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    Area of Science:

    • Computer Vision
    • Computational Geometry
    • Applied Mathematics

    Background:

    • Partial difference equations (PDEs) and variational methods are established for Euclidean image processing.
    • 3D sensors necessitate solving PDEs on non-Euclidean domains like surfaces and point clouds.

    Purpose of the Study:

    • To propose a simple method for solving PDEs on surfaces and point clouds.
    • To adapt existing image processing models and algorithms for 3D data using graph-based PDEs.

    Main Methods:

    • Utilizing the framework of partial difference equations on graphs.
    • Transcribing established image processing models onto graph structures.

    Main Results:

    • Demonstrated applicability to p-Laplacian restoration and inpainting.
    • Showcased utility in partial difference equations mathematical morphology.
    • Illustrated effectiveness in active contours segmentation on 3D data.

    Conclusions:

    • The proposed graph-based PDE method effectively extends image processing to 3D surfaces and point clouds.
    • This framework provides a unified approach for diverse computer vision tasks on complex geometric data.