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    We introduce incomplete gamma kernels, a novel generalization of Locally Optimal Projection (LOP) operators. These kernels enhance point cloud denoising, density estimation, and robust loss functions, offering improved performance in various applications.

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    Area of Science:

    • Computer Vision
    • Geometric Data Processing
    • Machine Learning

    Background:

    • Locally Optimal Projection (LOP) operators are crucial for point cloud processing.
    • Existing LOP methods, like the localized L1 estimator, have limitations in certain frameworks.
    • The Mean Shift framework is a widely used technique in data analysis.

    Purpose of the Study:

    • To generalize Locally Optimal Projection (LOP) operators using incomplete gamma kernels.
    • To establish a connection between the localized L1 estimator and the Mean Shift framework.
    • To explore the properties and applications of the novel incomplete gamma kernel family.

    Main Methods:

    • Development of a novel kernel based on the incomplete gamma function.
    • Generalization to a family of localized Lp estimators.
    • Analysis of kernel properties including distributional and Mean Shift induced aspects.
    • Derivation of theoretical insights into the operator's projection behavior.

    Main Results:

    • A novel kernel generalizing LOP operators and linking L1 estimation to Mean Shift.
    • A family of incomplete gamma kernels representing localized Lp estimators.
    • Demonstrated applications in Weighted LOP (WLOP) density weighting and Continuous LOP (CLOP) kernel approximation.
    • Introduction of robust incomplete gamma losses, encompassing Gaussian and LOP losses.
    • Successful integration of novel kernels as priors in neural networks.

    Conclusions:

    • Incomplete gamma kernels offer a powerful generalization of LOP operators.
    • The novel kernels provide enhanced performance in density estimation, filtering, and robust loss functions.
    • The framework facilitates the development of more accurate and robust algorithms in geometric data processing and machine learning.