Iterated Function Systems and the Inverse Problem of Fractal Construction Using Moments

Let K be a compact metric space and w _i :K→K, 1≤i≤N, be a set of contraction maps, with assigned probabilities pi. This contractive iterated function system (IFS) possesses a unique and invariant attractor set A. Given a target set S, the inverse problem consists in finding an IFS {K,w,p} whose attractor A approximates S as closely as possible. We examine a numerical method of approximating a (fractal) target set S by minimizing the distance between the moments of S and A. This amounts to a nonlinear optimization of the parameters defining the IFS. In this way, both the geometry and shading measure encoded in S may be simultaneously approximated in a quantified procedure.