Reparameterization invariant metric on the space of curves

TL;DR

This paper develops a reparameterization invariant Riemannian metric for open curves in manifolds using the SRVF framework, enabling better analysis of shape and trajectory data in curved spaces.

Contribution

It introduces a new Sobolev-type metric on the space of curves that accounts for origins and SRV representations, generalizing the SRVF framework to manifolds.

Findings

Defines a reparameterization invariant metric on the space of curves

Derives geodesic equations and exponential map for the new metric

Provides a theoretical framework for analyzing curves in general manifolds

Abstract

This paper focuses on the study of open curves in a manifold M, and proposes a reparameterization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [11] to define a reparameterization invariant metric on the space of immersions M' = Imm([0,1], M) by pullback of a metric on the tangent bundle TM' derived from the Sasaki metric. We observe that such a natural choice of Riemannian metric on TM' induces a first-order Sobolev metric on M' with an extra term involving the origins, and leads to a distance which takes into account the distance between the origins and the distance between the SRV representations of the curves. The geodesic equations for this metric are given, as well as an idea of how to compute the exponential map for observed trajectories in applications. This provides a generalized theoretical SRV framework for curves lying in a general manifold M .