Fitting smooth paths on riemannian manifolds

Abstract

In this paper we formulate a least squares problem on a Riemannian manifold M, in order to generate smoothing spline curves fitting a given data set of points in M, q0, q1, . . . , qN, at given instants of time t0 < t1 < • • • < tN. Using tools from Riemannian geometry, we derive the Euler-Lagrange equations associated to this variational problem and prove that its solutions are Riemannian cubic polynomials defined at each interval [ti, ti+1[, i = 0, . . . ,N −1, and satisfying some smoothing constraints at the knot points ti. The geodesic that best fits the data, arises as a limiting process of the above. When M is replaced by the Euclidean space IRn, the proposed problem has a unique solution which is a natural cubic spline given explicitly in terms of the data. We prove that, in this case, the straight line obtained from the limiting process is precisely the linear regression line associated to the data. Using tools from optimization on Riemannian manifolds we also present a direct procedure to generate geodesics fitting a given data set of time labelled points for the particular cases when M is the Lie group SO(n) and the unitary n−sphere Sn.