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