# Wavelet Theory: An Elementary Approach with Applications

# Wavelet Theory: An Elementary Approach with Applications

ISBN: 978-1-118-16565-2

Sep 2011

504 pages

## Description

A self-contained, elementary introduction to wavelet theory and applicationsExploring 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 the relevance of wavelet theory in the digital age, the book includes two in-depth 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 multi-part 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 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.

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

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)

- Provides comprehensive treatment of the topic using two real-life 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