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University of St. Thomas Center for Applied Mathematics

The Discrete Wavelet Transformations: An Elementary Approach With Applications

1. Introduction - Why Wavelets?: This short essay serves as a brief introduction to the topic - some historical perspectives are given, as well as motivation for why and how the book constructs and uses discrete wavelet transformations.
2. Linear Algebra Basics: 2.1 - Vectors, Inner Products, and Norms; 2.2 - Basic Matrix Theory; 2.3 - Block Matrix Arithmetic
3. An Introduction to Digital Images: 3.1 - Digital Image Basics; 3.2 - Color Images and Color Spaces; 3.3 - Qualitative/Quantitative Measures; 3.4 - Huffman Encoding
4. Complex Numbers and Fourier Series: 4.1 - The Complex Plane; 4.2 - The Complex Exponential Function; 4.3 - Fourier Series
5. Filters and Convolution: 5.1 - Convolution; 5.2 - Filters; 5.3 - Convolution as a Matrix Product
6. The Haar Wavelet Transform: 6.1 - Constructing the Haar Wavelet Transform; 6.2 - Iterating the Process; 6.3 - Two-Dimensional Haar Wavelet Transform; 6.4 - Data Compression and Edge Detection
7. Orthogonal Wavelet Transforms: 7.1 - Daubechies Filters of Length 4 and 6; 7.2 - Daubechies Filters of Even Length; 7.3 - Daubechies Transforms - Algorithms
8. Orthogonality and Fourier Series: 8.1 - Fourier Series and Lowpass Filters; 8.2 - Building G(ω) from H(ω); 8.3 - Coiflet Filters
9. Denoising Via Wavelet Shrinkage: 9.1 - An Overview of Wavelet Shrinkage; 9.2 - VisuShrink; 9.3 - SureShrink
10. Biorthogonal Filters: 10.1 - Constructing Biorthogonal Filters; 10.2 - Biorthogonal Spline Filters; 10.3 - The CDF97 Filter
11. Computing Biorthogonal Transforms: 11.1 - Computing the Transform; 11.2 - Computing the Inverse Transform; 11.3 - Symmetry and Image Boundary Effects
12. JPEG2000: 12.1 - An Overview of JPEG; 12.2 - The Basic JPEG2000 Standard; 12.3 - Lifting and Lossless Compression; 12.4 - Examples
Appendix A. Basic Statistics: A.1 - Descriptive Statistics; A.2 - Sample Spaces, Probability, and Random Variables; A.3 - Continuous Distributions; A.4 - Expectation; A.5 - Two Special Distributions