Reparameterization invariant metric on the space of curves

Alice Le Brigant (1, 2) , Marc Arnaudon (1) , Frédéric Barbaresco (2)

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

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 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 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 .

Mots clés: trajectories on manifolds; square-root velocity function; shape analysis; reparameterization invariance

Domaines: Géométrie différentielle [math.DG]

Alice Le Brigant, Marc Arnaudon, Frédéric Barbaresco. Reparameterization invariant metric on the space of curves. 2015. ⟨hal-01179508v1⟩