Computing distances and geodesics between manifold-valued curves in the SRV framework

TL;DR

This paper develops a Riemannian framework for computing distances and geodesics between manifold-valued curves using the SRVF, enabling shape analysis and deformation modeling in curved spaces.

Contribution

It introduces a reparametrization invariant metric on the space of manifold-valued curves using SRVF, and derives geodesic equations and exponential maps for this setting.

Findings

Defined a Sobolev metric on the space of curves in a Riemannian manifold

Derived geodesic equations and exponential maps for the proposed metric

Applied the framework to curves in the hyperbolic half-plane for radar signal processing

Abstract

This paper focuses on the study of open curves in a Riemannian manifold M, and proposes a reparametrization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. to define a Riemannian metric on the space of immersions M'=Imm([0,1],M) by pullback of a natural metric on the tangent bundle TM'. This induces a first-order Sobolev metric on M' and leads to a distance which takes into account the distance between the origins in M and the L2-distance between the SRV representations of the curves. The geodesic equations for this metric are given and exploited to define an exponential map on M'. The optimal deformation of one curve into another can then be constructed using geodesic shooting, which requires to characterize the Jacobi fields of M'. The particular case of curves lying in the hyperbolic half-plane is considered as an example, in the setting of radar signal processing.