Overcomplete Wavelet Transform

Abstract

This chapter deals with discrete wavelet transforms that are formed from the general samples of a continuous wavelet transform. Conceptually, there are few constraints on the spacing between sample points throughout the time—scale plane; however, computational consideration is restricted here to an interesting subclass of sampling sets that allow for the fast computation of the forward and inverse transforms.¹ In this case, the freedom of choice of analyzing wavelets remains nearly unrestricted by the implementation. In general, the resulting transform is one that has underlying atoms that are nonorthogonal, and even more important, may be overcomplete. Consequently, such atomic functions have associated redundant (inner product) representations. For these reasons, the term overcomplete wavelet transform (OCWT) is used to describe the transform.