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.
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. Contents
Why Wavelets? | Vectors and Matrices | Vectors, Inner Products, and Norms | Basic Matrix Theory | Block Matrix Arithmetic | An Introduction to Digital Images | The Basics of Grayscale Digital Images | Color Images and Color Spaces | Qualitative and Quantitative Measures | Huffman Encoding | Complex Numbers and Fourier Series | The Complex Plane and Arithmetic | Complex Exponential Functions | Fourier Series | Convolution and Filters | Convolution | Filters | Convolution as a Matrix Product | The Haar Wavelet Transformation | Constructing the Haar Wavelet Transformation | Iterating the Process | The Two-Dimensional Haar Wavelet Transformation | Applications: Image Compression and Edge Detection | Daubechies Wavelet Transformations | Daubechies Filters of Length 4 and 6 | Daubechies Filters of Even Length | Algorithms for Daubechies Wavelet Transformations | Orthogonality and Fourier Series | Fourier Series and Lowpass Filters | Building G([omega]) from H([omega]) | Coiflet Filters | Wavelet Shrinkage: An Application to Denoising | An Overview of Wavelet Shrinkage | VisuShrink | SureShrink | Biorthogonal Filters | Constructing Biorthogonal Filters | Biorthogonal Spline Filters | The Cohen-Daubechies-Feauveau 9/7 Filter | Computing Biorthogonal Wavelet Transformations | Computing the Biorthogonal Wavelet Transformation | Computing the Inverse Biorthogonal Wavelet Transformation | Symmetry and Boundary Effects | The JPEG2000 Image Compression Standard | An Overview of JPEG | The Basic JPEG2000 Algorithm | Lifting and Lossless Compression | Basic Statistics | Descriptive Statistics | Sample Spaces, Probability, and Random Variables | Continuous Distributions | Expectation | Two Special Distributions Related Topics
Algebra, Calculus and Analysis