Ten Lectures on Wavelets

Abstract

xix 4-x 4-35 7pp. Price $3 Z 50 paper.Wavelets are used for signal or image decomposition, compression, smoothing, and reconstruction. They form solutions to integral equations and transform linear systems. Even spectrum analysis of band-and timelimited signals is more efficient using the wavelet transform rather than the Fourier transform. In addition, wavelet theory establishes a mathematical reference frame from which sub-band coding in signal processing, Heisenberg-Weyl coherent states in quantum physics, and seismic data analysis in geophysics can be developed as special cases. Ten Lectures on Wavelets provides a graduate level tutorial on the mathematical aspects of wavelet theory that is behind these applications. The book contains an introduction, preliminaries and notation, ten chapters, references, subject index, and author index. Each chapter includes notes and remarks.The author presents in this book her ten lectures delivered at the Conference Board of Mathematical Sciences (CBMS) conference at the University of Lowell, MA in June 1990. Each lecture corresponds to a chapter. An excellent introduction puts this book in perspective with other recent publications on wavelets and also reviews its contents.The first chapter motivates the reader by explaining "The What, Why and How of Wavelets." Two examples serve to achieve this objective. One example, done in great detail, is the wavelet decomposition of a function using the Haar wavelet. Another example compares the arrival time and spectral analysis of transient signals embedded in a sum of harmonic signals by the wavelet transform using the Morlet wavelet to the windowed Fourier transform of the same function using three different window intervals. The results are an incentive to continue. The comparison between the windowed Fourier transform and the wavelet transform is carried on throughout the text with the clear conclusion that the final analysis has yet to be done.Chapters 2-4 discuss the continuous and discrete wavelet transforms. The important concept of frames is introduced. Frames are similar to and sometimes are orthonormal bases, but are considerably more general. Chapter 4 also introduces time-frequency lattice density and the preliminaries for orthonormal wavelet bases. Comparisons with the Fourier trans-form are made frequently.The next five chapters deal with various aspects of orthonormal wavelet bases. Also discussed are the sub-band coding connection to wavelets, symmetrical and asymmetrical wavelets in image processing and numerical analysis, and connections to functional spaces. Another important concept of multiresolution analysis and its uses as a tool for constructing bases is developed. Multiresolution analysis builds wavelet bases by applying selfsimilar sets of wavelets each scaled, modulated, and translated copies of the "mother wavelet." An extension, called voices, is developed by using multiple mother wavelets. While Chap. 9 is the most challenging, the reader is compensated with a demonstration of why the wavelet transfo...

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