MEAN SIZE-AND-SHAPES AND MEAN SHAPES - A GEOMETRIC POINT-OF-VIEW

Unlike the means of distributions on a euclidean space, it is not enti rely clear how one should define the means of distributions on the siz e-and-shape or shape spaces of k labelled points in R(m), since these spaces are all curved. In this paper, we discuss, from a shape-theoret ic point of view, some questions which arise in practice while using p rocrustean methods to define mean size-and-shapes or shapes. We obtain sufficient conditions for such means to be unique and for the corresp onding generalized procrustean algorithms to converge to them. These c onditions involve the curvature of the size-and-shape or shape spaces and are much less restrictive than asking for the data to be concentra ted.