Graph based k-means clustering

Abstract

An original approach to cluster multi-component data sets is proposed that includes an estimation of the number of clusters. Using Prim's algorithm to construct a minimal spanning tree (MST) we show that, under the assumption that the vertices are approximately distributed according to a spatial homogeneous Poisson process, the number of clusters can be accurately estimated by thresholding the sequence of edge lengths added to the MST by Prim's algorithm. This sequence, called the Prim trajectory, contains sufficient information to determine both the number of clusters and the approximate locations of the cluster centroids. The estimated number of clusters and cluster centroids are used to initialize the generalized Lloyd algorithm, also known as k-means, which circumvents its well known initialization problems. We evaluate the false positive rate of our cluster detection algorithm, using Poisson approximations in Euclidean spaces. Applications of this method in the multi/hyper-spectral imagery domain to a satellite view of Paris and to an image of Mars are also presented.