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.
Author(s): David K. Ruch, Patrick J. Van Fleet
Edition: 1
Publisher: Wiley-Interscience
Year: 2009
Language: English
Pages: 502
Tags: Приборостроение;Обработка сигналов;Вейвлет-анализ;
Wavelet Theory: An Elementary Approach with Applications......Page 5
CONTENTS......Page 9
Preface......Page 13
Acknowledgments......Page 21
1.1 Complex Numbers and Basic Operations......Page 23
Problems......Page 27
1.2 The Space L2(R)......Page 29
Problems......Page 38
1.3 Inner Products......Page 40
Problems......Page 47
1.4 Bases and Projections......Page 48
Problems......Page 50
2 Fourier Series and Fourier Transformations......Page 53
2.1 Euler's Formula and the Complex Exponential Function......Page 54
Problems......Page 58
2.2 Fourier Series......Page 59
Problems......Page 71
2.3 The Fourier Transform......Page 75
Problems......Page 88
2.4 Convolution and 5-Splines......Page 94
Problems......Page 104
3 Haar Spaces......Page 107
3.1 The Haar Space Vo......Page 108
3.2 The General Haar Space Vj......Page 115
Problems......Page 129
3.3 The Haar Wavelet Space W0......Page 130
Problems......Page 141
3.4 The General Haar Wavelet Space Wj......Page 142
Problems......Page 155
3.5 Decomposition and Reconstruction......Page 156
Problems......Page 162
3.6 Summary......Page 163
4 The Discrete Haar Wavelet Transform and Applications......Page 167
4.1 The One-Dimensional Transform......Page 168
Problems......Page 181
4.2 The Two-Dimensional Transform......Page 185
Problems......Page 193
4.3 Edge Detection and Naive Image Compression......Page 194
5 Multiresolution Analysis......Page 201
5.1 Multiresolution Analysis......Page 202
Problems......Page 218
5.2 The View from the Transform Domain......Page 222
Problems......Page 234
5.3 Examples of Multiresolution Analyses......Page 238
Problems......Page 246
5.4 Summary......Page 247
6 Daubechies Scaling Functions and Wavelets......Page 255
6.1 Constructing the Daubechies Scaling Functions......Page 256
Problems......Page 268
6.2 The Cascade Algorithm......Page 273
Problems......Page 287
6.3 Orthogonal Translates, Coding, and Projections......Page 290
Problems......Page 298
7 The Discrete Daubechies Transformation and Applications......Page 299
7.1 The Discrete Daubechies Wavelet Transform......Page 300
Problems......Page 312
7.2 Projections and Signal and Image Compression......Page 315
Problems......Page 332
7.3 Naive Image Segmentation......Page 336
Problems......Page 344
8 Biorthogonal Scaling Functions and Wavelets......Page 347
8.1 A Biorthogonal Example and Duality......Page 348
Problems......Page 355
8.2 Biorthogonality Conditions for Symbols and Wavelet Spaces......Page 356
Problems......Page 372
8.3 Biorthogonal Spline Filter Pairs and the CDF97 Filter Pair......Page 375
Problems......Page 390
8.4 Decomposition and Reconstruction......Page 392
8.5 The Discrete Biorthogonal Wavelet Transform......Page 397
Problems......Page 410
8.6 Riesz Basis Theory......Page 412
Problems......Page 419
9 Wavelet Packets......Page 421
9.1 Constructing Wavelet Packet Functions......Page 422
Problems......Page 435
9.2 Wavelet Packet Spaces......Page 436
9.3 The Discrete Packet Transform and Best Basis Algorithm......Page 446
Problems......Page 461
9.4 The FBI Fingerprint Compression Standard......Page 462
Appendix A: Huffman Coding......Page 477
Problems......Page 484
References......Page 487
Topic Index......Page 491
Author Index......Page 501
