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Poisson Coordinates.

Xian-Ying Li, Shi-Min Hu

    IEEE Transactions on Visualization and Computer Graphics
    |April 18, 2012
    PubMed
    Summary
    This summary is machine-generated.

    We introduce Poisson coordinates, a fast method for estimating harmonic functions. This novel approach offers lower energy and superior performance compared to Mean Value Coordinates, benefiting computer graphics and geometric computation.

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    Area of Science:

    • Mathematics
    • Computer Graphics
    • Geometric Computation

    Background:

    • Harmonic functions are crucial in computer graphics for image processing and geometric computation.
    • They are critical points of Dirichlet energy and projections of conformal maps.

    Purpose of the Study:

    • To introduce Poisson coordinates, a novel transfinite interpolation scheme for rapid harmonic function estimation.
    • To extend Mean Value Coordinates (MVCs) with enhanced properties.

    Main Methods:

    • Developed Poisson coordinates based on the Poisson integral formula.
    • Provided continuous and 2D discrete formulas for Poisson coordinates.
    • Demonstrated pseudoharmonic properties and compared Dirichlet energies with MVCs.

    Main Results:

    • Poisson coordinates inherit linear precision, smoothness, and kernel positivity from MVCs.
    • Poisson coordinates are proven to be pseudoharmonic, reproducing harmonic functions on n-dimensional balls.
    • Experimental results show lower Dirichlet energies for Poisson coordinates than MVCs on 2D domains.

    Conclusions:

    • Poisson coordinates offer a rapid and efficient method for estimating harmonic functions.
    • The approach provides valuable insights for coordinate-based interpolation and harmonic function estimation.
    • Poisson coordinates represent a significant improvement over MVCs, particularly for convex domains.