Faculty of Science
http://hdl.handle.net/10294/147
2019-05-20T11:22:14ZApproximation and Visualization of Sets Defined by Iterated Function Systems
http://hdl.handle.net/10294/8564
Approximation and Visualization of Sets Defined by Iterated Function Systems
Hepting, Daryl H.
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.
1991-03-01T00:00:00ZThe Escape Buffer: Efficient Computation of Escape Time for Linear Fractals
http://hdl.handle.net/10294/8412
The Escape Buffer: Efficient Computation of Escape Time for Linear Fractals
Hepting, Daryl; Hart, John
The study of linear fractals has gained a great deal from the study of quadratic fractals, despite important differences. Methods for classifying points in the complement of a fractal shape were originally developed for quadratic fractals, to provide insight into their underlying dynamics. These methods were later modified for use with linear fractals. This paper reconsiders one such classification, called escape time, and presents a new algorithm for its computation that is significantly faster and conceptually simpler. Previous methods worked backwards, by mapping pixels into classified regions, whereas the new forward algorithm uses an "escape buffer" to map classified regions onto pixels. The efficiency of the escape buffer is justified by a careful analysis of its performance on linear fractals with various properties.
1995-05-17T00:00:00ZRendering Methods for Iterated Function Systems
http://hdl.handle.net/10294/8411
Rendering Methods for Iterated Function Systems
Hepting, Daryl; Prusinkiewicz, Przemyslaw; Saupe, Dietmar
This paper describes rendering methods for iterated function systems (IFS’s). The rendering process consists of the generation of a field of data using an IFS and its visualization by means of computer graphics. Two groups of methods are presented: 1. Rendering of the attractor A of an IFS. These attracting methods may visualize the geometry and additionally the invariant measure supported by the attractor. 2. Rendering the complement of the attractor. There are three approaches, namely methods representing Euclidean distance from A; repelling methods, computing the escape time of a point from A, and methods using (electrostatic) potential functions of the attractor. The last of these methods calculates integrals with respect to the invariant measure of the attractor. An algorithm which generates an approximation of such integrals with prescribed tolerance is presented. This provides an alternative to the usual approach based on Elton's ergodic theorem and time average of trajectories generated by the “chaos game", where no error bound is available. Algorithms specifying the details of all methods are presented, some of them in the form of pseudocode. Examples of images obtained using these algorithms are given. The relationship to previously developed methods for visualizing Mandelbrot and Julia sets is also discussed.
1991-12-01T00:00:00ZA Linear Model for Three-Way Analysis of Facial Similarity
http://hdl.handle.net/10294/8407
A Linear Model for Three-Way Analysis of Facial Similarity
Hepting, Daryl H.; Bin Amer, Hadeel Hatim; Yao, Yiyu
Card sorting was used to gather information about facial similarity judgments. A group of raters put a set of facial photos into an unrestricted number of different piles according to each rater’s judgment of similarity. This paper proposes a linear model for 3-way analysis of similarity. An overall rating function is a weighted linear combination of ratings from individual raters. A pair of photos is considered to be similar, dissimilar, or divided, respectively, if the overall rating function is greater than or equal to a certain threshold, is less than or equal to another threshold, or is between the two thresholds. The proposed framework for 3-way analysis of similarity is complementary to studies of similarity based on features of photos.
2018-05-18T00:00:00Z