Approximation and Visualization of Sets Defined by Iterated Function Systems

Abstract

An iterated function system (IFS) is defined to be a set of contractive affine transformations. When iterated, these transformations define a closed set, called the attractor of an IFS, which has fractal characteristics. Fractals of any sort are currently a topic of great popular appeal, largely due to the exciting images to which they lend themselves. Iterated function systems represent one of the newest sources of fractal images. Research to date has focused on exploiting IFS techniques for the generation of fractals and for use in modelling applications. Both areas of this research are well suited to computer graphics, and this thesis examine the IFS techniques from a computer graphics perspective. As a source of fractals, iterated function systems have some relationship to other methods of fractal generation. In particular, the relationship between IFS attractors and Julia sets will be examined throughout the thesis. Many insights can be gained from the previous work done by Peitgen, Richter and Saupe [32, 33] both in terms of methods for the generation of the fractal sets and methods for their visualization. The differences between the linear transformations which compose an IFS and the quadratic polynomials which define Julia sets are significant, but not moreso than their similarities. This thesis deals with the related questions of approximation and visualization. The method of constructing the approximating set of points is dependent upon the visualization method in use. Methods have been developed both to visualize the attractor and its complement. The two techniques used to examine the complement set are based on the distance and escape-time functions. The modelling power of standard IFS techniques is limited in that they cannot be used to model any object which is not strictly self-affine. To combat this, methods for controlling transformation application are examined which allow objects without strict self-affinity to be modelled. As part of this research, an extensible software system was developed to allow experimentation with the various concepts discussed. A description of that system is included in Chapter 6.