Wavelet Theory

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 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.

A self-contained, elementary introduction to wavelet theory andapplicationsExploring the growing relevance of wavelets in the field ofmathematics, Wavelet Theory: An Elementary Approach withApplications provides an introduction to the topic, detailing thefundamental concepts and presenting its major impacts in the worldbeyond academia. Drawing on concepts from calculus and linearalgebra, this book helps readers sharpen their mathematical proofwriting and reading skills through interesting, real-worldapplications.The book begins with a brief introduction to the fundamentals ofcomplex numbers and the space of square-integrable functions. NextFourier series and the Fourier transform are presented as tools forunderstanding wavelet analysis and the study of wavelets in thetransform domain. Subsequent chapters provide a comprehensivetreatment of various types of wavelets and their related conceptssuch as Haar spaces, multiresolution analysis, Daubechies waveletsand biorthogonal wavelets. In addition, the authors include twochapters that carefully detail the transition from wavelet theoryto the discrete wavelet transformations. To illustrate therelevance of wavelet theory in the digital age, the book includestwo in-depth sections on current applications: the FBI WaveletScalar Quantization Standard and image segmentation.In order to facilitate mastery of the content, the book featuresmore than 400 exercises that range from theoretical tocomputational in nature and are structured in a multi-part formatin order to assist readers with the correct proof or solution.These problems provide an opportunity for readers to furtherinvestigate various applications of wavelets. All problems arecompatible with software packages and computer labs that areavailable on the book's related Web site, allowing readers toperform various imaging/audio tasks, explore computer wavelettransformations and their inverses, and visualize the applicationsdiscussed throughout the book.Requiring only a prerequisite knowledge of linear algebra andcalculus, Wavelet Theory is an excellent book for courses inmathematics, engineering, and physics at the upper-undergraduatelevel. It is also a valuable resource for mathematiciansengineers, and scientists who wish to learn about wavelet theory onan elementary level.

²Preface xiAcknowledgments xix1 The Complex Plane and the Space L²(R) 11.1 Complex Numbers and Basic Operations 1Problems 51.2 The Space L²(R) 7Problems 161.3 Inner Products 18Problems 251.4 Bases and Projections 26Problems 282 Fourier Series and Fourier Transformations 312.1 Euler's Formula and the Complex Exponential Function 32Problems 362.2 Fourier Series 37Problems 492.3 The Fourier Transform 53Problems 662.4 Convolution and 5-Splines 72Problems 823 Haar Spaces 853.1 The Haar Space Vo 86Problems 933.2 The General Haar Space Vj 93Problems 1073.3 The Haar Wavelet Space W0 108Problems 1193.4 The General Haar Wavelet Space Wj 120Problems 1333.5 Decomposition and Reconstruction 134Problems 1403.6 Summary 1414 The Discrete Haar Wavelet Transform and Applications 1454.1 The One-Dimensional Transform 146Problems 1594.2 The Two-Dimensional Transform 163Problems 1714.3 Edge Detection and Naive Image Compression 1725 Multiresolution Analysis 1795.1 Multiresolution Analysis 180Problems 1965.2 The View from the Transform Domain 200Problems 2125.3 Examples of Multiresolution Analyses 216Problems 2245.4 Summary 2256 Daubechies Scaling Functions and Wavelets 2336.1 Constructing the Daubechies Scaling Functions 234Problems 2466.2 The Cascade Algorithm 251Problems 2656.3 Orthogonal Translates, Coding, and Projections 268Problems 2767 The Discrete Daubechies Transformation and Applications 2777.1 The Discrete Daubechies Wavelet Transform 278Problems 2907.2 Projections and Signal and Image Compression 293Problems 3107.3 Naive Image Segmentation 314Problems 3228 Biorthogonal Scaling Functions and Wavelets 3258.1 A Biorthogonal Example and Duality 326Problems 3338.2 Biorthogonality Conditions for Symbols and Wavelet Spaces 334Problems 3508.3 Biorthogonal Spline Filter Pairs and the CDF97 Filter Pair 353Problems 3688.4 Decomposition and Reconstruction 370Problems 3758.5 The Discrete Biorthogonal Wavelet Transform 375Problems 3888.6 Riesz Basis Theory 390Problems 3979 Wavelet Packets 3999.1 Constructing Wavelet Packet Functions 400Problems 4139.2 Wavelet Packet Spaces 414Problems 4249.3 The Discrete Packet Transform and Best Basis Algorithm 424Problems 4399.4 The FBI Fingerprint Compression Standard 440Appendix A: Huffman Coding 455Problems 462References 465Topic Index 469Author Index 479