Issue No.02 - Feb. (2013 vol.19)
Xian-Ying Li , Dept. of Comput. Sci. & Technol., Tsinghua Univ., Beijing, China
Shi-Min Hu , Dept. of Comput. Sci. & Technol., Tsinghua Univ., Beijing, China
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TVCG.2012.109
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.
stochastic processes, computational geometry, computer graphics, gradient methods, interpolation, coordinates-based interpolation, Poisson coordinates, harmonic functions, Dirichlet energy functional, linear projections, conformal maps, computer graphics, gradient-domain image processing, shape-preserving geometric computation, transfinite interpolation scheme, Poisson integral formula, mean value coordinates, MVC, 2D discrete forms, Dirichlet energies, Interpolation, Harmonic analysis, Kernel, Equations, Integral equations, Closed-form solutions, Image processing, pseudoharmonic, Poisson integral formula, transfinite interpolation, barycentric coordinates
Xian-Ying Li, Shi-Min Hu, "Poisson Coordinates", IEEE Transactions on Visualization & Computer Graphics, vol.19, no. 2, pp. 344-352, Feb. 2013, doi:10.1109/TVCG.2012.109