Wavelets

Abstract

Wavelet analysis is a particular time-scale or space-scale representation of signals that has become popular in physics, mathematics, and engineering in the last few years. The genesis of the method is interesting for the present book, so we will spend a paragraph outlining it. After the empirical discovery by Jean Morlet (who was analyzing microseismic data in the context of oil exploration [152]), it was recognized from the very beginning by Grossmann, Morlet, and Paul [156]–[160] that wavelets are simply coherent states associated to the affine group of the line (dilations and translations). Thus, immediately the stage was set for a far-reaching generalization, using the formalism developed in Chapter 8 (it is revealing to note that two out of those three authors are mathematical physicists). But then the wind changed. Meyer [217] and Mallat [212] made the crucial discovery that orthonormal bases of regular wavelets could be built, and even with compact support, as shown by Daubechies [99], by changing the perspective (of course, the orthonormal basis of the Haar wavelets was known since the beginning of the century, but these are piecewise constant, discontinuous functions). Group theory lost its priority in favor of the so-called multires-olution analysis (more about this in Section 13.1.1), which made contact with the world of signal processing and engineering. The theory then really caught the attention of practitioners, and it started to grow explosively.