The Square Root Velocity Framework for Curves in a Homogeneous Space

In this paper we study the shape space of curves with values in a homogeneous space M = G/K, where G is a Lie group and K is a compact Lie subgroup. We generalize the square root velocity framework to obtain a reparametrization invariant metric on the space of curves in M. By identifying curves in M with their horizontal lifts in G, geodesics then can be computed. We can also mod out by reparametrizations and by rigid motions of M. In each of these quotient spaces, we can compute Karcher means, geodesics, and perform principal component analysis. We present numerical examples including the analysis of a set of hurricane paths.