Combinatorial Grouping of Edges using Geometric Consistency in a Lagrangian Framework

The grouping of edges into long meaningful curves is a goal of perceptual grouping. The typical approach involves a first stage of salience computation based on some pairwise affinity measure, followed by a second stage of linking salient edges into long curves. We argue that the two stages cannot be separated because the curve hypothesis each edge is representing must be explicitly represented to avoid "cross-talk" among various groupings. We also advocate reasoning directly with subpixel edges which does not rely on a Euclidean lattice of pre-determined positions, orientations, and curve labels. Specifically, we propose that a discrete combination of edges represent a curve hypothesis and that a first-order curvature variation can effectively measure consistency among edges making up this curve hypothesis. These combinations are constructed by forming viable edge pairs, then triplets, and finally quadruplets. This results in a hierarchical organization of an edge map into grouped quadruplets, triplets, pairs, and singleton edges with associated measures of curvature variation, curvature, tangents, and position, which we expect can be used for linking into longer curve segments as well as for mid-level vision tasks such as tracking and stereo.