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

Brigant Alice Le

doi:10.3934/jgm.2017005

Article Contents

2017, Volume 9, Issue 2: 131-156. Doi: 10.3934/jgm.2017005

This issue Previous Article Preface Next Article The Madelung transform as a momentum map

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

Brigant Alice Le^1,2,

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: December 31, 2015

Revised: October 31, 2016

Published: October 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 58B20, 58D10, 49Q10; Secondary: 62B10.

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Figure 1. Illustration of the distance between two curves $c_0$ and $c_1$ in the space of curves $\mathcal{M}$

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Figure 2. Geodesic shooting in the space of curves $\mathcal M$

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Figure 3. Steps of the first iteration of the geodesic shooting algorithm applied to a pair of geodesics of the upper half-plane $\mathbb H$

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Figure 4. 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

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Figure 5. Geodesics of the hyperbolic half-plane

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Figure 6. 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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