ABSTRACT
The triumphant progress wavelets have made in a great variety of applications is based in the first place on the so-called “fast algorithms” (fast wavelet transform, FWT), and these in turn owe their existence to a careful choice of the mother wavelet ψ. So far in this book the particular mother wavelet chosen only had to fulfill some “technical” conditions, such as trψ £ L1 or ψ £ Cr for some r > 0 and, of course, ψ(0 ) = 0 or, even better, ψ should be of a certain order N > 1. The trigonometric basis functions ea : t elat are distinguished by the follow ing linear reproducing property: If such a function is subject to a translation Th , it simply picks up a constant factor:
Contrary to this, in the realm of wavelets the operation of scaling is the central theme, i.e., for arbitrary a £ R* the operation
With respect to this operation, the wavelets considered so far did not behave in a special way (except t/>Haar)· OK, their graph became flattened out or got compressed in the ^-direction, depending on the value of a, but there was no reproduction property in the sense that the scaled version of a φ could be related to the original ψ in some other way. In the discrete case only the integer iterates of a single scaling operation Da , σ > 1 denoting the zoom step, enter the picture. From now until the end of the book we choose σ := 2; by the way, this is also the value most commonly used in practice. If we now adopt a mother wavelet that in a certain way “reproduces itself” when it is subject to the scaling D<i, then novel and highly desirable effects develop. That’s what “multiresolution analysis” is all about. To be more specific, things are arranged in such a way that the mother wavelet ψ satisfies a linear identity having the following structure:
This identity carries in its wake analogous linear formulas between the scalar products (/, ψη,)ζ) and (/, tpn+i,k}, so that these scalar products (called the wavelet coefficients of / ) need not be computed by tedious integrations over and over again when going from one zoom level to the next one. The definitive formulas will look somewhat different, but this is the general idea.
