Fundamentals of Wavelets: Theory, Algorithms, and Applications

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ISBN: 9780471197485 | 0471197483
Cover: Hardcover
Copyright: 2/1/1999

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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.

Preface

1 What This Book Is About

(4)

2 Mathematical Preliminaries

(26)

2.1 Linear Spaces

(2)

2.2 Vectors and Vector Spaces

(2)

2.3 Basis Functions

(4)

2.3.1 Orthogonality and Biorthogonality

(3)

2.4 Local Basis and Riesz Basis

(2)

2.5 Discrete Linear Normed Space

(1)

2.6 Approximation by Orthogonal Projection

(2)

2.7 Matrix Algebra and Linear Transformation

(5)

2.7.1 Elements of Matrix Algebra

(1)

2.7.2 Eigenmatrix

(1)

2.7.3 Linear Transformation

(1)

2.7.4 Change of Basis

(1)

2.7.5 Hermitian Matrix, Unitary Matrix, and Orthogonal Transformation

(1)

2.8 Digital Signals

(6)

2.8.1 Sampling of Signal

(1)

2.8.2 Linear Shift-Invariant Systems

(1)

2.8.3 Convolution

(1)

2.8.4 z-Transform

(1)

2.8.5 Region of Convergence

(2)

2.8.6 Inverse z-Transform

(1)

2.9 Exercises

(1)

References

(1)

3 Fourier Analysis

(26)

3.1 Fourier Series

(1)

3.2 Examples

(3)

3.2.1 Rectified Sine Wave

(1)

3.2.2 Comb Function and the Fourier Series Kernel K(N)(t)

(2)

3.3 Fourier Transform

(2)

3.4 Properties of the Fourier Transform

(3)

3.4.1 Linearity

(1)

3.4.2 Time Shifting and Time Scaling

(1)

3.4.3 Frequency Shifting and Frequency Scaling

(1)

3.4.4 Moments

(1)

3.4.5 Convolution

(1)

3.4.6 Parseval's Theorem

(1)

3.5 Examples of the Fourier Transform

(3)

3.5.1 Rectangular Pulse

(1)

3.5.2 Triangular Pulse

(1)

3.5.3 Gaussian Function

(1)

3.6 Poisson's Sum

(3)

3.6.1 Partition of Unity

(1)

3.7 Sampling Theorem

(3)

3.8 Partial Sum and the Gibbs Phenomenon

(1)

3.9 Fourier Analysis of Discrete-Time Signals

(4)

3.9.1 Discrete Fourier Basis and Discrete Fourier Series

(2)

3.9.2 Discrete-Time Fourier Transform

(2)

3.10 Discrete Fourier Transform

(1)

3.11 Exercises

(1)

References

(1)

4 Time-Frequency Analysis

(32)

4.1 Window Function

(2)

4.2 Short-Time Fourier Transform

(4)

4.2.1 Inversion Formula

(1)

4.2.2 Gabor Transform

(1)

4.2.3 Time-Frequency Window

(1)

4.2.4 Properties of STFT

(2)

4.3 Discrete Short-Time Fourier Transform

(1)

4.3.1 Examples of STFT

(1)

4.4 Discrete Gabor Representation

(2)

4.5 Continuous Wavelet Transform

(5)

4.5.1 Inverse Wavelet Transform

(1)

4.5.2 Time-Frequency Window

(2)

4.6 Discrete Wavelet Transform

(1)

4.7 Wavelet Series

(1)

4.8 Interpretations of the Time-Frequency Plot

(2)

4.9 Wigner-Ville Distribution

(4)

4.10 Properties of the Wigner-Ville Distribution

(1)

4.10.1 A Real Quantity

(1)

4.10.2 Marginal Properties

(1)

4.10.3 Correlation Function

(1)

4.11 Quadratic Superposition Principle

(2)

4.12 Ambiguity Function

(1)

4.13 Exercises

(1)

4.14 Computer Programs

(3)

4.14.1 Short-Time Fourier Transform

(1)

4.14.2 Wigner-Ville Distribution

(2)

References

(1)

5 Multiresolution Analysis

(19)

5.1 Multiresolution Spaces

(3)

5.2 Orthogonal, Biorthogonal, and Semiorthogonal Decomposition

(4)

5.3 Two-Scale Relations

(1)

5.4 Decomposition Relation

(1)

5.5 Spline Functions

(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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Fundamentals of Wavelets: Theory, Algorithms, and Applications

Summary

Author Biography

Table of Contents

Supplemental Materials

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Fundamentals of Wavelets: Theory, Algorithms, and Applications

Summary

Author Biography

Table of Contents

Supplemental Materials

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