Computing distances and geodesics between manifold-valued curves in the SRV framework
Computing distances and geodesics between manifold-valued curves in the SRV framework
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在 SRV 框架中计算流形值曲线之间的距离和测地线
DOI：
10.3934/jgm.2017005
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Alice Le Brigant
中科院分区：
文献类型：
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作者：
Alice Le Brigant
本文主要研究黎曼流形M中的开曲线，并在这类路径空间上提出了一种重参数不变度量。我们使用Sriastava等人提出的平方根速度函数(SRVF)。通过拉回切丛TM‘上的自然度量来定义浸入空间M’=imm([0，1]，M)上的黎曼度量。这引入了M‘上的一阶Soblev度量，并得到了一个距离，该距离考虑了M中原点之间的距离和曲线的SRV表示之间的L2距离。给出了该度量的测地方程，并利用该方程定义了M‘上的指数映射。一条曲线到另一条曲线的最优变形可以用测地线打靶法来构造，这需要刻画M‘的Jacobi场。在雷达信号处理的背景下，以双曲半平面上曲线的特殊情况为例。
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 L2-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'. The optimal deformation of one curve into another can then be constructed using geodesic shooting, which requires to characterize the Jacobi fields of M'. The particular case of curves lying in the hyperbolic half-plane is considered as an example, in the setting of radar signal processing.