6. Orthonormal Bases of Compactly Supported Wavelets
Society for Industrial and Applied Mathematics eBooks · 1992
Except for the Haar basis, all the examples of orthonormal wavelet bases in the previous chapter consisted of infinitely supported functions, as a result of the orthogonalization trick (5.3.3). To construct orthonormal examples in which ψ is compactly supported, it pays to start from (or, equivalently, from the subband filtering scheme—see §5.6) rather than from ϕ or the . In §6.1 we show how to construct so that (5.1.20) is satisfied as well as (5.5.5) for some (a necessary condition to have some regularity for ψ). Not every such is associated to an orthonormal wavelet basis, however, an issue addressed in §§6.2 and 6.3. The main results of these two sections are summarized in Theorem 6.3.6, at the end of §6.3. Section 6.4 contains examples of compactly supported wavelets generating orthonormal bases. The orthonormal wavelet bases thus obtained cannot, in general, be written in a closed analytic form. Their graph can be computed with arbitrarily high precision, via an algorithm that I call the “cascade algorithm,” which is in fact a “refinement scheme” as used in computer aided design. All this is discussed in §6.5. A lot of this material goes back to Daubechies (1988b); for many of the results, better, simpler, or more general proofs have been found since, and I have given preference to these new ways of looking at things. These different approaches are borrowed mainly from Mallat (1989), Cohen (1990), Lawton (1990, 1991), Meyer (1990), and Cohen, Daubechies, and Feauveau (1992); for the link with refinement equations the references are Cavaretta, Dahmen, and Micchelli (1991) and Dyn and Levin (1990), as well as earlier papers by these authors (see §6.5).