The Square Root Velocity Framework for Curves in a Homogeneous Space

Abstract

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.