Generalizing the Motzkin-Straus theorem to edge-weighted graphs, with applications to image segmentation

Abstract

The Motzkin-Straus theorem is a remarkable result from graph theory that has recently found various applications in computer vision and pattern recognition. Given an unweighted undirected graph G with adjacency matrix A, it establishes a connection between the local/global solutions of the following quadratic program: maximize cursive Greek chiTA cursive Greek chi/2 subject to eTcursive Greek chi = 1, cursive Greek chi ∈ IR+n where e = (1, ..., 1)T, and the maximal/maximum cliques of G. Given an edge-weighted undirected graph G and the corresponding weight matrix A, in this paper we address the following question: What kind of (combinatorial) structures of G are associated to the (continuous) local solutions of our quadratic program? We show that these structures correspond to a "weighted" generalization of maximal cliques, thereby providing a first step towards an edge-weighted generalization of the Motzkin-Straus theorem. Moreover, we show how these structures can be relevant in clustering as well as image segmentation problems. We present experimental results on real-world images which show the effectiveness of the proposed approach. © Springer-Verlag Berlin Heidelberg 2003.