论文标题
使用未标记的路径或循环长度进行三重量
Trilateration using Unlabeled Path or Loop Lengths
论文作者
论文摘要
令$ \ mathbf {p} $为$ \ mathbb {r}^d $中的$ n $点的配置，对于某些$ n $，以及一些$ d \ ge 2 $。每对点都定义一个边缘，该边缘在配置中具有欧几里得长度。路径是点的有序序列，循环是在同一点开始并结束的路径。路径或循环作为边序，也具有欧几里得长度，仅是其欧几里得边缘长度的总和。我们有兴趣在给定一组边缘，路径和环长度的情况下重建$ \ Mathbf {p} $。特别是，我们考虑了简单地给出一组实数的未标记的设置，并且没有标记与组合数据的标记，以描述哪些路径或循环产生这些长度。在本文中，我们研究了$ \ mathbf {p} $何时将通过详尽的三重征服过程从某些给定的路径或循环长度来确定（直至不可知的欧几里得变换）。 这种过程已经用于使用未标记的边缘长度进行更简单的重建问题。本文还提供了一个完整的证明，即当给出足够丰富的边缘测量值时，该过程必须在边缘设置中起作用，并假设$ \ mathbf {p} $是通用的。
Let $\mathbf{p}$ be a configuration of $n$ points in $\mathbb{R}^d$ for some $n$ and some $d \ge 2$. Each pair of points defines an edge, which has a Euclideanlength in the configuration. A path is an ordered sequence of the points, and a loop is a path that begins and ends at the same point. A path or loop, as a sequence of edges, also has a Euclidean length, which is simply the sum of its Euclidean edge lengths. We are interested in reconstructing $\mathbf{p}$ given a set of edge, path and loop lengths. In particular, we consider the unlabeled setting where the lengths are given simply as a set of real numbers, and are not labeled with the combinatorial data describing which paths or loops gave rise to these lengths. In this paper, we study the question of when $\mathbf{p}$ will be uniquely determined (up to an unknowable Euclidean transform) from some given set of path or loop lengths through an exhaustive trilateration process. Such a process has already been used for the simpler problem of reconstruction using unlabeled edge lengths. This paper also provides a complete proof that this process must work in that edge-setting when given a sufficiently rich set of edge measurements and assuming that $\mathbf{p}$ is generic.
