# Wavelet Theory. An Elementary Approach with Applications

- ID: 2171917
- November 2009
- 504 Pages
- John Wiley and Sons Ltd

A self–contained, elementary introduction to wavelet theory and applications

Exploring the growing relevance of wavelets in the field of mathematics, Wavelet Theory: An Elementary Approach with Applications provides an introduction to the topic, detailing the fundamental concepts and presenting its major impacts in the world beyond academia. Drawing on concepts from calculus and linear algebra, this book helps readers sharpen their mathematical proof writing and reading skills through interesting, real–world applications.

The book begins with a brief introduction to the fundamentals of complex numbers and the space of square–integrable functions. Next, Fourier series and the Fourier transform are presented as tools for understanding wavelet analysis and the study of wavelets in the transform domain. Subsequent chapters provide a comprehensive treatment of various types of wavelets and their related concepts, such as Haar spaces, multiresolution analysis, Daubechies wavelets, and biorthogonal wavelets. In addition, the authors include two chapters that carefully detail the transition from wavelet theory to the discrete wavelet transformations. To illustrate
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²Preface xi

Acknowledgments xix

1 The Complex Plane and the Space L²(R) 1

1.1 Complex Numbers and Basic Operations 1

Problems 5

1.2 The Space L²(R) 7

Problems 16

1.3 Inner Products 18

Problems 25

1.4 Bases and Projections 26

Problems 28

2 Fourier Series and Fourier Transformations 31

2.1 Euler's Formula and the Complex Exponential Function 32

Problems 36

2.2 Fourier Series 37

Problems 49

2.3 The Fourier Transform 53

Problems 66

2.4 Convolution and 5–Splines 72

Problems 82

3 Haar Spaces 85

3.1 The Haar Space Vo 86

Problems 93

3.2 The General Haar Space Vj 93

Problems 107

3.3 The Haar Wavelet Space W0 108

Problems 119

3.4 The General Haar Wavelet Space Wj 120

Problems 133

3.5 Decomposition and Reconstruction 134

Problems 140

3.6 Summary 141

4 The Discrete Haar Wavelet Transform and Applications 145

4.1 The One–Dimensional Transform 146

Problems 159

4.2 The Two–Dimensional Transform 163

Problems 171

4.3 Edge Detection and Naive Image Compression 172

5 Multiresolution Analysis 179

5.1 Multiresolution Analysis 180

Problems 196

5.2 The View from the Transform Domain 200

Problems 212

5.3 Examples of Multiresolution Analyses 216

Problems 224

5.4 Summary 225

6 Daubechies Scaling Functions and Wavelets 233

6.1 Constructing the Daubechies Scaling Functions 234

Problems 246

6.2 The Cascade Algorithm 251

Problems 265

6.3 Orthogonal Translates, Coding, and Projections 268

Problems 276

7 The Discrete Daubechies Transformation and Applications 277

7.1 The Discrete Daubechies Wavelet Transform 278

Problems 290

7.2 Projections and Signal and Image Compression 293

Problems 310

7.3 Naive Image Segmentation 314

Problems 322

8 Biorthogonal Scaling Functions and Wavelets 325

8.1 A Biorthogonal Example and Duality 326

Problems 333

8.2 Biorthogonality Conditions for Symbols and Wavelet Spaces 334

Problems 350

8.3 Biorthogonal Spline Filter Pairs and the CDF97 Filter Pair 353

Problems 368

8.4 Decomposition and Reconstruction 370

Problems 375

8.5 The Discrete Biorthogonal Wavelet Transform 375

Problems 388

8.6 Riesz Basis Theory 390

Problems 397

9 Wavelet Packets 399

9.1 Constructing Wavelet Packet Functions 400

Problems 413

9.2 Wavelet Packet Spaces 414

Problems 424

9.3 The Discrete Packet Transform and Best Basis Algorithm 424

Problems 439

9.4 The FBI Fingerprint Compression Standard 440

Appendix A: Huffman Coding 455

Problems 462

References 465

Topic Index 469

Author Index 479

"The book, putting emphasize on an analytic facet of wavelets, can be seen as complementary

. to the previous Patrick J. Van Fleet's book, DiscreteWavelet Transformations: An Elementary

. Approach with Applications, focused on their algebraic properties." (Zentralblatt MATH, 2011)

"Requiring only a prerequisite knowledge of calculus and linear algebra, Wavelet theory is an excellent book for courses in mathematics, engineering, and physics at the upper–undergraduate level. It is also a valuable resource for mathematicians, engineers, and scientists who wish to learn about wavelet theory on an elementary level." (Mathematical Reviews, 2011)