Iterated Function Systems

Abstract

In the early 1980's, Michael Barnsley introduced the idea of synthesizing a predetermined image as the attractor of a chaotic process. Other researchers had previously shown that chaotic systems were capable of producing fascinating images known as strange attractors. Barnsley, however, was the first to take a step toward solving the inverse problem: Given a specified image, can one come up with a chaotic system that has the given image as its strange attractor? Barnsley used a particular system of mappings which he called an iterated function system (IFS). IFS's are, at best, only a crude form of image compression. It should be stressed that IFS's in their original form, and as they are presented in this chapter, are not the basis of current approaches to fractal image compression (a misconception held by some of the detractors of fractal image compression). However, IFS's were the inspiration for fractal approaches to image compression. And while IFS's are not viable themselves as complete image compression systems, an understanding of IFS theory is essential to understanding how fractal image compression works. In this chapter, we will develop the mathematical background of IFS theory and see how to implement such a system on a computer.