Item Open AccessApproximation and Visualization of Sets Defined by Iterated Function Systems(University of Regina, 1991-03) 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. Item Open AccessThe Escape Buffer: Efficient Computation of Escape Time for Linear Fractals(Canadian Human Computer Communications Society, 1995-05-17) Hepting, Daryl; Hart, JohnThe 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. Item Open AccessRendering Methods for Iterated Function Systems(North-Holland, 1991-12) Hepting, Daryl; Prusinkiewicz, Przemyslaw; Saupe, DietmarThis 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. Item Open AccessA Linear Model for Three-Way Analysis of Facial Similarity(Springer, Cham, 2018-05-18) Hepting, Daryl H.; Bin Amer, Hadeel Hatim; Yao, YiyuCard 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. Item Open AccessThree-Way Analysis of Facial Similarity Judgments(2017-10-23) Hepting, Daryl H.; Bin Amer, Hadeel Hatim; Yao, YiyuThe card sorting problem involves the similarity judgments of pairs of photos, taken from a set of photos, by a group of participants. Given the lack of an objective standard for judging similarity, different participants may be using different strategies in judging the similarity of photos. It could be very useful to identify and study these strategies. In this paper, we present a framework for three-way analysis of judgments of similarity. Based on judgments by the set of participants, we divide all pairs of photos into three classes: a set of similar pairs that are judged by at least 60% of participants as similar; a set of dissimilar pairs that are judged by at least 60% of participants as dissimilar; and a set of undecidable pairs that have conflicting judgments. A more refined three-way classification method is also suggested based on a quantitative description of the quality of similarity judgments. The classification in terms of three classes provides an effective method to examine the notions of similarity, dissimilarity, and disagreement. Item Open AccessDiscernibility in the Analysis of Binary Card Sort Data(Springer, 2013-10-11) Hepting, DarylIn an open card sorting study of 356 facial photographs, each of 25 participants created an unconstrained number of piles. We consider all 63,190 possible pairs of photos: if both photos are in the same pile for a participant, we consider them as rated similar; otherwise we consider them as rated dissimilar. Each pair of photos is an attribute in an information system where the participants are the objects. We consider whether the attribute values permit accurate classification of the objects according to binary decision classes, without loss of generality. We propose a discernibility coefficient to measure the support of an attribute for classification according to a given decision class pair. We hypothesize that decision class pairs with the support of many attributes are more representative of the data than those with the support of few attributes. We present some computational experiments and discuss opportunities for future work. Item Open AccessOperationalizing Ethics in Food Choice Decisions(Springer, 2014-06) Hepting, Daryl; Jaffe, JoAnn; Maciag, TimothyThere is a large gap between attitude and action when it comes to consumer purchases of ethical food. Amongst the various aspects of this gap, this paper focuses on the difficulty in knowing enough about the various dimensions of food production, distribution and consumption to make an ethical food purchasing decision. There is neither one universal definition of ethical food. We suggest that it is possible to support consumers in operationalizing their own ethics of food with the use of appropriate information and communication technology. We consider eggs as an example because locally produced options are available to many people on every continent. We consider the dimensions upon which food ethics may be constructed, then discuss the information required to assess it and the tools that can support it. We then present an overview of opportunities for design of a new software tool. Finally, we offer some points for discussion and future work.