Untangling Cycles for Contour Grouping

We introduce a novel topological formulation for contour grouping. Our grouping criterion, called untangling cycles, exploits the inherent topological 1D structure of salient contours to extract them from the otherwise 2D image clutter. To define a measure for topological classification robust to clutter and broken edges, we use a graph formulation instead of the standard computational topology. The key insight is that a pronounced 1D contour should have a clear ordering of edgels, to which all graph edges adhere, and no long range entanglements persist. Finding the contour grouping by optimizing these topological criteria is challenging. We introduce a novel concept of circular embedding to encode this combinatorial task. Our solution leads to computing the dominant complex eigenvectors/ eigenvalues of the random walk matrix of the contour grouping graph. We demonstrate major improvements over state-of-the-art approaches on challenging real images.