Distance measures play an important role in shape classification and data analysis problems. Topological distances based on Reeb graphs and persistence diagrams have been employed to obtain effective algorithms in shape matching and scalar data analysis. In the current paper, we propose an improved distance measure between two multi-fields by computing a multi-dimensional Reeb graph (MDRG) each of which captures the topology of a multi-field through a hierarchy of Reeb graphs in different dimensions. A hierarchy of persistence diagrams is then constructed by computing a persistence diagram corresponding to each Reeb graph of the MDRG. Based on this representation, we propose a novel distance measure between two MDRGs by extending the bottleneck distance between two Reeb graphs. We show that the proposed measure satisfies the pseudo-metric and stability properties. We examine the effectiveness of the proposed multi-field topology-based measure on two different applications: (1) shape classification and (2) detection of topological features in a time-varying multi-field data. In the shape classification problem, the performance of the proposed measure is compared with the well-known topology-based measures in shape matching. In the second application, we consider a time-varying volumetric multi-field data from the field of computational chemistry where the goal is to detect the site of stable bond formation between Pt and CO molecules. We demonstrate the ability of the proposed distance in classifying each of the sites as occurring before and after the bond stabilization.

翻译：距离度量在形状分类和数据分析问题中扮演着重要角色。基于Reeb图和持续同调图的拓扑距离已被用于形状匹配和标量数据分析中的高效算法。本文通过计算多维Reeb图（MDRG）提出了一种改进的多场间距离度量，其中每个MDRG通过不同维度上的Reeb图层次结构捕捉多场的拓扑特性。随后，通过计算与MDRG中每个Reeb图对应的持续同调图，构建了一个持续同调图的层次结构。基于这一表示，我们通过扩展两个Reeb图之间的瓶颈距离，提出了一种新颖的MDRG间距离度量。我们证明了该度量满足伪度量性和稳定性性质。我们在两个不同应用中检验了所提出的基于多场拓扑度量的有效性：（1）形状分类；（2）时变多场数据中拓扑特征的检测。在形状分类问题中，我们将所提出度量的性能与形状匹配中著名的基于拓扑的度量进行了比较。在第二个应用中，我们考虑了来自计算化学领域的时变体积多场数据，其目标是检测Pt与CO分子之间稳定键形成的位置。我们展示了所提出距离在区分每个位点（键稳定化前后）的能力。