Faculty of Sciencehttp://hdl.handle.net/10294/1472018-11-09T18:51:43Z2018-11-09T18:51:43ZThe Escape Buffer: Efficient Computation of Escape Time for Linear FractalsHepting, DarylHart, Johnhttp://hdl.handle.net/10294/84122018-10-20T09:00:16Z1995-05-17T00:00:00ZThe 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 SystemsHepting, DarylPrusinkiewicz, PrzemyslawSaupe, Dietmarhttp://hdl.handle.net/10294/84112018-10-16T09:00:15Z1991-12-01T00:00:00ZRendering 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 SimilarityHepting, Daryl H.Bin Amer, Hadeel HatimYao, Yiyuhttp://hdl.handle.net/10294/84072018-10-10T09:00:16Z2018-05-18T00:00:00ZA 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.
2018-05-18T00:00:00ZAn Erdős-Ko-Rado theorem for the derangement graph of \(PGL_3(q)\) acting on the projective planeMeagher, KarenSpiga, Pablohttp://hdl.handle.net/10294/82892018-04-18T21:59:57Z2014-01-01T00:00:00ZAn Erdős-Ko-Rado theorem for the derangement graph of \(PGL_3(q)\) acting on the projective plane
Meagher, Karen; Spiga, Pablo
In this paper we prove an Erdős-Ko-Rado-type theorem for
intersecting sets of permutations. We show that an intersecting set
of maximal size in the projective general linear group \(PGL_3(q)\),
in its natural action on the points of the projective line, is
either a coset of the stabilizer of a point or a coset of the
stabilizer of a line. This gives the first evidence to the veracity
of Conjecture~\(2\) from K.~Meagher, P.~Spiga, An Erdős-Ko-Rado theorem for
the derangement graph of \(\mathrm{PGL}(2,q)\) acting on the
projective line, \( \textit{J. Comb. Theory Series A} \textbf{118} \)
(2011), 532--544.
2014-01-01T00:00:00Z