Daubechies’ Orthonormal Wavelet Bases
Gerald Kaiser · Birkhäuser Boston eBooks · 2010
In this chapter we construct finite filter sequences leading to a family of scaling functions øN and wavelets ΨN where N = 1,2 … N = 1 gives the Haar system, and N = 2,3… give multiresolution analyses with orthonormal bases of continuous, compactly supported wavelets of increasing support width and increasing regularity. Each of these systems, first obtained by Daubechies, is optimal in a certain sense. We then examine some ways in which the dilation equations with the above filter sequences determine the scaling functions, both in the frequency domain and in the time domain. In the last section we propose a new algorithm for computing the scaling function in the time domain, based on the statistical concept of cumulants. We show that this method also provides an alternative scheme for the construction of filter sequences.