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Discrete Wavelet Transformations: An Elementary Approach with Applications

Book Description

An "applications first" approach to discrete wavelet transformations

Discrete 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.

Table of Contents

  1. Coverpage
  2. Titlepage
  3. Copyright
  4. Dedication
  5. Contents
  6. Preface
  7. Acknowledgments
  8. 1 Introduction: Why Wavelets?
  9. 2 Vectors and Matrices
    1. 2.1 Vectors, Inner Products, and Norms
      1. Problems
      2. Computer Lab
    2. 2.2 Basic Matrix Theory
      1. Problems
      2. Computer Lab
    3. 2.3 Block Matrix Arithmetic
      1. Problems
      2. Computer Lab
  10. 3 An Introduction to Digital Images
    1. 3.1 The Basics of Grayscale Digital Images
      1. Problems
      2. Computer Labs
    2. 3.2 Color Images and Color Spaces
      1. Problems
      2. Computer Labs
    3. 3.3 Qualitative and Quantitative Measures
      1. Problems
      2. Computer Labs
    4. 3.4 Huffman Encoding
      1. Problems
      2. Computer Labs
  11. 4 Complex Numbers and Fourier Series
    1. 4.1 The Complex Plane and Arithmetic
      1. Problems
      2. Computer Lab
    2. 4.2 Complex Exponential Functions
      1. Problems
    3. 4.3 Fourier Series
      1. Problems
      2. Computer Lab
  12. 5 Convolution and Filters
    1. 5.1 Convolution
      1. Problems
      2. Computer Lab
    2. 5.2 Filters
      1. Problems
      2. Computer Lab
    3. 5.3 Convolution as a Matrix Product
      1. Problems
  13. 6 The Haar Wavelet Transformation
    1. 6.1 Constructing the Haar Wavelet Transformation
      1. Problems
      2. Computer Labs
    2. 6.2 Iterating the Process
      1. Problems
      2. Computer Labs
    3. 6.3 The Two-Dimensional Haar Wavelet Transformation
      1. Problems
      2. Computer Labs
    4. 6.4 Applications: Image Compression and Edge Detection
      1. Problems
      2. Computer Labs
  14. 7 Daubechies Wavelet Transformations
    1. 7.1 Daubechies Filters of Length 4 and 6
      1. Problems
      2. Computer Labs
    2. 7.2 Daubechies Filters of Even Length
      1. Problems
      2. Computer Labs
    3. 7.3 Algorithms for Daubechies Wavelet Transformations
      1. Problems
      2. Computer Labs
  15. 8 Orthogonality and Fourier Series
    1. 8.1 Fourier Series and Lowpass Filters
      1. Problems
    2. 8.2 Building G(ω) from H(ω)
      1. Problems
    3. 8.3 Coiflet Filters
      1. Problems
      2. Computer Labs
  16. 9 Wavelet Shrinkage: An Application to Denoising
    1. 9.1 An Overview of Wavelet Shrinkage
      1. Problems
      2. Computer Labs
    2. 9.2 VisuShrink
      1. Problems
      2. Computer Labs
    3. 9.3 SureShrink
      1. Problems
      2. Computer Labs
  17. 10 Biorthogonal Filters
    1. 10.1 Constructing Biorthogonal Filters
      1. Problems
    2. 10.2 Biorthogonal Spline Filters
      1. Problems
      2. Computer Lab
    3. 10.3 The Cohen-Daubechies-Feauveau 9/7 Filter
      1. Problems
      2. Computer Lab
  18. 11 Computing Biorthogonal Wavelet Transformations
    1. 11.1 Computing the Biorthogonal Wavelet Transformation
      1. Problems
      2. Computer Labs
    2. 11.2 Computing the Inverse Biorthogonal Wavelet Transformation
      1. Problems
      2. Computer Labs
    3. 11.3 Symmetry and Boundary Effects
      1. Problems
      2. Computer Labs
  19. 12 The JPEG2000 Image Compression Standard
    1. 12.1 An Overview of JPEG
      1. Problems
      2. Computer Lab
    2. 12.2 The Basic JPEG2000 Algorithm
      1. Problems
    3. 12.3 Lifting and Lossless Compression
      1. Problems
      2. Computer Lab
    4. 12.4 Examples
      1. Computer Lab
  20. Appendix A: Basic Statistics
    1. A.1 Descriptive Statistics
      1. Problems
    2. A.2 Sample Spaces, Probability, and Random Variables
      1. Problems
    3. A.3 Continuous Distributions
      1. Problems
    4. A.4 Expectation
      1. Problems
    5. A.5 Two Special Distributions
      1. Problems
  21. References
  22. Index