Segmentation via Graph-Spectral Methods and Riemannian Geometry

In this paper, we describe the use of graph-spectral techniques and their relationship to Riemannian geometry for the purposes of segmentation and grouping. We pose the problem of segmenting a set of tokens as that of partitioning the set of nodes in a graph whose edge weights are given by the geodesic distances between points in a manifold. To do this, we commence by explaining the relationship between the graph Laplacian, the incidence mapping of the graph and a Gram matrix of scalar products. This treatment permits the recovery of the embedding coordinates in a closed form and opens up the possibility of improving the segmentation results by modifying the metric of the space in which the manifold is defined. With the set of embedding coordinates at hand, we find the partition of the embedding space which maximises both, the inter-cluster distance and the intra-cluster affinity. The utility of the method for purposes of grouping is illustrated on a set of shape silhouettes.