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…mehr
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.Hinweis: Dieser Artikel kann nur an eine deutsche Lieferadresse ausgeliefert werden.
David K. Ruch, PhD, is Professor in the Department of Mathematical and Computer Sciences at the Metropolitan State College of Denver. He has authored more than twenty journal articles in his areas of research interest, which include wavelets and functional analysis. 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.
Inhaltsangabe
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
