Persistent homology is a tool that can be employed to summarize the shape of data by quantifying homological features. When the data is an object in $\mathbb{R}^d$, the (augmented) persistent homology transform ((A)PHT) is a family of persistence diagrams, parameterized by directions in the ambient space. A recent advance in understanding the PHT used the framework of reconstruction in order to find finite a set of directions to faithfully represent the shape, a result that is of both theoretical and practical interest. In this paper, we improve upon this result and present an improved algorithm for graph -- and, more generally one-skeleton -- reconstruction. The improvement comes in reconstructing the edges, where we use a radial binary (multi-)search. The binary search employed takes advantage of the fact that the edges can be ordered radially with respect to a reference plane, a feature unique to graphs.

翻译：持久性同质学是用来通过量化同质特征来总结数据形状的工具。 当数据是以$$\mathb{R ⁇ d$为对象时, (强化的) 持久性同质变换(((A)PHT) 是一个由持久性图解组成的组合, 以环境空间的方向为参数。 最近在理解 PHT 时, 利用了重建框架来找到一组有限的方向, 以忠实地代表形状。 这个结果既有理论意义也有实际意义。 在本文中, 我们改进了这个结果, 并提出了一个改进的图表算法 -- -- 更一般地说来是一skeleton -- 重建。 在重建边缘的过程中, 我们使用一个半径双向( 多重) 搜索。 所使用的二进式搜索利用了以下事实, 即边缘可以对引用的平面进行直线排列, 这是图表独有的特征 。