×

Wavelets and multiwavelets. (English) Zbl 1058.65150

Boca Raton, FL: Chapman and Hall/CRC (ISBN 1-58488-304-9/hbk; 978-0-203-01159-1/ebook). xii, 275 p. (2004).
Orthonormal multiwavelets associated with certain refinable function vectors (also called scaling function vectors) have several advantages in comparison with scalar orthonormal wavelets. For example, an orthonormal multiwavelet can possess all the desirable properties of orthogonality, short support, high order of smoothness and vanishing moments, and symmetry/antisymmetry. Thus, orthonormal multiwavelets offer the possibility of superior performance for signal/image processing and numerical analysis applications. There are many papers on the multiwavelet theory and its applications. However, there has not been a full exposition of multiwavelet theory in the book form. This book contains a detailed study of multiwavelet theory as well as a comprehensive treatment of the scalar wavelets.
This book is organized into two parts. The first part (Chapter 1–Chapter 5) deals with the scalar wavelet theory, while the second part (Chapter 6–Chapter 11) is devoted to the multiwavelet theory. Most sections of the two parts are in parallel, so it is easy to check how a scalar result generalizes to the multiwavelet case.
Chapter 1 discusses some basic topics of the scalar wavelets such as refinable (two-scaling) functions, multiresolution analysis (MRA), moments, approximation orders and symmetry. Chapter 2 deals with the discrete wavelet transform (DWT), the pre/postprocessing of the input signals for the DWT, the extension of the finite signal for the DWT. Other topics such as lifting scheme are also discussed in this chapter. Chapter 3 is on the construction of compactly supported wavelets. In particular, the constructions of Daubechies’ wavelets, Coiflets and Cohen-Daubechies-Feauveau’s biorthogonal wavelets are presented in this chapter. Chapter 4 discusses the applications of wavelets to signal compression, signal denoising and numerical analysis. The fifth and the final chapter of Part One is denoted to the study of the existence of the refinable functions, Sobolev and Hölder smoothness estimates of the refinable functions. The convergence of the cascade algorithm, the stability of the refinable functions and the wavelet systems are also discussed in this chapter.
Chapter 6, the first chapter of Part Two, discusses the basic theory on multiwavelets, which includes refinable function vectors, MRAs of multiplicity \(r\), approximation order of the refinable function vectors. Chapter 7 deals with the discrete multiwavelet transform (DMWT), the pre/postprocessing of the input signals for the DMWT, balanced multiwavelets, the extension of the finite signal for the DMWT and modulation/polyphase formulations of the multifilter banks. Chapter 8 is on the two-scale similarity transform. Chapter 9 discusses the factorization of polyphase matrices; and Chapter 10 deals with the construction of the multiwavelets corresponding to a given refinable function vector. Chapter 11 is the last chapter of the book, and it is devoted to the study of the existence of the refinable function vectors, Sobolev and Hölder smoothness estimates of the refinable function vectors. The author also provides an appendix containing a list of of some wavelet and multiwavelet filters, a section on the mathematical ground and a list of web and software resources.
In summary, this is a clearly written introduction to multiwavelets as well as to the scalar wavelets which provides background materials on most of major topics of the current multiwavelet theory. It could be used for an advanced undergraduate course and a graduate course.

MSC:

65T60 Numerical methods for wavelets
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
94A12 Signal theory (characterization, reconstruction, filtering, etc.)

Software:

Matlab
PDFBibTeX XMLCite
Full Text: Link