A relaxed approach for curve matching with elastic metrics
A relaxed approach for curve matching with elastic metrics
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DOI：
10.1051/cocv/2018053
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发表时间：
2019-11-27
影响因子：
1.4
通讯作者：
Moller-Andersen, Jakob
中科院分区：
文献类型：
--
作者：
Bauer, Martin;Bruveris, Martins;Moller-Andersen, Jakob
本文研究了非参数曲线空间上的一类黎曼度量，并给出了一种在给定边界条件下计算测地线的方法。它以几种重要的方式扩展了以前关于这一主题的工作。该模型和由此产生的匹配算法集成在一个共同的设置家庭的H-2度量与常系数和尺度不变的H-2度量开放和封闭的沉浸曲线。这些家庭包括作为特殊情况下的类的一阶弹性度量。与以前的方法的一个本质区别是边界约束的处理方式。通过利用基于变量的相似性度量，我们提出了一个宽松的变分公式的匹配问题，避免了优化的重新参数化组的必要性。此外，我们表明，我们也可以商出有限维的相似性组，如平移，旋转和缩放组。不同的属性和优势，说明通过数值例子中，我们还提供了一个比较与相关的几何形状配准方法。
In this paper, we study a class of Riemannian metrics on the space of unparametrized curves and develop a method to compute geodesics with given boundary conditions. It extends previous works on this topic in several important ways. The model and resulting matching algorithm integrate within one common setting both the family of H-2-metrics with constant coefficients and scale-invariant H-2-metrics on both open and closed immersed curves. These families include as particular cases the class of first-order elastic metrics. An essential difference with prior approaches is the way that boundary constraints are dealt with. By leveraging varifold-based similarity metrics we propose a relaxed variational formulation for the matching problem that avoids the necessity of optimizing over the reparametrization group. Furthermore, we show that we can also quotient out finite-dimensional similarity groups such as translation, rotation and scaling groups. The different properties and advantages are illustrated through numerical examples in which we also provide a comparison with related diffeomorphic methods used in shape registration.