Searching topological similarity between a pair of shapes or data is an important problem in data analysis and visualization. The problem of computing similarity measures using scalar topology has been studied extensively and proven useful in shape and data matching. Even though multi-field (or multivariate) topology-based techniques reveal richer topological features, research on computing similarity measures using multi-field topology is still in its infancy. In the current paper, we propose a novel similarity measure between two piecewise-linear multi-fields based on their multi-resolution Reeb spaces - a newly developed data-structure that captures the topology of a multi-field. Overall, our method consists of two steps: (i) building a multi-resolution Reeb space corresponding to each of the multi-fields and (ii) proposing a similarity measure for a list of matching pairs (of nodes), obtained by comparing the multi-resolution Reeb spaces. We demonstrate an application of the proposed similarity measure by detecting the nuclear scission point in a time-varying multi-field data from computational physics.

中文翻译：

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使用多分辨率 Reeb 空间的多场数据之间的拓扑相似性度量

搜索一对形状或数据之间的拓扑相似性是数据分析和可视化中的一个重要问题。使用标量拓扑计算相似性度量的问题已被广泛研究并证明在形状和数据匹配中很有用。尽管基于多场（或多元）拓扑的技术揭示了更丰富的拓扑特征，但使用多场拓扑计算相似性度量的研究仍处于起步阶段。在当前的论文中，我们提出了一种基于多分辨率 Reeb 空间的两个分段线性多场之间的新相似性度量——一种新开发的数据结构，可以捕捉多场的拓扑结构。总的来说，我们的方法包括两个步骤：(i) 构建与每个多域对应的多分辨率 Reeb 空间，以及 (ii) 为匹配对（节点的）列表提出相似性度量，通过比较多分辨率 Reeb 空间获得。我们通过检测来自计算物理学的时变多场数据中的核断裂点来展示所提出的相似性度量的应用。