American Institute of Mathematical Sciences

June  2017, 9(2): 131-156. doi: 10.3934/jgm.2017005

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

 1 Institut Mathématique de Bordeaux, UMR 5251, Université de Bordeaux and CNRS, France 2 Thales Air Systems, Surface Radar Domain, Technical Directorate, Voie Pierre-Gilles de Gennes, 91470 Limours, France

Received  January 2016 Revised  November 2016 Published  May 2017

This paper focuses on the study of open curves in a Riemannian manifold $M$, and proposes a reparameterization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [29] to define a Riemannian metric on the space of immersions $\mathcal{M}=\text{Imm}([0,1],M)$ by pullback of a natural metric on the tangent bundle $\text{T}\mathcal{M}$. This induces a first-order Sobolev metric on $\mathcal{M}$ 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 $\mathcal M$. The optimal deformation of one curve into another can then be constructed using geodesic shooting, which requires to characterize the Jacobi fields of $\mathcal M$. The particular case of curves lying in the hyperbolic half-plane $\mathbb H$ is considered as an example, in the setting of radar signal processing.

Citation: Alice Le Brigant. Computing distances and geodesics between manifold-valued curves in the SRV framework. Journal of Geometric Mechanics, 2017, 9 (2) : 131-156. doi: 10.3934/jgm.2017005
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References:
Illustration of the distance between two curves $c_0$ and $c_1$ in the space of curves $\mathcal{M}$
Geodesic shooting in the space of curves $\mathcal M$
Steps of the first iteration of the geodesic shooting algorithm applied to a pair of geodesics of the upper half-plane $\mathbb H$
Optimal deformations between pairs of geodesics (in black) of the upper half-plane $\mathbb H$, for our metric (in blue) and for the $L^2$-metric (in green). The orientation of the right-hand curve is inverted in the second image compared to the first, and in the fourth compared to the third
Geodesics of the hyperbolic half-plane
Computation of the mean curve (in black) for 4 sets of 11 curves in the hyperbolic half-plane, constructed from simulated helicopter radar data
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