Digital Compression And Iterated Function Systems

Abstract

Images that contain a high degree of complexity, such as natural landscapes, are difficult to compress well because of the large amount of information they contain. But, in some cases, this intricacy can be described by a simple set of rules. Such is the case with Fractal sets. Fractals with their properties of roughness and self-similarity, offer the best geometry for modeling certain highly-detailed images. A simple set of equations applied iteratively to themselves can generate a complicated digital image. Iterated Function Systems (IFS) [2] offers a method of describing complicated digital files with a small set of functions exhibiting fractal properties. The image to be coded with an IFS is first covered with affine transformations of itself. The coding is accomplished by saving the coefficients of the transformations. The decoding is performed by generating a dynamical system whose attractor is suitably close to the original image. The amount of distortion is dependent on the quality of the initial covering. This paper will describe the mathematics of IFS, the coding and decoding of a digital image with IFS, error analysis of IFS compression, and comparison to other compression techniques.