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    This study introduces an efficient algorithm for computing the Reeb space of bivariate functions, enabling practical analysis of complex data. The method offers significant speedups and parameter-free operation for scientific visualization.

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    Area of Science:

    • Computational geometry
    • Scientific visualization
    • Data analysis

    Background:

    • The Reeb space is a fundamental topological structure for analyzing scalar functions.
    • Previous methods for computing bivariate Reeb spaces were computationally expensive and impractical.
    • Univariate Reeb space computation techniques provided a foundation for bivariate generalization.

    Purpose of the Study:

    • To present the first practical, output-sensitive algorithm for computing the Reeb space of bivariate piecewise linear scalar functions.
    • To enable efficient and exact computation of bivariate Reeb spaces on tetrahedral meshes.
    • To demonstrate the utility of Reeb spaces for data segmentation and visualization.

    Main Methods:

    • Generalizing univariate Reeb space algorithms to the bivariate case.
    • Identifying the Jacobi set (bivariate analog of critical points).
    • Segmenting the mesh using Jacobi Fiber Surfaces (bivariate analog of critical contours).
    • Implementing a simplification heuristic for Reeb space coarsening.
    • Developing a parameter-free, output-sensitive algorithm.

    Main Results:

    • Achieved orders of magnitude speedup compared to previous approaches.
    • Enabled the first tractable computation of bivariate Reeb spaces in practice.
    • Demonstrated a parameter-free algorithm, unlike range-based quantization methods.
    • Introduced continuous scatterplot peeling for clutter reduction in scatterplots.
    • Provided a C++ implementation for reproducibility and further development.

    Conclusions:

    • The developed algorithm significantly advances the practical computation and application of bivariate Reeb spaces.
    • The parameter-free nature and efficiency make it suitable for diverse scientific visualization tasks.
    • Reeb space computation facilitates semi-automatic segmentation and improved data analysis through techniques like continuous scatterplot peeling.