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

Alice Le Brigant (1, 2)

1 Thales Research and Technology [Palaiseau]
2 IMB - Institut de Mathématiques de Bordeaux

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 L 2-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. Two possibilities are presented to effectively compute the optimal deformation of one curve into another – that is, the geodesic linking two elements of M' – geodesic shooting, and path straightening. The particular case of curves lying in the hyperbolic half-plane H is considered as an example.

Mots clés: Reparametrization invariance; Square Root Velocity Function; Geodesic Shooting; Path straightening

Domaines: Géométrie différentielle [math.DG]

Alice Le Brigant. Computing distances and geodesics between manifold-valued curves in the SRV framework. 2016. ⟨hal-01253495v2⟩