Poisson Coordinates

Li, Xian-Ying and Hu, Shi-Min ORCID: https://orcid.org/0000-0001-7507-6542 2013. Poisson Coordinates. IEEE Transactions on Visualization and Computer Graphics 19 (2) , pp. 344-352. 10.1109/TVCG.2012.109

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Abstract

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 formulas 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.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Computer Science & Informatics
Subjects:	Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Uncontrolled Keywords:	Poisson integral formula, barycentric coordinates, pseudoharmonic, transfinite interpolation
Publisher:	IEEE
ISSN:	1077-2626
Last Modified:	24 Oct 2022 10:45
URI:	https://orca.cardiff.ac.uk/id/eprint/45769

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