Correspondence Between Fractal-Wavelet Transforms and Iterated Function Systems With Grey Level Maps

Abstract

It is well known that the action of a “Fractal Transform” or (Local) Iterated Function System with Grey Level Maps (IFSM) on a function f(x) induces a very simple mapping on its expansion coefficients c ij in the Haar wavelet basis. This is the basis of the “discrete fractal-wavelet transform”: subtrees of the wavelet coefficient tree are scaled and copied to lower subtrees. Such transforms, which we shall also refer to as IFSW—IFS on wavelet coefficients—have been introduced into image processing with other (compactly supported) wavelet basis sets in an attempt to remove the blocking artifacts in the standard IFS block encoding algorithms. Although not as straightforward as in the Haar case, we show that there is a relationship between such wavelet transforms and IFSM. In fact, for most such transforms, there is an equivalent IFSM, which provides a further mathematical basis for their use in image processing. We also present results for the case of periodized wavelets, a common implementation in image processing. Finally, we prove some results on the fractal dimension of the graph of an attractor of IFSM or IFSW operators.