An "applications first" approach to discrete wavelet transformationsDiscrete Wavelet Transformations provides readers with a broad elementary introduction to discrete wavelet transformations and their applications. With extensive graphical displays, this self-contained book integrates concepts from calculus and linear algebra into the construction of wavelet transformations and their various applications, including data compression, edge detection in images, and signal and image denoising.The book begins with a cursory look at wavelet transformation development and illustrates its allure in digital signal and image applications. Next, a chapter on digital image basics, quantitative and qualitative measures, and Huffman coding equips readers with the tools necessary to develop a comprehensive understanding of the applications. Subsequent chapters discuss the Fourier series, convolution, and filtering, as well as the Haar wavelet transform to introduce image compression and image edge detection. The development of Daubechies filtersis presented in addition to coverage of wavelet shrinkage in the area of image and signal denoising. The book concludes with the construction of biorthogonal filters and also describes their incorporation in the JPEG2000 image compression standard.The author's "applications first" approach promotes a hands-on treatment of wavelet transforma-tion construction, and over 400 exercises are presented in a multi-part format that guide readers through the solution to each problem. Over sixty computer labs and software development projects provide opportunities for readers to write modules and experiment with the ideas discussed throughout the text. The author's software package, DiscreteWavelets, is used to perform various imaging and audio tasks, compute wavelet transformations and inverses, and visualize the output of the computations. Supplementary material is also available via the book's related Web site, which includes an audio and video repository, final project modules, and softwarefor reproducing examples from the book. All software, including the DiscreteWavelets package, is available for use with Mathematica®, MATLAB®, and Maple.Discrete Wavelet Transformations strongly reinforces the use of mathematics in digital data applications, sharpens programming skills, and provides a foundation for further study of more advanced topics, such as real analysis. This book is ideal for courses on discrete wavelet transforms and their applications at the undergraduate level and also serves as an excellent reference for mathematicians, engineers, and scientists who wish to learn about discrete wavelet transforms at an elementary level.
Author(s): Patrick Van Fleet
Edition: 1
Publisher: Wiley-Interscience
Year: 2008
Language: English
Pages: 571
Tags: Приборостроение;Обработка сигналов;Вейвлет-анализ;
Discrete Wavelet Transformations: An Elementary Approach with Applications......Page 6
Contents......Page 10
Preface......Page 16
Acknowledgments......Page 26
1 Introduction: Why Wavelets?......Page 28
2 Vectors and Matrices......Page 42
2.1 Vectors, Inner Products, and Norms......Page 43
Problems......Page 47
2.2 Basic Matrix Theory......Page 49
Problems......Page 63
2.3 Block Matrix Arithmetic......Page 65
Problems......Page 72
Computer Lab......Page 75
3 An Introduction to Digital Images......Page 76
3.1 The Basics of Grayscale Digital Images......Page 77
Problems......Page 93
3.2 Color Images and Color Spaces......Page 97
Problems......Page 102
3.3 Qualitative and Quantitative Measures......Page 105
Problems......Page 111
3.4 Huffman Encoding......Page 115
Problems......Page 121
Computer Labs......Page 122
4 Complex Numbers and Fourier Series......Page 124
4.1 The Complex Plane and Arithmetic......Page 125
Problems......Page 130
4.2 Complex Exponential Functions......Page 131
Problems......Page 136
4.3 Fourier Series......Page 137
Problems......Page 148
Computer Lab......Page 153
5 Convolution and Filters......Page 154
5.1 Convolution......Page 155
Problems......Page 162
Computer Lab......Page 164
5.2 Filters......Page 165
Problems......Page 174
5.3 Convolution as a Matrix Product......Page 177
Problems......Page 182
6 The Haar Wavelet Transformation......Page 184
6.1 Constructing the Haar Wavelet Transformation......Page 185
Problems......Page 198
6.2 Iterating the Process......Page 200
Problems......Page 209
6.3 The Two-Dimensional Haar Wavelet Transformation......Page 210
Problems......Page 225
6.4 Applications: Image Compression and Edge Detection......Page 226
Problems......Page 244
Computer Labs......Page 248
7 Daubechies Wavelet Transformations......Page 250
7.1 Daubechies Filters of Length 4 and 6......Page 251
Problems......Page 270
Computer Labs......Page 279
7.2 Daubechies Filters of Even Length......Page 280
Problems......Page 290
7.3 Algorithms for Daubechies Wavelet Transformations......Page 292
Problems......Page 305
Computer Labs......Page 306
8 Orthogonality and Fourier Series......Page 316
8.1 Fourier Series and Lowpass Filters......Page 317
8.2 Building G(w) from H(w)......Page 323
Problems......Page 332
8.3 Coiflet Filters......Page 333
Problems......Page 347
Computer Labs......Page 351
9 Wavelet Shrinkage: An Application to Denoising......Page 352
9.1 An Overview of Wavelet Shrinkage......Page 353
Problems......Page 358
Computer Labs......Page 359
9.2 VisuShrink......Page 360
Problems......Page 368
Computer Labs......Page 369
9.3 SureShrink......Page 370
Problems......Page 381
Computer Labs......Page 385
10 Biorthogonal Filters......Page 386
10.1 Constructing Biorthogonal Filters......Page 388
Problems......Page 403
10.2 Biorthogonal Spline Filters......Page 406
Problems......Page 426
Computer Lab......Page 428
10.3 The Cohen–Daubechies–Feauveau 9/7 Filter......Page 429
Problems......Page 438
Computer Lab......Page 441
11 Computing Biorthogonal Wavelet Transformations......Page 442
11.1 Computing the Biorthogonal Wavelet Transformation......Page 443
Computer Labs......Page 451
11.2 Computing the Inverse Biorthogonal Wavelet Transformation......Page 452
Problems......Page 469
11.3 Symmetry and Boundary Effects......Page 470
Problems......Page 488
Computer Labs......Page 492
12 The JPEG2000 Image Compression Standard......Page 494
12.1 An Overview of JPEG......Page 495
Computer Lab......Page 502
12.2 The Basic JPEG2000 Algorithm......Page 503
Problems......Page 510
12.3 Lifting and Lossless Compression......Page 511
Problems......Page 519
12.4 Examples......Page 521
Computer Lab......Page 527
A.1 Descriptive Statistics......Page 528
Problems......Page 530
A.2 Sample Spaces, Probability, and Random Variables......Page 531
A.3 Continuous Distributions......Page 534
Problems......Page 540
A.4 Expectation......Page 541
Problems......Page 546
A.5 Two Special Distributions......Page 547
Problems......Page 550
References......Page 554
Index......Page 560
