Genetic Algorithms for the 1-D Fractal Inverse Problem

We describe the basics of one-dimensional IFS type fractals including their generation, the forward problem, and their encoding, the inverse problem. We also describe the details of a solution of the latter by a Genetic Algorithm. The resultant algorithm converges remarkably well, and, when parallelized as described, achieves a 600 fold speed-up using 12 processors. 1 Fractal Sets Generated by Iterated Function Systems LetW = {w1, w2, . . . , wn} be a finite set of affine maps of the unit interval I = [0, 1] into itself, that is maps of the form w(x) = sx+ a, 0 ≤ x ≤ 1. Here the parameter s is the scale factor and the parameter a is the translation. Alternatively, putting l = a and r = s+ a, then w(x) = l(1− x) + rx. (1.1) In this form the unit interval is seen to map into the interval between l and r and so 0 ≤ l, r ≤ 1. We impose the condition that the image set w(I) be a strict subset of I, i.e. that w be a contraction map and |s| < 1. In this case such a map w has a fixed point in I. Associated with every such collectionW is the attractor A, that is the unique subset A ⊂ I characterized by the selfcovering property that