Reparameterization Invariant Metric on the Space of Curves

Abstract

This paper focuses on the study of open curves in a manifold M, and its aim is to define a reparameterization invariant distance 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 $\mathcal {M}=\text {Imm}([0,1],M)$ by pullback of a metric on the tangent bundle $\text {T}\mathcal {M}$ derived from the Sasaki metric. We observe that such a natural choice of Riemannian metric on $\text {T}\mathcal {M}$ induces a first-order Sobolev metric on $\mathcal {M}$ 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 image curves by the SRVF parallel transported to a same vector space, with an added curvature term. This provides a generalized theoretical SRV framework for curves lying in a general manifold M.