基于多维持续图的多场拓扑距离
A Topological Distance between Multi-fields based on Multi-Dimensional Persistence Diagrams
AI总结：
本文提出基于多维Reeb图和持续图计算多场拓扑距离的新方法，证明了其伪度量性质和稳定性，并在形状检索和化学键检测中验证了优于传统Reeb图方法的辨别力。
摘要：
基于Reeb图或轮廓树计算两个标量场之间拓扑距离的问题已被研究，并成功应用于拓扑形状匹配、数据分析和可视化中的各种问题。然而，将此类结果推广到基于其Reeb空间计算两个多场之间的距离测度仍处于起步阶段。为此，在本文中，我们提出了一种技术，通过计算对应于每个（量化）Reeb空间的新型多维持续图（MDPD）来计算两个多场之间的有效距离测度。首先，我们构建一个多维Reeb图（MDRG），它是将Reeb空间层次分解为Reeb图集合的结构。然后基于MDRG的组件Reeb图的持续图计算对应于每个MDRG的MDPD。我们的距离测度将Reeb图的两个持续图之间的Wasserstein距离扩展到MDRG的MDPD。我们证明了所提出的测度是伪度量并满足稳定性性质。所提出距离测度的有效性已在（i）形状检索竞赛数据——SHREC 2010，以及（ii）来自计算化学的Pt-CO键检测数据中得到证明。实验结果表明，与Reeb图之间的类似测度相比，基于Reeb空间的所提出距离测度在聚类形状和检测稳定Pt-CO键形成方面具有更强的辨别力。
英文摘要：
The problem of computing topological distance between two scalar fields based on Reeb graphs or contour trees has been studied and applied successfully to various problems in topological shape matching, data analysis, and visualization. However, generalizing such results for computing distance measures between two multi-fields based on their Reeb spaces is still in its infancy. Towards this, in the current paper we propose a technique to compute an effective distance measure between two multi-fields by computing a novel \emph{multi-dimensional persistence diagram} (MDPD) corresponding to each of the (quantized) Reeb spaces. First, we construct a multi-dimensional Reeb graph (MDRG), which is a hierarchical decomposition of the Reeb space into a collection of Reeb graphs. The MDPD corresponding to each MDRG is then computed based on the persistence diagrams of the component Reeb graphs of the MDRG. Our distance measure extends the Wasserstein distance between two persistence diagrams of Reeb graphs to MDPDs of MDRGs. We prove that the proposed measure is a pseudo-metric and satisfies a stability property. Effectiveness of the proposed distance measure has been demonstrated in (i) shape retrieval contest data - SHREC $2010$ and (ii) Pt-CO bond detection data from computational chemistry. Experimental results show that the proposed distance measure based on the Reeb spaces has more discriminating power in clustering the shapes and detecting the formation of a stable Pt-CO bond as compared to the similar measures between Reeb graphs.
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