Customer Reviews

Wavelets: A Primer

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Choosing a book on Wavelets to use as material for an introductory course is far from simple: there are many books on the subject and often the mathematical starting level of the audience is not uniform. But the book by Blatter is in my opinion very good if one wants to start from scratch (well, not quite of course: some knowledge of Lebesgue integration and infinite dimensional Banach and Hilbert spaces of functions is needed) in a mathematical or technical environment.

Starting with 2 chapters on Fourier analysis, going from windowed Fourier transform to a first introduction of continuous wavelet transform (CWT), and giving connections to several well known facts from signal analysis (the Heisenberg uncertainty priciple and the Shannon sampling theorem), the theoretical framework of the CWT is given in chapter 3. Interesting are the results on the decay of the wavelet transform in terms of the smoothness of the signal and the number of vanishing moments for the wavelet.

This is followed by a chapter on the important subject of 'frames' and one on multiresolution analysis (the foundation for the hierarchy of orthonormal bases for L^2(R)). The scaling function (time domain) is actuallly constructed from its properties in the Fourier domain (i.e. the frequency domain).

The final chapter treats the basic idea behind and constructions of orthonormal wavelets with compact support (including the Daubechies wavelets), also touching upon binary interpolation and so called spline wavelets.

For applications to a host of disciplines it is possible to consult one of the many applied books: in my opinion it is necessary to have a firm grasp of the underlying theoretical framework before really understanding what one is doing. I have nothing against applications, but once one has to go back to the roots: back to the valley to be able to reach the crest of the wavelet without accidents.

This book is a good tool for those who want to explore the mathematical rigors of wavelet theory. The "primer" assumes that the user is a fairly well versed mathematician, but the derivations are complete. There is little to nothing said about the practical applications of wavelets. For applications, look to other books.

This booklet, the translation of the original German version "Wavelets - Eine Einfuehrung" (published by Vieweg-Verlag), is indeed a real primer.

It removes old layers of thinking about the structure of signals and replaces them by an elegant formalism rooted in the theory of Hilbert spaces and orthonormal bases.

A graduate background on Hilbert spaces (L_2 etc.), the Fourier transform and some integration theory (Lebesgue integration) is recommended.

There are very few (easily corrected) typos and the book can be used as a basis for a one quarter course of 4 hours each week during 7 weeks (or a semester of 2 hours), with a backing of handout problems on several aspects of the theory.