平方根速度框架中的最优重参数化
Optimal reparametrizations in the square root velocity framework
摘要：
平方根速度框架是形状分析中定义曲线和函数数据之间的距离的方法。识别两条曲线，如果它们通过重新参数化而不同，则会导致非参数化曲线的商空间。在本文中，我们研究分析和拓扑方面的建设，这类绝对连续曲线。我们证明了平方根速度变换是一个同胚，并且重新参数化半群的作用是连续的。我们还表明，给定两个$C^1$ -曲线，存在最佳的reparametrisations实现由它们所代表的非参数化曲线之间的最小距离。此外，我们给出了两个Lipschitz曲线的例子，其中不存在最佳重新参数化对。
英文摘要：
The square root velocity framework is a method in shape analysis to define a distance between curves and functional data. Identifying two curves if they differ by a reparametrisation leads to the quotient space of unparametrised curves. In this paper we study analytical and topological aspects of this construction for the class of absolutely continuous curves. We show that the square root velocity transform is a homeomorphism and that the action of the reparametrisation semigroup is continuous. We also show that given two $C^1$-curves, there exist optimal reparametrisations realising the minimal distance between the unparametrised curves represented by them. Furthermore we give an example of two Lipschitz curves, for which no pair of optimal reparametrisations exists.
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