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To achieve precise distance measurements, especially in surveying and construction, certain corrections must be applied to account for potential sources of error like the standardization errors, temperature variations, and slope adjustments.Standardization error emerges when measurement equipment undergoes changes, such as wear, repairs, or weather impacts. To address this, surveyors compare the equipment’s readings to a standard. This process identifies any deviation that might lead to...
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Computing a Stable Distance on Merge Trees.

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    This study introduces a new distance measure for merge trees, balancing stability and discriminativity. This approach enables effective comparison of scalar fields, even with complex data, and proves useful in shape analysis and fluid dynamics.

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    Area of Science:

    • Computational topology
    • Data analysis
    • Scientific visualization

    Background:

    • Distances on merge trees are crucial for comparing scalar fields.
    • Existing methods often sacrifice stability or discriminativity for computational feasibility.
    • The need for computationally feasible yet robust distance measures persists.

    Purpose of the Study:

    • To develop a novel distance measure for merge trees that preserves both stability and discriminativity.
    • To explore practical applications of this distance measure in scenarios with small merge trees.
    • To provide theoretical justification for using persistence simplification to improve computational efficiency.

    Main Methods:

    • Construction of a new distance metric for merge trees.
    • Theoretical analysis of the distance's stability and discriminativity properties.
    • Proof that persistence simplification bounds the increase in distance.

    Main Results:

    • A distance measure is presented that retains both stability and discriminativity.
    • The method is demonstrated on small merge trees for shape comparison and periodicity detection.
    • A theoretical bound is established for distance increase after persistence simplification.

    Conclusions:

    • The developed distance measure offers a robust way to compare scalar fields using merge trees.
    • The findings suggest that trading stability and discriminativity may not always be necessary.
    • The approach has practical implications for analyzing complex datasets in fields like fluid dynamics and shape analysis.