A Topological Distance between Multi-fields based on Multi-Dimensional Persistence Diagrams

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.