Archetypal shapes based on landmarks and extension to handle missing data

Abstract

Archetype and archetypoid analysis are extended to shapes. The objective is to find representative shapes. Archetypal shapes are pure (extreme) shapes. We focus on the case where the shape of an object is represented by a configuration matrix of landmarks. As shape space is not a vectorial space, we work in the tangent space, the linearized space about the mean shape. Then, each observation is approximated by a convex combination of actual observations (archetypoids) or archetypes, which are a convex combination of observations in the data set. These tools can contribute to the understanding of shapes, as in the usual multivariate case, since they lie somewhere between clustering and matrix factorization methods. A new simplex visualization tool is also proposed to provide a picture of the archetypal analysis results. We also propose new algorithms for performing archetypal analysis with missing data and its extension to incomplete shapes. A well-known data set is used to illustrate the methodologies developed. The proposed methodology is applied to an apparel design problem in children.