On the inverse fractal problem for two–dimensional attractors

Abstract

We give a theoretical, geometric solution of the inverse fractal problem for a large class of two-dimensional attractors which we call polyhulled disjoint (PHD) attractors. These are attractors of iterated function systems (IFS) with affine maps and range over a wide spectrum of shapes, both abstract and natural. Encoding the objects of a two-dimensional image in terms of their IFS results in enormous data compression. Given a PHD attractor we present an algorithm to find its IFS code by performing a geometric analysis based on the extreme points theorem. We introduce an edge detection device, the springbar function, to extract the affine orbits entangled inside the attractor and then use the elementary properties of these orbits to determine the mappings which generated them. Our solution is amenable to numerical implementation.