The k-means algorithm for 3D shapes with an application to apparel design

Abstract

Clustering of objects according to shapes is of key importance in many scientific fields. In this paper we focus on the case where the shape of an object is represented by a configuration matrix of landmarks. It is well known that this shape space has a finite-dimensional Riemannian manifold structure (non-Euclidean) which makes it difficult to work with. Papers about clustering on this space are scarce in the literature. The basic foundation of the (Formula presented.) -means algorithm is the fact that the sample mean is the value that minimizes the Euclidean distance from each point to the centroid of the cluster to which it belongs, so, our idea is integrating the Procrustes type distances and Procrustes mean into the (Formula presented.) -means algorithm to adapt it to the shape analysis context. As far as we know, there have been just two attempts in that way. In this paper we propose to adapt the classical (Formula presented.) -means Lloyd algorithm to the context of Shape Analysis, focusing on the three dimensional case. We present a study comparing its performance with the Hartigan-Wong (Formula presented.) -means algorithm, one that was previously adapted to the field of Statistical Shape Analysis. We demonstrate the better performance of the Lloyd version and, finally, we propose to add a trimmed procedure. We apply both to a 3D database obtained from an anthropometric survey of the Spanish female population conducted in this country in 2006. The algorithms presented in this paper are available in the Anthropometry R package, whose most current version is always available from the Comprehensive R Archive Network.