Poisson Coordinates

Xian-Ying Li Tsinghua University Shi-Min Hu Tsinghua University

Geometric Modeling and Processing mathscidoc:1608.16040

IEEE Transactions on Visualization and Computer Graphics, 19, (2), 344, 2013.2
Harmonic functions are the critical points of a Dirichlet energy functional, the linear projections of conformal maps. They play an important role in computer graphics, particularly for gradient-domain image processing and shape-preserving geometric computation. We propose Poisson coordinates, a novel transfinite interpolation scheme based on the Poisson integral formula, as a rapid way to estimate a harmonic function on a certain domain with desired boundary values. Poisson coordinates are an extension of the Mean Value coordinates (MVCs) which inherit their linear precision, smoothness, and kernel positivity. We give explicit formulae for Poisson coordinates in both continuous and 2D discrete forms. Superior to MVCs, Poisson coordinates are proved to be pseudoharmonic (i.e., they reproduce harmonic functions on n-dimensional balls). Our experimental results show that Poisson coordinates have lower Dirichlet energies than MVCs on a number of typical 2D domains (particularly convex domains). As well as presenting a formula, our approach provides useful insights for further studies on coordinates-based interpolation and fast estimation of harmonic functions.
Poisson integral formula, transfinite interpolation, barycentric coordinates, pseudoharmonic
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  title={Poisson Coordinates},
  author={Xian-Ying Li, and Shi-Min Hu},
  booktitle={IEEE Transactions on Visualization and Computer Graphics},
Xian-Ying Li, and Shi-Min Hu. Poisson Coordinates. 2013. Vol. 19. In IEEE Transactions on Visualization and Computer Graphics. pp.344. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160819214913255939309.
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