使用无标签路径或环长度的三边测量
Trilateration using Unlabeled Path or Loop Lengths
AI总结：
本文研究了在无标签设置下，如何通过穷举三边测量过程，利用给定的路径或环长度重构点配置，并证明了在边设置下该方法的有效性。
摘要：
设 $\mathbf{p}$ 是 $\mathbb{R}^d$ 中 $n$ 个点的配置，其中 $n$ 为某个值，$d \ge 2$。每对点定义一条边，该边在配置中具有欧几里得长度。路径是点的有序序列，而环是起点和终点为同一点的路径。作为边序列的路径或环也具有欧几里得长度，即其欧几里得边长之和。我们感兴趣的是在给定一组边、路径和环长度的情况下重构 $\mathbf{p}$。特别地，我们考虑无标签设置，即长度仅作为一组实数给出，且未用描述哪些路径或环产生这些长度的组合数据进行标记。本文研究 $\mathbf{p}$ 何时能通过穷举三边测量过程从某组给定的路径或环长度中被唯一确定（至多相差一个不可知的欧几里得变换）。该过程已用于使用无标签边长度进行重构的较简单问题。本文还提供了完整证明，表明在给定足够丰富的边测量集并假设 $\mathbf{p}$ 是通用的时，该过程在边设置中必然有效。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
