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Diffusion Equations over Arbitrary Triangulated Surfaces for Filtering and Texture Applications
May/June 2008 (vol. 14 no. 3)
pp. 666-679
In computer graphics, triangular mesh representations of surfaces have become very popular. Compared with parametric and implicit forms of surfaces, triangular mesh surfaces have many advantages, such as easy to render, convenient to store and the ability to model geometric objects with arbitrary topology. In this paper, we are interested in data processing over triangular mesh surfaces through PDEs (partial differential equations). We study several diffusion equations over triangular mesh surfaces, and present corresponding numerical schemes to solve them. Our methods work for triangular mesh surfaces with arbitrary geometry (the angles of each triangle are arbitrary) and topology (open meshes or closed meshes of arbitrary genus). Besides the flexibility, our methods are efficient due to the implicit/semi-implicit time discretization. We finally apply our methods to several filtering and texture applications such as image processing, texture generating and regularization of harmonic maps over triangular mesh surfaces. The results demonstrate the flexibility and effectiveness of our methods.

[1] “Compatible Spatial Discretizations” The IMA Volumes in Mathematics and Its Applications, vol. 142, D.N. Arnold, P.B. Bochev, R.B.Lehoucq, R.A. Nicolaides, and M. Shashkov, eds., Springer, 2006.
[2] O.K.C. Au, C.L. Tai, L.G. Liu, and H.B. Fu, “Dual Laplacian Editing for Meshes,” IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 3, pp. 386-395, May/June 2006.
[3] J.F. Aujol, G. Aubert, L.B. Feraud, and A. Chambolle, “Image Decomposition into a Bounded Variation Component and an Oscillating Component,” J. Math. Imaging and Vision, vol. 22, no. 1, pp. 71-88, 2005.
[4] J.F. Aujol, G. Gilboa, T.F. Chan, and S. Osher, “Structure-Texture Image Decomposition-Modeling, Algorithms, and Parameter Selection,” Int'l J. Computer Vision, vol. 67, no. 1, pp. 111-136, 2006.
[5] C.L. Bajaj and G. Xu, “Anisotropic Diffusion of Surfaces andFunctions on Surfaces,” ACM Trans. Graphics, vol. 22, no. 1, pp. 4-32, 2003.
[6] C.A.Z. Barcelos and M.A. Batista, “Image Inpainting and Denoisingby Nonlinear Partial Differential Equations,” Proc. 16th Brazilian Symp. Computer Graphics and Image Processing (SIBGRAPI '03), pp. 287-293, 2003.
[7] C.A.Z. Barcelos, M.A. Batista, A.M. Martins, and A.C. Nogueira, “LevelLines Continuation Based Digital Inpainting,” Proc. 17thBrazilian Symp. Computer Graphics and Image Processing (SIBGRAPI'04), pp. 50-57, 2004.
[8] T. Barth and M. Ohlberger, “Finite Volume Methods: Foundation and Analysis,” Encyclopedia of Computational Mechanics. John Wiley & Sons, 2004.
[9] M. Bertalmio, A.L. Bertozzi, and G. Sapiro, “Navier-Stokes, Fluid Dynamics, and Image and Video Inpainting,” Proc. IEEE Conf. Computer Vision and Pattern Recognition (CVPR '01), pp. 355-362, 2001.
[10] M. Bertalmio, L.T. Cheng, S. Osher, and G. Sapiro, “Variational Problems and Partial Differential Equations on Implicit Surfaces,” J. Computational Physics, vol. 174, no. 2, pp. 759-780, 2001.
[11] M. Bertalmio, G. Sapiro, V. Caselles, and C. Ballester, “Image Inpainting,” Proc. ACM SIGGRAPH '00, pp. 417-424, 2000.
[12] A. Bossavit, “Generalized Finite Differences in Computational Electromagnetics,” Progress in Electromagnetics Research, vol. 32, pp. 45-64, 2001.
[13] A. Bossavit, Computational Electromagnetism. Academic Press, 2004.
[14] V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic Active Contours,” Int'l J. Computer Vision, vol. 22, pp. 61-79, 1997.
[15] E. Catmull and J. Clark, “Recursively Generated B-Spline Surfaces on Arbitrary Topological Meshes,” Computer-Aided Design, vol. 10, no. 6, pp. 350-355, 1978.
[16] T.F. Chan, S.H. Kang, and J. Shen, “Total Variation Denoising andEnhancement of Color Images Based on the CB and HSVColor Models,” J. Visual Comm. and Image Representation, vol. 12, pp. 422-435, 2001.
[17] T.F. Chan, S. Osher, and J. Shen, “The Digital TV Filter and Nonlinear Denoising,” IEEE Trans. Image Processing, vol. 10, no. 2, pp. 231-241, 2001.
[18] T.F. Chan and F. Park, “Data Dependent Multiscale Total Variation Based Image Decomposition and Contrast Preserving Denoising,” Technical Report UCLA CAM Report 04-15, UCLACAM, 2004.
[19] T.F. Chan and J. Shen, “Mathematical Models for Local NontextureInpaintings,” SIAM J. Applied Math., vol. 62, no. 3, pp. 1019-1043, 2001.
[20] T.F. Chan, J. Shen, and L. Vese, “Variational PDE Models in Image Processing,” Notice of Am. Math. Soc., vol. 50, pp. 14-26, 2003.
[21] T.F. Chan and L.A. Vese, “An Active Contour Model without Edges,” Lecture Notes in Computer Science, vol. 1682, pp. 141-151, Springer, 1999.
[22] M. Desbrun, A.N. Hirani, M. Leok, and J.E. Marsden, Discrete Exterior Calculus, http://arxiv.org/abs/math.DG0508341, 2005.
[23] M. Desbrun, M. Meyer, P. Schröder, and A.H. Barr, “Implicit Fairing of Irregular Meshes Using Diffusion and Curvature Flow,” Proc. ACM SIGGRAPH, 1999.
[24] Q. Du and L.L. Ju, “Finite Volume Methods on Spheres and Spherical Centroidal Voronoi Meshes,” SIAM J. Numerical Analysis, vol. 43, no. 4, pp. 1673-1692, 2005.
[25] S. Elcott, Y. Tong, E. Kanso, P. Schröder, and M. Desbrun, “Stable, Circulation-Preserving, Simplicial Fluids,” ACM Trans. Graphics, vol. 26, no. 1, 2007.
[26] P.M. Gandoin and O. Devillers, “Progressive Lossless Compression of Arbitrary Simplicial Complexes,” ACM Trans. Graphics, vol. 21, no. 3, pp. 372-379, 2002.
[27] G. Gilboa, N. Sochen, and Y.Y. Zeevi, “A Forward-and-Backward Diffusion Process for Adaptive Image Enhancement and Denoising,” IEEE Trans. Image Processing, vol. 11, no. 7, pp. 689-703, 2002.
[28] A.N. Hirani, “Discrete Exterior Calculus,” PhD dissertation, California Inst. Tech nology, 2003.
[29] A. Hummel, “Representations Based on Zero-Crossings in Scale-Space,” Proc. IEEE Conf. Computer Vision and Pattern Recognition (CVPR '86), pp. 204-209, 1986.
[30] R. Kimmel, “Intrinsic Scale Space for Images on Surfaces: The Geodesic Curvature Flow,” Graphical Models and Image Processing, vol. 59, no. 5, pp. 365-372, 1997.
[31] J. Koenderink, “The Structure of Images,” Biological Cybernetics, vol. 50, pp. 363-370, 1984.
[32] G. Lin and P.Y. Yu, “An Improved Vertex Caching Scheme for 3DMesh Rendering,” IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 4, pp. 640-648, July/Aug. 2006.
[33] M. Meyer, M. Desbrun, P. Schröder, and A. Barr, “Discrete Differential-Geometry Operator for Triangulated 2-Manifolds,” Visualization and Math. III, H.-C. Hege and K. Polthier, eds., Springer, 2002.
[34] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. Springer, 2002.
[35] S. Osher, A. Sole, and L. Vese, “Image Decomposition and Restoration Using Total Variation Minimization and the $H^{-1}$ Norm,” Multiscale Modeling and Simulation, vol. 1, pp. 349-370, 2003.
[36] P. Perona and J. Malik, “Scale-Space and Edge Detection Using Anisotropic Diffusion,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629-639, July 1990.
[37] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C, second ed. Cambrige Univ. Press, 1992.
[38] E. Radmoser, O. Scherzer, and J. Weickert, “Scale-Space Properties of Regularization Methods,” Lecture Notes in Computer Science, vol. 1682, pp. 211-222, Springer, 1999.
[39] J.R. Rommelse, H.X. Lin, and T.F. Chan, “A Robust Level Set Algorithm for Image Segmentation and Its Parallel Implementation,” Technical Report UCLA CAM Report 03-05, UCLA CAM, 2003.
[40] L. Rudin, S. Osher, and E. Fatemi, “Nonlinear Total Variation Based Noise Removal Algorithms,” Physica D, vol. 60, pp. 259-268, 1992.
[41] J. Shen, “Inpainting and the Fundamental Problem of Image Processing,” SIAM News, vol. 36, no. 5, 2003.
[42] L. Shi and Y. Yu, “Inviscid and Incompressible Fluid Simulation on Triangle Meshes,” J. Computer Animation and Virtual Worlds, vol. 15, pp. 173-181, 2004.
[43] O. Sorkine, D. Cohen-Or, R. Goldenthal, and D. Lischinski, “Bounded Distortion Piecewise Mesh Parameterization,” Proc. IEEE Conf. Visualization (VIS '02), pp. 355-362, 2002.
[44] A. Spira and R. Kimmel, “Enhancing Images Painted on Manifolds,” Lecture Notes in Computer Science, vol. 3459, pp. 492-502, Springer, 2005.
[45] A. Spira and R. Kimmel, “Segmentation of Images Painted on Parametric Manifolds,” Proc. European Signal Processing Conf. (EUSIPCO '05), Sept. 2005.
[46] J. Stam, “Flows on Surfaces of Arbitrary Topology,” ACM Trans. Graphics, vol. 22, no. 3, pp. 724-731, 2003.
[47] G. Taubin and J. Rossignac, “Geometric Compression throughTopological Surgery,” ACM Trans. Graphics, vol. 17, no. 2, pp. 84-115, 1998.
[48] A. Turing, “The Chemical Basis of Morphogenesis,” Philosophical Trans. Royal Soc. B, vol. 237, pp. 37-72, 1952.
[49] G. Turk, “Generating Textures on Arbitrary Surfaces Using Reaction-Diffusion,” Computer Graphics, vol. 25, no. 4, pp. 289-298, 1991.
[50] L.A. Vese and T.F. Chan, “A Multiphase Level Set Framework for Image Segmentation Using the Mumford-Shah Model,” Int'l J.Computer Vision, vol. 50, no. 3, pp. 271-293, 2002.
[51] J. Weickert and B. Benhamouda, “A Semidiscrete Nonlinear Scale-Space Theory and Its Relation to the Perona-Malik Paradox,” Advances in Computer Vision. Springer, pp. 1-10, 1997.
[52] A. Witkin and M. Kass, “Reaction-Diffusion Textures,” Computer Graphics (Proc. ACM SIGGRAPH '91), vol. 25, no. 4, pp. 299-308, 1991.
[53] C.L. Wu, J.S. Deng, and F.L. Chen, “Fast Data Extrapolating,” J.Computational and Applied Math., vol. 206, no. 1, pp. 146-157, 2007.
[54] C.L. Wu, J.S. Deng, W.M. Zhu, and F.L. Chen, “Inpainting Images on Implicit Surfaces,” Proc. 13th Pacific Conf. Computer Graphics and Applications (PG '05), pp. 142-144, 2005.
[55] G. Xu, Q. Pan, and C.L. Bajaj, “Discrete Surface Modelling Using Partial Differential Equations,” Computer Aided Geometric Design, vol. 23, no. 2, pp. 125-145, 2006.
[56] Y.Z. Yu, K. Zhou, D. Xu, X.H. Shi, H.J. Bao, B.N. Guo, and H.Y. Shum, “Mesh Editing with Poisson-Based Gradient Field Manipulation,” Proc. ACM SIGGRAPH '04, pp. 641-648, 2004.
[57] E. Zhang, K. Mischaikow, and G. Turk, “Feature-Based Surface Parameterization and Texture Mapping,” ACM Trans. Graphics, vol. 24, no. 1, pp. 1-27, 2005.
[58] H.K. Zhao, S. Osher, B. Merriman, and M. Kang, “Implicit and Nonparametric Shape Reconstruction from Unorganized Points Using a Variational Level Set Method,” Computer Vision and Image Understanding, vol. 80, no. 3, pp. 295-319, 2000.
[59] D. Zorin and P. Schroder, “Subdivision for Modeling and Animation,” ACM SIGGRAPH '00 Course Notes, 2000.

Index Terms:
Computer Graphics, Partial Differential Equations, Image Processing and Computer Vision
Citation:
Chunlin Wu, Jiansong Deng, Falai Chen, "Diffusion Equations over Arbitrary Triangulated Surfaces for Filtering and Texture Applications," IEEE Transactions on Visualization and Computer Graphics, vol. 14, no. 3, pp. 666-679, May-June 2008, doi:10.1109/TVCG.2008.10
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