The Community for Technology Leaders
RSS Icon
Issue No.02 - Feb. (2013 vol.19)
pp: 344-352
Xian-Ying Li , Dept. of Comput. Sci. & Technol., Tsinghua Univ., Beijing, China
Shi-Min Hu , Dept. of Comput. Sci. & Technol., Tsinghua Univ., Beijing, China
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
[1] R. Fattal, D. Lischinski, and M. Werman, "Gradient Domain High Dynamic Range Compression," ACM Trans. Graphics, vol. 21, no. 3, pp. 249-256, 2002.
[2] P. Pérez, M. Gangnet, and A. Blake, "Poisson Image Editing," ACM Trans. Graphics, vol. 22, no. 3, pp. 313-318, 2003.
[3] A. Levin, A. Zomet, S. Peleg, and Y. Weiss, "Seamless Image Stitching in the Gradient Domain," Proc. Eighth European Conf. Computer Vision (ECCV '04), pp. 377-389, 2004.
[4] X.D. Gu and S.T. Yau, Computational Conformal Geometry. Int'l Press, 2008.
[5] O. Weber and C. Gotsman, "Controllable Conformal Maps for Shape Deformation and Interpolation," ACM Trans. Graphics, vol. 29, no. 4, article 78, pp. 1-11, 2010.
[6] M.S. Floater and K. Hormann, "Surface Parameterization: A Tutorial and Survey," Advances in Multiresolution for Geometric Modelling, pp. 157-186, Springer, 2005.
[7] K. Hormann, B. Lévy, and A. Sheffer, "Mesh Parameterization: Theory and Practice," Proc. SIGGRAPH Course Notes, 2007.
[8] Y. Lipman, D. Levin, and D. Cohen-Or, "Green Coordinates," ACM Trans. Graphics, vol. 27, no. 3,article 78, pp. 1-10, 2008.
[9] M. Ben-Chen, O. Weber, and C. Gotsman, "Variational Harmonic Maps for Space Deformation," ACM Trans. Graphics, vol. 28, no. 3,article 34, pp. 1-11, 2009.
[10] S. Dong, S. Kircher, and M. Garland, "Harmonic Functions for Quadrilateral Remeshing of Arbitrary Manifolds," Computer Aided Geometric Design, vol. 22, no. 5, pp. 392-423, July 2005.
[11] L. Gårding, "The Dirichlet Problem," The Math. Intelligencer, vol. 2, no. 1, pp. 43-53, 1976.
[12] D. Hilbert, "Über Das Dirichletsche Prinzip," Math. Annalen, vol. 59, no. 1, pp. 161-186, 1904.
[13] O. Perron, "Eine Neue Behandlung Der Ersten Randwertaufgabe Für ${\Delta } u=0$ ," Math. Zeitschrift, vol. 18, no. 1, pp. 42-54, 1923.
[14] N. Zeev, Conformal Mapping. Dover, 1952.
[15] T. DeRose and M. Meyer, "Harmonic Coordinates," technical report, Pixar Animation Studios, 2006.
[16] P. Joshi, M. Meyer, T. DeRose, B. Green, and T. Sanocki, "Harmonic Coordinates for Character Articulation," ACM Trans. Graphics, vol. 26, no. 3,article 71, pp. 1-9, 2007.
[17] D. Greenspan, "On the Numerical Solution of Dirichlet Problems," The Quarterly J. Mechanics and Applied Math., vol. 12, no. 1, pp. 117-123, 1959.
[18] R.M. Rustamov, "Boundary Element Formulation of Harmonic Coordinates," technical report, Dept. of Math., Purdue Univ., 2008.
[19] M.S. Floater, "Mean Value Coordinates," Computer Aided Geometric Design, vol. 20, no. 1, pp. 19-27, Mar. 2003.
[20] T. Ju, S. Schaefer, and J. Warren, "Mean Value Coordinates for Closed Triangular Meshes," ACM Trans. Graphics, vol. 24, no. 3, pp. 561-566, 2005.
[21] M.S. Floater, G. Kós, and M. Reimers, "Mean Value Coordinates in 3D," Computer Aided Geometric Design, vol. 22, no. 7, pp. 623-631, Oct. 2005.
[22] K. Hormann and M.S. Floater, "Mean Value Coordinates for Arbitrary Planar Polygons," ACM Trans. Graphics, vol. 25, no. 4, pp. 1424-1441, 2006.
[23] C. Dyken and M.S. Floater, "Transfinite Mean Value Interpolation," Computer Aided Geometric Design, vol. 26, no. 1, pp. 117-134, Jan. 2009.
[24] A. Belyaev, "On Transfinite Barycentric Coordinates," Proc. Fourth Eurographics Symp. Geometry Processing (SGP '06), pp. 89-99, 2006.
[25] T. Ju, P. Liepa, and J. Warren, "A General Geometric Construction of Coordinates in a Convex Simplicial Polytope," Computer Aided Geometric Design, vol. 24, no. 3, pp. 161-178, Apr. 2007.
[26] S. Bruvolla and M.S. Floater, "Transfinite Mean Value Interpolation in General Dimension," J. Computational and Applied Math., vol. 233, no. 7, pp. 1631-1639, 2010.
[27] Y. Lipman, J. Kopf, D. Cohen-Or, and D. Levin, "Gpu-Assisted Positive Mean Value Coordinates for Mesh Deformations," Proc. Fifth Eurographics Symp. Geometry Processing (SGP '07), pp. 117-123, 2007.
[28] E. Wachspress, A Rational Finite Element Basis. Academic Press, 1975.
[29] M. Meyer, H. Lee, A. Barr, and M. Desbrun, "Generalized Barycentric Coordinates on Irregular Polygons," J. Graphics Tools, vol. 7, no. 1, pp. 13-22, 2002.
[30] J. Warren, "Barycentric Coordinates for Convex Polytopes," Advances in Computational Math., vol. 6, no. 1, pp. 97-108, 1996.
[31] J. Warren, S. Schaefer, A.N. Hirani, and M. Desbrun, "Barycentric Coordinates for Convex Sets," Advances in Computational Math., vol. 27, no. 3, pp. 319-338, 2007.
[32] S. Schaefer, T. Ju, and J. Warren, "A Unified, Integral Construction for Coordinates over Closed Curves," Computer Aided Geometric Design, vol. 24, nos. 8/9, pp. 481-493, Nov. 2007.
[33] N.H. Christ, R. Friedberg, and T.D. Lee, "Weights of Links and Plaquettes in a Random Lattice," Nuclear Physics B, vol. 240, no. 3, pp. 337-346, 1982.
[34] U. Pinkall and K. Polthier, "Computing Discrete Minimal Surfaces and Their Conjugates," Experimental Math., vol. 2, no. 1, pp. 15-36, 1993.
[35] M. Eck, T.D. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and W. Stuetzle, "Multiresolution Analysis of Arbitrary Meshes," Proc. SIGGRAPH '95, pp. 173-182, 1995.
[36] D. Shepard, "A Two-Dimensional Interpolation Function for Irregularly-Spaced Data," Proc. 23rd ACM Nat'l Conf., pp. 517-524, 1968.
[37] V. Caselles, J.M. Morel, and C. Sbert, "An Axiomatic Approach to Image Interpolation," IEEE Trans. Image Processing, vol. 7, no. 3, pp. 376-386, Mar. 1998.
[38] W.J. Gordon and J.A. Wixom, "Pseudo-Harmonic Interpolation on Convex Domains," SIAM J. Numerical Analysis, vol. 11, no. 5, pp. 909-933, Oct. 1974.
[39] J. Manson, K. Li, and S. Schaefer, "Positive Gordon-Wixom Coordinates," Computer Aided Design, vol. 43, no. 11, pp. 1422-1426, 2011.
[40] K. Hormann and N. Sukumar, "Maximum Entropy Coordinates for Arbitrary Polytopes," Computer Graphics Forum, vol. 27, no. 5, pp. 1513-1520, 2008.
[41] J. Manson and S. Schaefer, "Moving Least Squares Coordinates," Proc. Symp. Geometry Processing, pp. 1517-1524, 2010.
95 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool