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    Area of Science:

    • Computational topology
    • Data analysis
    • Scientific visualization

    Background:

    • Topological distance computation using Reeb graphs/contour trees is established for scalar fields.
    • Generalizing distance measures for multi-fields using Reeb spaces remains a challenge.

    Purpose of the Study:

    • To propose a novel technique for computing distance measures between multi-fields.
    • To introduce multi-dimensional persistence diagrams (MDPDs) derived from Reeb spaces for this purpose.

    Main Methods:

    • Construction of a multi-dimensional Reeb graph (MDRG) as a hierarchical decomposition of the Reeb space.
    • Computation of MDPDs from the persistence diagrams of component Reeb graphs within the MDRG.
    • Extension of the Wasserstein distance to MDPDs of MDRGs, proving it as a pseudo-metric with stability.

    Main Results:

    • The proposed distance measure demonstrates greater discriminating power compared to Reeb graph-based measures.
    • Effectiveness validated on shape retrieval data (SHREC 2010) and computational chemistry data (Pt-CO bond detection).

    Conclusions:

    • The novel distance measure based on Reeb spaces offers improved capabilities for shape clustering and chemical bond analysis.
    • This work advances the field of topological data analysis for complex multi-field datasets.