Mean size-and-shapes and mean shapes: a geometric point of view

Unlike the means of distributions on a euclidean space, it is not entirely clear how one should define the means of distributions on the size-and-shape or shape spaces of k labelled points in .m since these spaces are all curved. In this paper, we discuss, from a shape-theoretic point of view, some questions which arise in practice while using procrustean methods to define mean size-and-shapes or shapes. We obtain sufficient conditions for such means to be unique and for the corresponding generalized procrustean algorithms to converge to them. These conditions involve the curvature of the size-and-shape or shape spaces and are much less restrictive than asking for the data to be concentrated.