Constructing reparameterization invariant metrics on spaces of plane curves
Constructing reparameterization invariant metrics on spaces of plane curves
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DOI：
10.1016/j.difgeo.2014.04.008
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发表时间：
2014-06-01
影响因子：
0.5
通讯作者：
Michor, Peter W.
中科院分区：
文献类型：
--
作者：
Bauer, Martin;Bruveris, Martins;Michor, Peter W.
三角形形状空间用于描述从一个形状到另一个形状的变形，并定义形状之间的距离。我们研究了一个家庭的度量空间的曲线，其中包括几个最近提出的度量，度量的特点是映射到向量空间，测地线可以很容易地计算。这个族由参数化平面曲线空间Imm（S-1，R-2）和非参数化曲线商空间Imm（S-1，R-2）/Diff（S-1）上的一阶Sobolev型黎曼度量组成。对于开参数曲线空间，我们得到了测地距离的显式公式，并证明了在参数曲线空间上截面曲率为零，在非参数曲线空间上截面曲率为非负.对于一个特定的度量，我们提供了一个数值算法，计算非参数化，封闭曲线之间的测地线，利用一个约束的制定，实现数值使用RATTLE算法。我们说明了一些形状之间的数值测试的算法。(C)2014爱思唯尔有限公司版权所有。
Metrics on shape spaces are used to describe deformations that take one shape to another, and to define a distance between shapes. We study a family of metrics on the space of curves, which includes several recently proposed metrics, for which the metrics are characterised by mappings into vector spaces where geodesics can be easily computed. This family consists of Sobolev-type Riemannian metrics of order one on the space Imm(S-1, R-2) of parameterized plane curves and the quotient space Imm(S-1,R-2)/Diff (S-1) of unparameterized curves. For the space of open parameterized curves we find an explicit formula for the geodesic distance and show that the sectional curvatures vanish on the space of parameterized open curves and are non-negative on the space of unparameterized open curves. For one particular metric we provide a numerical algorithm that computes geodesics between unparameterized, closed curves, making use of a constrained formulation that is implemented numerically using the RATTLE algorithm. We illustrate the algorithm with some numerical tests between shapes. (C) 2014 Elsevier B.V. All rights reserved.