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 ametric 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 terminvolving the origins, and leads to a distancewhich 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.
CITATION STYLE
Brigant, A. L., Arnaudon, M., & Barbaresco, F. (2015). Reparameterization invariant metric on the space of curves. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9389, pp. 140–149). Springer Verlag. https://doi.org/10.1007/978-3-319-25040-3_16
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