Wavelet Transforms: Introduction to Theory and Applications

Written by researchers Raghuveer M. Rao and Ajit S. Bopardikar, Wavelet Transforms provides engineers, scientists, and students with a practical understanding of wavelet transforms and their properties. The authors introduce the underlying theory of the transform by presenting a wide range of applications, such as signal processing, image processing, and communications. This book identifies problems for which wavelet transform techniques are well-suited, shows how to implement wavelet transforms efficiently, and explains how to choose or design appropriate wavelets for a given application. In Chapter 1, basic linear filtering principles are utilized to introduce the reader to continuous wavelet transform. In Chapter 2, the basics of discrete wavelet transforms and multiresolution analysis are presented. Multiresolution analysis is then further explored in Chapter 3. Chapter 4 contains alternative wavelet representations, such as biorthogonal bases, wavelet packets, and multiresolution analysis of images. Chapter 5 provides a detailed treatment of the use of wavelet transform techniques in signal and image compression. In Chapter 6, applications to areas such as denoising, object isolation, and detection are presented. Chapter 7 addresses several more advanced topics, including: choice or design of wavelets for a given application, projection relations for the continuous wavelet transform, biorthogonal bandlimited wavelets, matched wavelet construction, self-similar signals, and linear scale-invariant systems. The supporting disk contains MATLAB routines that enable the reader to experiment with various algorithms and techniques presented in the book. Practical in their approach, Rao and Bopardikar present the material in a visual and comprehensive manner, using geometric analogies and filtering concepts. The book is written in a language familiar to readers with a basic undergraduate engineering degree.

Table of Contents:
Preface. Acknowledgments. 1. Continuous Wavelet Transform. Introduction. Continuous-Time Wavelets. Definition of the CWT. The CWT as a Correlation. Constant Q-Factor Filtering Interpretation and Time-Frequency Resolution. The CWT as an Operator. Inverse CWT. Problems. 2. Introduction to the Discrete Wavelet Transform and Orthogonal Wavelet Decomposition. Introduction. Approximations of Vectors in Nested Linear Vector Subspaces. Example of Approximating Vectors in Nested Subspaces of a Finite-Dimensional Linear Vector Space. Example of Approximating Vectors in Nested Subspaces of an Infinite-Dimensional Linear Vector Space. Example of an MRA. Bases for the Approximation Subspaces and Haar Scaling Function. Bases for the Detail Subspaces and Haar Wavelet. Digital Filter Implementation of the Haar Wavelet Decomposition. Problems. 3. MRA, Orthonormal Wavelets, and Their Relationship to Filter Banks. Introduction. Formal Definition of an MRA. Construction of a General Orthonormal MRA. Scaling Function and Subspaces. Implications of the Dilation Equation and Orthogonality. A Wavelet Basis for the MRA. Two-scale Relation for psi (t). Basis for the Detail Subspaces. Direct Sum Decomposition. Digital Filtering Interpretation. Decomposition Filters. Reconstructing the Signal. Examples of Orthogonal Basis-Generating Wavelets. Daubechies D4 Scaling Function and Wavelet. Bandlimited Wavelets. Interpreting Orthonormal MRAs for Discrete-Time Signals. Continuous-time MRA Interpretation for the DTWT. Discrete-Time MRA. Basis Functions for the DTWT. Miscellaneous Issues Related to PRQMF Filter Banks. Generating Scaling Functions and Wavelets from Filter Coefficients. Problems. 4. Alternative Wavelet Representations. Introduction. Biorthogonal Wavelet Bases. Filtering Relationship for Biorthogonal Filters. Examples of Biorthogonal Scaling Functions and Wavelets. Two-Dimensional Wavelets. Nonseparable Multidimensional Wavelets. Wavelet Packets. Problems. 5. Wavelet Transform and Data Compression. Introduction. Transform Coding. DTWT for Image Compression. Image Compression Using DTWT and Run-length Encoding. Embedded Tree Image Coding. Comparison with JPEG. Audio Compression. Audio Masking. Standards Specifying Subband Implementation: ISO/MPEG Coding for Audio. Wavelet-Based Audio Coding. Video Coding Using Multiresolution Techniques: A Brief Introduction. 6. Other Applications of Wavelet Transforms. Introduction. Wavelet Denoising. Speckle Removal. Edge Detection and Object Isolation. Image Fusion. Object Detection by Wavelet Transforms of Projections. Communication Applications. Scaling Functions as Signaling Pulses. Discrete Wavelet Multitone Modulation. 7. Advanced Topics. Introduction. CWTs Revisited. Parseval's Identity for the CWT. Inverse CWT Is a Many-to-One Operation. Wavelet Inner Product as a Projection Operation. Bridging the Gap Between CWTs and DWTs. CWT with an Orthonormal Basis-Generating Wavelet. A Trous Algorithm. Regularity and Convergence. Daubechies Construction of Orthonormal Scaling Functions. Bandlimited Biorthogonal Decomposition. Scaling Function Pair Construction. Wavelet Pair Construction. Design and Selection of Wavelets. The Lifting Scheme. Best Basis Selection. Wavelet Matching. Perfect Reconstruction Circular Convolution Filter Banks. Downsampling. Upsampling. Procedure for Implementation. Conditions for Perfect Reconstruction. Procedure for Constructing PRCC Filter Banks. Interpolators Matched to the Input Process. Interpolation Sampling. Frequency-Sampled Implementation of Bandlimited DWTs. The Scaling Operation and Self-Similar Signals. LTI Systems and Eigenfunctions. Continuous-Time Linear Scale-Invariant System. Scaling in Discrete Time. Discrete-time LSI Systems. Appendix A. Fundamentals of Multirate Systems. The Downsampler. The Upsampler. Noble Identities. Appendix B. Linear Algebra and Vector Spaces. Brief Review of Vector Spaces. Vector Subspace. Linear Independence and Bases. Inner Product Spaces. Hilbert Space and Riesz Bases. Index.