In the previous chapter, we described a procedure for decomposing a signal into its Haas wavelet components of varying frequency (see Theorem 4.12). The Haar wavelet scheme relied on two functions: the Haar scaling function ϕ and the Haar wavelet ψ. Both are simple to describe and lead to an easy decomposition algorithm. The drawback with the Haar decomposition algorithm is that both of these functions are discontinuous (ϕ at x = 0, 1 and ψ at x = 0, 1/2, 1). As a result, the Haar decomposition algorithm provides only crude approximations to a continuously varying signal (as already mentioned, Figure 4.4 does not approximate Figure 4.3 very well). What is needed is a theory similar to what has been described in the past sections but with continuous versions of our building blocks, ϕ and ψ. In this chapter, we present a framework for creating more general ϕ and ψ. The resulting theory, due to Stéphane Mallat (Mallat, 1989, 1998), is called a multiresolution analysis. In the sections that follow, this theory will be used together with a continuous ϕ and ψ (to be constructed later) that will improve the performance of the signal decomposition algorithm with the Haar wavelets described in a previous section.
5.1 THE MULTIRESOLUTION FRAMEWORK
Before we define the notion of a multiresolution analysis, we need to provide some background. Recall that the Sampling Theorem (see Theorem 2.23) approximately reconstructs a signal f from samples taken ...