Description
Book SynopsisA 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,
Trade Review
"The book, putting emphasize on an analytic facet of wavelets, can be seen as complementary
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)
Table of Contents²Preface 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
