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.

翻译：基于Reeb图或等高线的标量场拓扑距离计算问题已被研究，并成功应用于拓扑形状匹配、数据分析和可视化等多个领域。然而，将此类结果推广到基于Reeb空间的多域拓扑距离度量仍处于初级阶段。为此，本文提出一种通过计算对应各（量化）Reeb空间的新型多维持久性图（MDPD），从而有效计算两个多域间距离度量的技术。首先，我们构建多维Reeb图（MDRG），将Reeb空间层次分解为一系列Reeb图。随后，基于MDRG各分量Reeb图的持久性图计算对应各MDRG的MDPD。本文所提距离度量将Reeb图持久性图之间的Wasserstein距离扩展至MDRG的MDPD。我们证明该度量满足伪度量性质及稳定性。该距离度量的有效性已在以下任务中得到验证：（i）形状检索竞赛数据集SHREC 2010；（ii）计算化学中的Pt-CO键检测数据。实验结果表明，基于Reeb空间的所提距离度量，相较于Reeb图间的类似度量，在形状聚类及稳定Pt-CO键形成检测方面具有更强的区分能力。