Locating Fréchet means with application to shape spaces

Abstract

We use Jacobi field arguments and the contraction mapping theorem to locate Fréchet means of a class of probability measures on locally symmetric Riemannian manifolds with non-negative sectional curvatures. This leads, in particular, to a method for estimating Fréchet mean shapes, with respect to the distance function 𝜌 determined by the induced Riemannian metric, of a class of probability measures on Kendall's shape spaces. We then combine this with the technique of `horizontally lifting' to the pre-shape spheres to obtain an algorithm for finding Fréchet mean shapes, with respect to 𝜌, of a class of probability measures on Kendall's shape spaces in terms of the vertices of random shapes. This gives us, for example, an algorithm for finding Fréchet mean shapes of samples of configurations on the plane which is expressed directly in terms of the vertices.