The perceptual organization of point constellations

The Perceptual Organization of Point Constellations Matthew J. Dry (matt.dry@psy.kuleuven.be) Department of Psychology, K.U. Leuven, Tiensestraat 102, B-3000 Leuven Daniel J. Navarro (daniel.navarro@adelaide.edu.au) Kym Preiss (kym.preiss@unisa.edu.au) School of Psychology, University of Adelaide, Adelaide, SA, 5005 Australia Michael D. Lee (mdlee@uci.edu) Department of Cognitive Sciences, 3151 Social Sciences Plaza A, University of California, Irvine, CA 92697, USA from the environment. The constellation Cassiopeia is an example, the Dipper is another. For ages people have seen the same groups as units, and at the present time children need no instructions to see the same units” (p. 141-142). As noted by previous researchers (e.g., Compton & Logan, 1993), while there is a large body of experimental and phenomenological evidence suggesting that the Gestalt organizational principles play an important role in early visual perception little effort has been made to develop formal models of the cognitive or perceptual processes underlying these principles. In this paper we outline a formal approach to describing the perception of structure in constellation (and constellation-like) stimuli. The approach is based upon Delaunay triangulation, a powerful measure of relational structure. In the following sections we describe Delaunay triangulation and outline the aims of the paper. Abstract In this paper we present observers with point patterns based on 30 major star constellations and ask them to connect the points to show the structure they perceive. The resulting empirical structures had a high inter-rater reliability and a high degree of overlap with constellation structures recorded in star atlases, suggesting that the perception of structure in point patterns is largely invariant across individuals. Further, we demonstrate that the empirical structures correspond closely with the structures developed in the field of relational geometry. We discuss the results of the experiment in light of previous findings and suggest a number of potential approaches to formally modeling human performance on clustering tasks. Keywords: Gestalt organizational principles; visual perception; relational structure; Delaunay triangulation; perceptual organization; perceptual modeling. Introduction Delaunay Triangulation One of the fundamental tasks of early visual perception is the spatial organization of an image (e.g., Marr, 1982; Ullman, 1984). Researchers from the Gestalt school (e.g., Koffka, 1935; Kohler, 1929; Wertheimer, 1938) demonstrated that the organization of many visual stimuli appeared to be dictated by a number of simple principles such as relative proximity, similarity, good continuation, and common fate. Figure 1a is a replication of one of the examples given by Kohler (1929) as demonstration of organizational structure based on the grouping-by-proximity principle: specifically (all other things being equal) elements with a greater relative proximity tend to be grouped together. Hence, Figure 1a is generally seen as being organized into two groups of three objects rather than one group of two objects and one group of four objects (or any other possible configuration). Kohler suggested that this principle holds equally well for more complex stimuli, citing the constellations in the night sky as an ecologically plausible example. He noted that “If on a clear night we look up at the sky, some stars are immediately seen as belonging together and, as detached Given any group of co-planar points it is possible to obtain a cell structure that defines the regions within the plane that are closer to each point than any other point. This is known as the Voronoi tessellation of the point set, an example of which is shown in Figure 1b. Joining the points in the set that share common Voronoi edges gives the Delaunay triangulation of the set, an example of which can be seen in Figure 1c. An important property of the Delaunay triangulation is that it is a super-graph of a number of relational structures: The Gabriel graph (Gabriel & Sokal, 1969) is the set of the Delaunay edges that intersect with only one Voronoi edge. The relative neighborhood graph (Toussaint, 1980) connects points if no other point is closer to both of them than their inter-point distance. A spanning tree is a structure that connects all of the points in a set with n-1 edges and contains no circuits. The minimum spanning tree is the structure that minimizes the total length of the edges connecting points (Zahn, 1971). Finally, two points are joined as nearest neighbors if one of the points lies closer to the other than to any other point within the set. Examples