Theory of Generalized Fractal Transforms

The most popular \fractal-based" algorithms for both the representation as well as compression of computer images have involved some implementation of the method of Iterated Function Systems (IFS) on complete metric spaces, e.g. IFS with probabilities (IFSP), Iterated Fuzzy Set Systems (IFZS), Fractal Transforms (FT), the Bath Fractal Transform (BFT) and IFS with grey-level maps (IFSM). (FT and BFT are special cases of IFSM.) The \IFS component" of these methods is a set of N contraction maps w = fw 1 ; w 2 \base space" representing the computer screen. Most discussions of these methods, both practical as well as theoretical in nature, assume that the sets w i (X) are nonoverlapping (or at least ignore any overlapping), i.e. that w ?1 i (x) exists for only one value i 2 f1; 2; : : :Ng. As such, given an image function u, its so-called fractal transform (Tu)(x) at any point x 2 X is given by i (u(w ?1 i (x))) (where i : R ! R with appropriate restrictions). In the spirit of our earlier works, we consider the more general overlapping case, when x has more than one preimage, i.e. w ?1 i k (x); 1 < k n(x), and the question of how to combine the n(x) fractal components i k (u(w ?1 i k (x)))) to form a generalized fractal transform (Tu)(x). In this paper, we specify a set of rules for constructing such generalized fractal transforms, among them the condition that T be contractive with respect to the appropriate metric. We also provide a unifying treatment of the various IFS-type methods over function spaces (IFS, IFZS, IFSM) and nally link them with IFSP over the probability measure space M(X). Firstly, we consider \traditional IFS" on H(X) (the set of nonempty compact subsets of X) as a fractal transform method over an appropriate function space. This provides the connection between IFS and IFZS which, in turn, leads to IFSM. The nal link with IFSP on the probability measure space M(X) is provided by a formulation of fractal transforms over D 0 (X), the space of distributions on X. The IFSP and IFSM methods may then be considered as special cases of the distributional fractal transform.