Solving the Inverse Problem for Function and Image Approximation Using Iterated Function Systems

This paper is concerned with function approximation and image representation using a new formulation of Iterated Function Systems (IFS) over the general function spaces L p (X;): An N-map IFS with grey level maps (IFSM), to be denoted as (w;), is a set w of N contraction maps w i : X ! X over a compact metric space (X; d) (the \base space") with an associated set of maps i : R ! R. Associated with each IFSM is an operator T which, under certain conditions, may be contractive with unique xed point u 2 L p (X;). A rigorous solution to the following inverse problem is provided: Given a target v 2 L p (X;) and an > 0, nd an IFSM whose attractor satisses k u ? v k p <. An algorithm for the construction of IFSM approximations of arbitary accuracy to a target set in L 2 (X;), where X R D and = m (D) (Lebesgue measure), is also given. The IFSM formulation can easily be generalized to include the \local IFSM" (LIFSM) which considers the actions of contraction maps on subsets of X to produce smaller subsets. Some applications to function approximation on 0,1] and image representation on 0; 1] 2 are presented.