Perceptual organization via the symmetry map and symmetry transforms

Variations in the projection of objects on a 2D image, e.g., due to occlusion and articulation, lead to edge maps which are noisy, contain gaps and spurious elements, and which are deformed. These variations in turn cause variations in the edge map which are typically regularized by the use of a salient measure for each edge element. The use of edge salience, however, typically faced with two drawback. First, salience measures take advantage of boundary continuity, but not of shape continuity, which includes continuity of the interior. Second, while each edge element can only belong to one object boundary, in the computation of salience measures, it often freely contributes to the salience of edges in competing grouping hypotheses as well. We identify both drawbacks with the lack of an explicit intermediate representation between the edge map and grouped object boundaries. We propose that (i) a symmetry map can fully represent the initial edge map so that both boundary and regional continuities can be represented via skeletal/shock continuity; (ii) a re-organization of the edge map in the form of completing gaps, discarding spurious elements, smoothing, and partitioning a contour (grouped set of edge elements) can be represented by transformations on the symmetry map; (iii) the optimal grouping corresponds to the least action path consisting of a sequence of symmetry transforms. The focus of this paper is to define transformations on the symmetry map and illustrate results for them. Specifically, we illustrate how spurious elements can be removed, gaps completed, and parts computed despite significant noise.