  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.
