The Haar Wavelet

Abstract

Introduction In the previous chapters, we have developed the general ideas necessary to be able to speak with precision about wavelets and wavelet series. The key concepts were those of Hilbert spaces and of orthonormal bases and Fourier series in Hilbert spaces, and we have seen that Haar wavelet series are examples of such series. But although we have developed a substantial body of relevant background material, we have so far developed no general theory of wavelets. Indeed, the simple and rather primitive Haar wavelet is the only wavelet we have encountered, and we have not even hinted at ways of constructing other wavelets, perhaps with ‘better’ properties than the Haar wavelet, or at the desirability of being able to do so. These are the issues that we turn to in the final chapters of the book. Specifically, our major remaining goals are as follows: • to develop a general framework, called a multiresolution analysis , for the construction of wavelets; • to develop within this framework as much of the general theory of wavelets as our available methods allow; and • to use that theory to show how other wavelets can be constructed with certain specified properties, culminating in the construction of the infinite family of wavelets known as the Daubechies wavelets , of which the Haar wavelet is just the first member. The main aim of the present short chapter is to introduce the idea of a multiresolution analysis by defining and studying the multiresolution analysis that corresponds to the Haar wavelet. The Haar Wavelet Multiresolution Analysis The concept of a multiresolution analysis was developed in about 1986 by the French mathematicians Stephane Mallat and Yves Meyer, providing a framework for the systematic creation of wavelets, a role which it still retains. Mirroring all of our discussion of wavelets up to this point, we will introduce the idea of a multiresolution analysis by first examining it carefully in the specific case of the Haar wavelet. As we have found in our previous discussion, the Haar wavelet is simple enough for us to see in a very concrete fashion how the various constructions and arguments work.