Overview

Wavelets are everywhere nowadays. Be it in signal or image processing, in astronomy, in fluid dynamics (turbulence), in condensed matter physics, wavelets have found applications in almost every corner of physics. In addition, wavelet methods have become standard in applied mathematics, numerical analysis, approximation theory, etc. It is hardly possible to attend a conference on any of these fields without encountering several contributions dealing with them. Correspondingly, hundreds of papers appear every year and new books on the topic get published at a sustained pace, with publishers strongly competing with each other. So, why bother to publish an additional one?

The answer lies in the finer distinction between various types of wavelet transforms. There is, indeed, a crucial difference between two approaches, namely, the continuous wavelet transform (CWT) and the discrete wavelet transform (DWT). Furthermore, one has to distinguish between problems in one dimension (signal analysis) and problems in two dimensions (image processing), since the status of the literature is very different in the two cases.

Take first the one-dimensional case. Beginning with the classic textbook of Ingrid Daubechies [Dau92], several books, such as those of M. Holschneider [Hol95], B. Torr sani [Tor95] or A. Arn odo et al. [Arn95], cover the continuous wavelet transform, in a more or less mathematically oriented approach. On the other hand, the discrete wavelet transform is treated in many textbooks, more in the signal processing style, such as M. V. Wickerhauser [Wic94], M. Vetterli and J. Kova?evi? [Vet95], P. Wojtaszczyk [Woj97], or...