论文标题
有界变化曲线的平方根速度框架
A square root velocity framework for curves of bounded variation
论文作者
论文摘要
平方根速度变换是有效计算曲线之间距离的强大工具。同样，在考虑重新测量后，它定义了形状之间的距离，这仅取决于其内在的几何形状，而不是具体的参数化。尽管最初是为平滑曲线配制的，但平方根速度变换和所得形状距离已通过使用轻松的重新构度概念进行了彻底分析，以确定绝对连续曲线的设置。在本文中，我们将更远地将平方根速度距离推广到一类不连续曲线。我们将提供一个明确的公式，以将该距离的自然扩展到有界变化的曲线，并分析未参数曲线空间上所得的商距离。特别是，我们将讨论实现商空间上最小距离的最佳修复的存在。
The square root velocity transform is a powerful tool for the efficient computation of distances between curves. Also, after factoring out reparametrisations, it defines a distance between shapes that only depends on their intrinsic geometry but not the concrete parametrisation. Though originally formulated for smooth curves, the square root velocity transform and the resulting shape distance have been thoroughly analysed for the setting of absolutely continuous curves using a relaxed notion of reparametrisations. In this paper, we will generalise the square root velocity distance even further to a class of discontinuous curves. We will provide an explicit formula for the natural extension of this distance to curves of bounded variation and analyse the resulting quotient distance on the space of unparametrised curves. In particular, we will discuss the existence of optimal reparametrisations for which the minimal distance on the quotient space is realised.
