On the computation of the Karcher mean on spheres and special orthogonal groups

Abstract

This paper is concerned with computation of the Karcher mean on the unit sphere Sn and the special orthogonal group SO(n). The Karcher mean, or the Riemannian centre of mass, is defined as the point minimising the sum of the squared distances from that point to each of the given points. By its definition, the mean always belongs to the same space as the given points, however, it may not be unique. Motivated by applications in control, vision and robotics, this paper studies the numerical computation of the Karcher mean. We propose simpler and computationally more efficient gradient-like and Newton-like algorithms. We give explicit forms of these algorithms and show that if the set of points lie within a particular open ball, the algorithms are guaranteed to converge to the Karcher mean.