Computing the Riemannian center of mass on meshes

Abstract

The Riemannian center of mass (a.k.a. Karcher mean or Fréchet mean) provides the equivalent to the Euclidean affine average on manifolds. In spite of its many potential applications in computer graphics and geometric modeling, there exist surprisingly few algorithms to compute it. We present a direct method for computing the Riemannian center of mass on a triangle mesh. Our method works in the polyhedral metric and uses a piecewise-linear interpolation of gradients of the distance fields from a set of control points. We present applications for tracing splines on a surface, comparing to other methods at the state of the art, and showing that we produce quality results while supporting user interaction.