When the boundary of a familiar object is shown by a series of isolated dots, humans can often recognize the object with ease. This ability can be sustained with addition of distracting dots around the object. However, such capability has not been reproduced algorithmically on computers. We introduce a new algorithm that groups a set of dots into multiple non-disjoint subsets. It connects the dots into a spanning tree using the proximity cue. It then applies the straight polygon transformation to an initial polygon derived from the spanning tree. The straight polygon divides the space into polygons recursively and each polygon can be viewed as grouping of a subset of the dots. The number of polygons generated is O( @math ). We also introduce simple shape selection and recognition algorithms that can be applied to the grouping result. We used both natural and synthetic images to show effectiveness of these algorithms.
Kubovy investigated grouping of dots arranged on a periodic rectilinear lattice and studied stability of the grouping based on proximity @cite_1 . The multi-stability of dot grouping was later modeled with an exponential decay function of the distance between nearby pairs @cite_11 . Good fit of the model advocates proximity based groping. However, in @cite_17 , dots arranged on a curvilinear lattice showed grouping over smooth curves instead of over more proximal straight curves. The study appears to contradict the pure proximity based model and suggests more complex interplay between proximity and curvature cues in our perception. Nevertheless, the patterns used in @cite_17 are rather unnatural as dots are aligned perfectly along parallel curves providing distinct texture and they are viewed through a small aperture without any notion of boundaries. Hence, we think that the proximity cues are more dominant than the curvature ones, and stay focused on the proximity cues alone in our current study.
@cite_1 @cite_17 @cite_11
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Bravais (1850 1949) demonstrated that there are five types of periodic dot patterns (or lattices): oblique, rectangular, centered rectangular, square, and hexagonal. Gestalt psychologists studied grouping by proximity in rectangular and square dot patterns. In the first part of the present paper, I (1) describe the geometry of the five types of lattices, and (2) explain why, for the study of perception, centered rectangular lattices must be divided into two classes (centered rectangular andrhombic). I also show how all lattices can be located in a two-dimensional space. In the second part of the paper, I show how the geometry of these lattices determines their grouping and their multistability. I introduce the notion ofdegree of instability and explain how to order lattices from most stable to least stable (hexagonal). In the third part of the paper, I explore the effect of replacing the dots in a lattice with less symmetric motifs, thus creating wallpaper patterns. When a dot pattern is turned into a wallpaper pattern, its perceptual organization can be altered radically, overcoming grouping by proximity. I conclude the paper with an introduction to the implications of motif selection and placement for the perception of the ensuing patterns. Gestalt phenomena have long resisted quantification In the spirit of Gestalt field theory, we propose a theory that predicts the probability of grouping by proximity in the six kinds of dot lattices (hexagonal, rhombic, square, rectangular, centered rectangular, and oblique) We claim that the unstable perceptual organization of dot lattices is caused by competing forces that attract each dot to other dots in its neighborhood We model the decline of these forces as a function of distance with an exponential decay function This attraction function has one parameter, the attraction constant Simple assumptions allow us to predict the entropy of the perceptual organization of different dot lattices We showed dot lattices tachistoscopically to 7 subjects, and from the probabilities of the perceived organizations, we calculated the entropy of each lattice for each subject The model fit the data exceedingly well The attraction constant did not vary much over subjects Abstract This paper presents a computational approach to extracting basic perceptual structure, or the lowest level grouping in dot patterns. The goal is to extract the perceptual segments of dots that group together because of their relative locations. The dots are interpreted as belonging to the interior or the border of a perceptual segment, or being along a perceived curve, or being isolated. To perform the lowest level grouping, first the geometric structure of the dot pattern is represented in terms of certain geometric properties of the Voronoi neighborhoods of the dots. The grouping is accomplished through independent modules that posses narrow expertise for recognition of typical interior dots, border dots, curve dots, and isolated dots, from the properties of the Voronoi neighborhoods. The results of the modules are allowed to influence and change each other so as to result in perceptual components that satisfy global, Gestalt criteria such as border and curve smoothness and component compactness. Such latera; communication among the modules makes feasible a perceptual interpretation of the local structure in a manner that best meets the global expectations. Thus, an integration is performed of multiple constraints, active at different perceptual levels and having different scopes in the dot pattern, to infer the lowest level perceptual structure. The local interpretations as well as lateral corrections are performed through constraint propagation using a probabilistic relaxation process. The result is a partitioning of the dot pattern into different perceptual segments or tokens. Unlike dots, these segments posses size and shape properties in addition to locations. It is believed that certain contour attributes, specifically orientation, curvature and linear extent, provide essential cues for object (shape) recognition. The present experiment examined this hypothesis by comparing stimulus conditions that differentially provided such cues. A spaced array of dots was used to mark the outside boundary of namable objects, and subsets were chosen that contained either contiguous strings of dots or randomly positioned dots. These subsets were briefly and successively displayed using an MTDC information persistence paradigm. Across the major range of temporal separation of the subsets, it was found that contiguity of boundary dots did not provide more effective shape recognition cues. This is at odds with the concept that encoding and recognition of shapes is predicated on the encoding of contour attributes such as orientation, curvature and linear extent.