Wavelet Theory: An Elementary Approach with ApplicationsISBN: 9780470388402
504 pages
October 2009

Description
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, realworld applications.
The book begins with a brief introduction to the fundamentals of complex numbers and the space of squareintegrable 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 the relevance of wavelet theory in the digital age, the book includes two indepth sections on current applications: the FBI Wavelet Scalar Quantization Standard and image segmentation.
In order to facilitate mastery of the content, the book features more than 400 exercises that range from theoretical to computational in nature and are structured in a multipart format in order to assist readers with the correct proof or solution. These problems provide an opportunity for readers to further investigate various applications of wavelets. All problems are compatible with software packages and computer labs that are available on the book's related Web site, allowing readers to perform various imaging/audio tasks, explore computer wavelet transformations and their inverses, and visualize the applications discussed throughout the book.
Requiring only a prerequisite knowledge of linear algebra and calculus, Wavelet Theory is an excellent book for courses in mathematics, engineering, and physics at the upperundergraduate level. It is also a valuable resource for mathematicians, engineers, and scientists who wish to learn about wavelet theory on an elementary level.
Table of Contents
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 5Splines 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 OneDimensional Transform 146
Problems 159
4.2 The TwoDimensional 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
Author Information
Patrick J. Van Fleet, PhD, is Professor of Mathematics and Director of the Center for Applied Mathematics at the University of St. Thomas in St. Paul, Minnesota. He has written numerous journal articles in the areas of wavelets and spline theory. Dr. Van Fleet is the author of Discrete Wavelet Transformations: An Elementary Approach with Applications, also published by Wiley.
The Wiley Advantage

Provides comprehensive treatment of the topic using two reallife applications: the FBI Wavelet Scalar Quantization Standard and image segmentation. Both applications are developed in their entirety in the book, and readers will learn how to take a digital fingerprint and perform every step necessary to compress it via the FBI standard.

Supplemented with a software package called DiscreteWavelets, which helps both students and professionals to visualize the applications addressed throughout the book

Contains over 200 exercises that vary in type including rote drill, proof completion, and 'Further Discovery'. Extensive details and examples are provided via the exercises, which help readers to gain a full understanding of the material.

Provides readers with motivation and the needed background to pursue further study in both mathematical theory and applications

Contains a solid balance of rigor with details and applications
Reviews
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 upperundergraduate 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)
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