
6.5 Construction of Biorthogonal Wavelets
129(3)
6.6 Graphical Display of Wavelets
132(2)
6.6.1 Iteration Method
132(1)
6.6.2 Spectral Method
132(2)
6.6.3 Eigenvalue Method
134(1)
6.7 Exercises
134(4)
6.8 Computer Programs
138(1)
6.8.1 Daubechies Wavelet
138(1)
6.8.2 Iteration Method
139(1)
References
139(2)
7 Discrete Wavelet Transform and Filter Bank Algorithms
141(46)
7.1 Decimation and Interpolation
141(7)
7.1.1 Decimation
142(2)
7.1.2 Interpolation
144(3)
7.1.3 Convolution Followed by Decimation
147(1)
7.1.4 Interpolation Followed by Convolution
147(1)
7.2 Signal Representation in the Approximation Subspace
148(1)
7.3 Wavelet Decomposition Algorithm
149(4)
7.4 Reconstruction Algorithm
153(1)
7.5 Change of Bases
154(2)
7.6 Signal Reconstruction in Semiorthogonal Subspaces
156(7)
7.6.1 Change of Basis for Spline Functions
157(3)
7.6.2 Change of Basis for Spline Wavelets
160(3)
7.7 Examples
163(2)
7.8 Two-Channel Perfect Reconstruction Filter Bank
165(15)
7.8.1 Spectral-Domain Analysis of a Two-Channel PR Filter Bank
168(8)
7.8.2 Time-Domain Analysis
176(4)
7.9 Polyphase Representation for Filter Banks
180(2)
7.9.1 Signal Representation in the Polyphase Domain
180(1)
7.9.2 Filter Bank in the Polyphase Domain
181(1)
7.10 Comments on DWT and PR Filter Banks
182(1)
7.11 Exercises
183(1)
7.12 Computer Programs
184(2)
7.12.1 Algorithms
184(2)
References
186(1)
8 Fast Integral Transform and Applications
187(23)
8.1 Finer Time Resolution
188(2)
8.2 Finer Scale Resolution
190(4)
8.3 Function Mapping into the Interoctave Approximation Subspaces
194(2)
8.4 Examples
196(13)
8.4.1 IWT of a Linear Function
197(5)
8.4.2 Crack Detection
202(1)
8.4.3 Decomposition of Signals with Nonoctave Frequency Components
203(1)
8.4.4 Perturbed Sinusoidal Signal
203(1)
8.4.5 Chirp Signal
204(1)
8.4.6 Music Signal with Noise
204(1)
8.4.7 Dispersive Nature of the Waveguide Mode
205(4)
References
209(1)
9 Digital Signal Processing Applications
210(57)
9.1 Wavelet Packets
211(1)
9.2 Wavelet Packet Algorithms
212(2)
9.3 Thresholding
214(5)
9.3.1 Hard Thresholding
216(1)
9.3.2 Soft Thresholding
217(1)
9.3.3 Percentage Thresholding
218(1)
9.3.4 Implementation
218(1)
9.4 Interference Suppression
219(2)
9.5 Faulty Bearing Signature Identification
221(7)
9.5.1 Pattern Recognition of Acoustic Signals
221(5)
9.5.2 Wavelets, Wavelet Packets, and FFT Features
226(2)
9.6 Two-Dimensional Wavelets and Wavelet Packets
228(5)
9.6.1 Two-Dimensional Wavelets
228(3)
9.6.2 Two-Dimensional Wavelet Packets
231(2)
9.7 Wavelet and Wavelet Packet Algorithms for Two-Dimensional Signals
233(2)
9.7.1 Two-Dimensional Wavelet Algorithm
233(1)
9.7.2 Wavelet Packet Algorithm
234(1)
9.8 Image Compression
235(9)
9.8.1 Image Coding
235(1)
9.8.2 Wavelet Tree Coder
236(2)
9.8.3 EZW Code
238(1)
9.8.4 EZW Example
239(3)
9.8.5 Spatial-Oriented Tree
242(2)
9.8.6 Generalized Self-Similarity Tree
244(1)
9.9 Microcalcification Cluster Detection
244(5)
9.9.1 CAD Algorithm Structure
244(1)
9.9.2 Partitioning of Image and Nonlinear Contrast Enhancement
245(1)
9.9.3 Wavelet Decomposition of the Subimages
245(1)
9.9.4 Wavelet Coefficient Domain Processing
246(2)
9.9.5 Histogram Thresholding and Dark Pixel Removal
248(1)
9.9.6 Parametric ART2 Clustering
248(1)
9.9.7 Results
249(1)
9.10 Multicarrier Communication Systems
249(3)
9.10.1 OFDM Multicarrier Communication Systems
250(2)
9.10.2 Wavelet Packet-Based MCCS
252(1)
9.11 Three-Dimensional Medical Image Visualization
252(6)
9.11.1 Three-Dimensional Wavelets and Algorithms
255(1)
9.11.2 Rendering Techniques
256(2)
9.11.3 Region of Interest
258(1)
9.11.4 Summary
258(1)
9.12 Computer Programs
258(7)
9.12.1 Two-Dimensional Wavelet Algorithms
258(5)
9.12.2 Wavelet Packets Algorithms
263(2)
References
265(2)
10 Wavelets in Boundary Value Problems
267(34)
10.1 Integral Equations
268(4)
10.2 Method of Moments
272(1)
10.3 Wavelet Techniques
273(7)
10.3.1 Use of Fast Wavelet Algorithm
273(1)
10.3.2 Direct Application of Wavelets
274(1)
10.3.3 Wavelets in Spectral Domain
275(5)
10.3.4 Wavelet Packets
280(1)
10.4 Wavelets on the Bounded Interval
280(2)
10.5 Sparsity and Error Considerations
282(3)
10.6 Numerical Examples
285(6)
10.7 Semiorthogonal Versus Orthogonal Wavelets
291(3)
10.8 Differential Equations
294(1)
10.9 Expressions for Splines and Wavelets
295(2)
References
297(4)
Index 301

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