Mathematical Background

Hutchinson [36] introduced the theory of iterated function systems (a term coined by M. Barnsley) to model collections of contractive transformations in a metric space as dynamical systems. His idea was to use the Contractive Mapping Fixed-Point Theorem to show the existence and uniqueness of fractal sets that arise as fixed points of such systems. It was Barnsley’s observation, however, that led to the idea of using iterated function systems (IFS’s) to encode images. He noted that many fractals that can be very compactly specified by iterated function systems have a “natural” appearance. Given an IFS, it is easy to generate the fractal that it defines, but Barnsley posed the opposite question: given an image, is it possible to find an IFS that defines it?