Discrete Wavelet Transforms

Abstract

As we have seen in Section 12.5, the discretization of the CWT leads, among other things, to the theory of frames. For many practical purposes of signal processing, a tight frame is almost as good as an orthonormal basis. Actually, if one stays with the standard wavelets, as we have done so far, one cannot do better, since these wavelets do not generate any orthonormal basis (like the usual coherent states). There are cases, however, in which an orthonormal basis is really required. A typical example is data compression, which is performed (in the simplest case) by removing all wavelet expansion coefficients below a fixed threshhold. In order to not introduce any bias in this operation, the coefficients have to be as decorrelated as possible, and, of course, an orthonormal basis is ideal in this respect.