Subscribe

Issue No.03 - May/June (2008 vol.14)

pp: 666-679

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TVCG.2008.10

ABSTRACT

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.

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 & Computer Graphics*, vol.14, no. 3, pp. 666-679, May/June 2008, doi:10.1109/TVCG.2008.10REFERENCES

- [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.- [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.- [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.- [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.
- [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.- [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.- [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.- [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.- [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.
- [41] J. Shen, “Inpainting and the Fundamental Problem of Image Processing,”
SIAM News, vol. 36, no. 5, 2003.- [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.- [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.- [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.- [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.- [59] D. Zorin and P. Schroder, “Subdivision for Modeling and Animation,”
ACM SIGGRAPH '00 Course Notes, 2000. |