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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 multidimensional 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 topologybased 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.

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 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. Preliminary implementation results of the proposed measure have been reported by computing the distance matrix from the shape retrieval contest data -SHREC 2010 and SHREC 2018. Experimental results and evaluation measures show that the proposed distance measure based on the Reeb spaces has more discriminating power in clustering the shapes as compared to the similar measures between Reeb graphs.

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