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    This study introduces geometry-aware merge tree comparisons using labeled interleaving distances. This efficient method analyzes time-varying data by preserving topological features and reducing computational complexity.

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    Area of Science:

    • Topological Data Analysis
    • Scientific Visualization
    • Computational Geometry

    Background:

    • Merge trees are topological descriptors for scalar fields, useful for analyzing time-varying data.
    • Existing merge tree comparison methods are computationally expensive due to considering all critical point correspondences.
    • There is a need for efficient and geometry-aware merge tree comparison techniques.

    Purpose of the Study:

    • To develop a geometry-aware merge tree comparison method using labeled interleaving distances.
    • To improve the computational efficiency and practical utility of merge tree comparisons.
    • To enable the integration of geometric information into the analysis of time-varying datasets.

    Main Methods:

    • Decoupling merge tree comparison into a labeling step and a comparison step.
    • Generating correspondences between critical points of merge trees using geometry-aware labeling.
    • Encoding labeled merge trees as matrices for distance computation.

    Main Results:

    • The proposed method is general, computationally efficient, and practically useful.
    • The framework successfully integrates geometric information into the labeling process.
    • Reduced computational complexity compared to traditional methods by avoiding exhaustive correspondence checks.

    Conclusions:

    • Geometry-aware merge tree comparisons effectively detect transitions, clusters, and periodicities in time-varying datasets.
    • The method aids in diagnosing and highlighting topological changes between adjacent data instances.
    • This approach offers a significant advancement in the analysis and visualization of complex, time-varying data.