InterJournal Complex Systems, 133
Status: Submitted
Manuscript Number: [133]
Submission Date: 971022
Modeling fractal patterns with Genetic Algorithm solutions to a variant of the Inverse Problem for Iterated Function Systems (IFS)
Author(s): Olympia Lilly Bakalis

Subject(s): CX.00, CX.09, CX.06, CX.16, CX.66, CX.64, CX.07

Category: Report

Abstract:

We investigate the use of Iterated Function Systems (IFS) for modeling 2 dimensional fractal structures by seeking solutions to a variant of the Inverse IFS Problem: Given a fractal pattern we are looking for parameters in 24 dimensions for a small set of contractive affine maps and their associated probabilities which constitute the IFS. Upon iteration the IFS solution produces an attractor with the characteristics which describe the image under consideration. We define the ``Mandelbrot set" for a 3-map IFS family, and demonstrate the complexity of the error hypersurface on a cross section within it. We therefore chose, for an automated search, a Genetic Algorithm (GA). We designed a general objective function, incorporating the specified errors, to accomodate two different classes of IFS attractors, namely ``just touching" and ``minimally overlapping". Solutions obtained with the GA, in the 24 dimensional parameter space, meet the desired specifications which characterize the given pattern to within the prescribed discretization. The solutions are applicable to modeling spatial or temporal fractal structures, which exhibit scale invariance and power laws, such as those encountered during critical phenomena. An IFS model allows us to study the system under consideration which belongs to the same universality class as the model.

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