The World According To Wavelets

The World According To Wavelets. Barbara Burke Hubbard. A. K. Peters, Ltd., 1996. 264 pp. $34.

Wavelet theory is based on the idea of multi-scale analysis-a concept that is already familiar to many nonscientists through the popularization of fractals. Wavelets are sets of functions that can be transformed into each other by translation (shifting) and dilation (stretching or squeezing). In less than 10 years, wavelet analysis has had a major impact on such diverse fields as signal processing, where the applications include image compression, edge detection, and separation of noise from signal; numerical analysis, where it has led to more efficient algorithms; the study of turbulence; astronomy, where it has been used to identify new structures in clusters of galaxies; and medical imaging. When the FBI adopts a wavelet-based standard for digital coding of fingerprints, the time is ripe for a semipopular account of the subject.

In wavelet theory, the traditional Fourier-based approach of time-frequency analysis is replaced by one of time-scale analysis. The result is a tool, which has been described as a mathematical microscope, that uses closely spaced fine-scale functions to zoom in on areas of detail while concisely analyzing the rest with a few broad-scale functions. Much of the progress in wavelet theory is due to the development by Stephane Mallat and Yves Meyer of a unifying framework known as multiresolution analysis. Interest then exploded when Ingrid Daubechies succeeded in constructing a set of wavelets in which the basic "mother" wavelet was smooth and so "compactly supported" that it was zero on all but a small interval, yet the different self-similar wavelets in the set were orthogonal, that is they canceled out in a way that greatly simplified many of the associated mathematical formulas. It was as if major advances in optics suddenly made highpowered microscopes widely available.

But wavelets are more than a framework for...