The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space. Recent work has shown that well-chosen (finite) sets of PDs can differentiate between geometric simplicial complexes, providing a method for representing complex shapes using a finite set of descriptors. A related inverse problem is the following: given a set of PDs (or an oracle we can query for persistence diagrams), what is underlying geometric simplicial complex? In this paper, we present an algorithm for reconstructing embedded graphs in ${\mathbb{R}}^d$ (plane graphs in ${\mathbb{R}}^2$) with n vertices from $n^2-n+d+1$ directional (augmented) PDs. Additionally, we empirically validate the correctness and time-complexity of our algorithm in ${\mathbb{R}}^2$ on randomly generated plane graphs using our implementation, and explain the numerical limitations of implementing our algorithm.

持久图（PD）是一种越来越流行的拓扑描述符。通过以不同比例对拓扑特征的大小和突出程度进行编码，PD可以提供有关空间的重要几何和拓扑信息。最近的工作表明，精心挑选的（有限）PD集合可以区分几何简单复形，从而提供了一种使用有限描述符集合表示复杂形状的方法。一个相关的逆问题如下：给定一组PD（或我们可以查询持久性图的Oracle），什么是基本的几何简单复形？在本文中，我们提出了一种用于重建嵌入式图形的算法${\mathbb{[R}}^d$ （中的平面图 ${\mathbb{[R}}^2$）的n个顶点${ñ}^2-ñ+d+\mathrm{1个}$定向（增强）PD。此外，我们通过经验验证了算法在以下方面的正确性和时间复杂性：${\mathbb{[R}}^2$ 使用我们的实现对随机生成的平面图进行分析，并解释了实现我们的算法的数值限制。