Comparing Curves in Homogeneous Spaces

TL;DR

This paper generalizes the SRVF framework to curves in homogeneous spaces, enabling efficient shape analysis by defining a distance invariant to reparametrization and rigid motions, with proven existence of optimal reparametrizations.

Contribution

It introduces a generalized SRVF for homogeneous spaces, proving the existence of optimal reparametrizations and demonstrating computational efficiency.

Findings

Existence of optimal reparametrizations under mild conditions

Efficient computation of quotient space distances

Successful numerical demonstrations

Abstract

Of concern is the study of the space of curves in homogeneous spaces. Motivated by applications in shape analysis we identify two curves if they only differ by their parametrization and/or a rigid motion. For curves in Euclidean space the Square-Root-Velocity-Function (SRVF) allows to define and efficiently compute a distance on this infinite dimensional quotient space. In this article we present a generalization of the SRVF to curves in homogeneous spaces. We prove that, under mild conditions on the curves, there always exist optimal reparametrizations realizing the quotient distance and demonstrate the efficiency of our framework in selected numerical examples.