曲线空间上的重参数化不变度量
Reparameterization invariant metric on the space of curves
摘要：
本文研究了流形M中的开曲线，并给出了这类开曲线空间上的一个重参数化不变度量。本文利用Srivastava等人在[11]中引入的平方根速度函数（SRVF），通过拉回由Sasaki度量导出的切丛TM'上的度量，在浸入空间M' = Imm（[0，1]，M）上定义了一个重参数化不变度量.我们观察到，这样一个自然的选择TM'上的黎曼度量诱导M'上的一阶Sobolev度量与一个额外的条款涉及的起源，并导致一个距离，考虑到原点之间的距离和之间的距离的曲线的SRV表示。给出了该度量的测地线方程，以及在应用中如何计算观测轨迹的指数映射的想法。这为一般流形M中的曲线提供了一个广义的理论SRV框架。
英文摘要：
This paper focuses on the study of open curves in a 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 [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 SRV representations of the curves. The geodesic equations for this metric are given, as well as an idea of how to compute the exponential map for observed trajectories in applications. This provides a generalized theoretical SRV framework for curves lying in a general manifold M .
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