×

Triangular surface mesh fairing via Gaussian curvature flow. (English) Zbl 1096.65023

Summary: Surface mesh fairing by the mean curvature flow and its various modifications have become a popular topic. However, very few researches have been attempted on using the Gaussian curvature flow in surface fairing. The aim of this paper is to investigate such a problem. We find that Gaussian curvature flow can only be used to smooth convex meshes. Hence, it cannot be used to smooth noisy surface meshes because a noisy surface mesh is not convex. To overcome this difficulty, we design a new diffusion equation whose evolution direction depends on the mean curvature normal and the magnitude is a properly defined function of the Gaussian curvature. Experimental results show that the designed fairing scheme can effectively remove the noise and simultaneously preserve the sharp features, such as corners and edges.

MSC:

65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alvarez, L.; Lions, P.-L.; Morel, J.-M., Image selective smoothing and edge detection by nonlinear diffusion: II, SIAM J. Numer. Anal., 29, 3, 845-866 (1992) · Zbl 0766.65117
[2] Andrews, B., Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations, 2, 151-171 (1994) · Zbl 0805.35048
[3] Andrews, B., Motion of hypersurfaces by Gauss curvature, Pacific J. Math., 195, 1-34 (2000) · Zbl 1028.53072
[4] Bajaj, C. B.; Xu, G., Anisotropic diffusion of surface and functions on surfaces, ACM Trans. Graphics, 22, 1, 4-32 (2003)
[5] do Carmo, M. P., Differential Geometry of Curves and Surfaces (1976), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0326.53001
[6] Catte, F.; Lions, P.-L.; Morel, J.-M.; Coll, T., Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29, 1, 182-193 (1992) · Zbl 0746.65091
[7] Chen, C. Y.; Cheng, K. Y.; Liao, H. Y., Fairing of polygon meshes via Bayesian discriminant analysis, J. WSCG, 12, 1-3 (2004)
[8] Chou, K. S., Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Math. Appl., 38, 867-882 (1985) · Zbl 0612.53005
[9] Chow, B., Deforming convex hypersurfaces by the \(n\) th root of the Gaussian curvature, J. Differential Geom., 22, 117-138 (1985) · Zbl 0589.53005
[10] M. Desbrun, M. Meyer, P. Schroder, A.H. Barr, Implicit fairing of irregular meshes using diffusion and curvature flow, ACM SIGGRAPH ’99, pp. 317-324.; M. Desbrun, M. Meyer, P. Schroder, A.H. Barr, Implicit fairing of irregular meshes using diffusion and curvature flow, ACM SIGGRAPH ’99, pp. 317-324.
[11] Firey, W. J., On the shapes of worn stones, Mathematika, 21, 1-11 (1974) · Zbl 0311.52003
[12] I. Guskov, W. Sweldens, P. Schroder, Multiresolution signal processing for meshes, ACM SIGGRAPH’99, pp. 325-334.; I. Guskov, W. Sweldens, P. Schroder, Multiresolution signal processing for meshes, ACM SIGGRAPH’99, pp. 325-334.
[13] Ishii, H.; Mikami, T., Convexified Gauss curvature flow of sets: a stochastic approximation, SIAM J. Math. Anal., 36, 2, 552-579 (2002) · Zbl 1129.53048
[14] Ishii, H.; Mikami, T., A level set approach to the wearing process of a nonconvex stone, Calc. Var. Partial Differential Equations, 19, 1, 53-93 (2004) · Zbl 1114.53060
[15] Kimia, B. B.; Siddiqi, K., Geometric heat equation and nonlinear diffusion of shapes and images, Comput. Vision Image Understanding, 64, 305-322 (1996)
[16] M. Meyer, M. Desbrun, P. Schrder, A.H. Barr, Discrete differential-geometry operators for triangulated 2-manifolds, in: VisMath, 2002.; M. Meyer, M. Desbrun, P. Schrder, A.H. Barr, Discrete differential-geometry operators for triangulated 2-manifolds, in: VisMath, 2002.
[17] Morvan, J. M.; Thibert, B., Smooth surface and triangular mesh: comparison of the area, the normals and the unfolding, (Proceedings of the Seventh ACM Symposium on Solid Modeling and Applications (2002), ACM Press: ACM Press New York), 147-158
[18] Y. Ohtake, A.G. Belyaev, I.A. Bogaevski, Polyhedral surface smoothing with simultaneous mesh regularization, Proceedings of the Geometric Modeling and Processing 2000, GMP2000, pp. 229-237.; Y. Ohtake, A.G. Belyaev, I.A. Bogaevski, Polyhedral surface smoothing with simultaneous mesh regularization, Proceedings of the Geometric Modeling and Processing 2000, GMP2000, pp. 229-237.
[19] Sethian, J. A., Level Set Methods and Fast Marching Methods (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0929.65066
[20] G. Taubin, A signal processing approach to fair surface design, in: R. Cook (Ed.), SIGGRPH 95 Conference Proceedings, Annual Conference Series, ACM SIGGRAPH’95, pp. 351-358.; G. Taubin, A signal processing approach to fair surface design, in: R. Cook (Ed.), SIGGRPH 95 Conference Proceedings, Annual Conference Series, ACM SIGGRAPH’95, pp. 351-358.
[21] Thurmer, G.; Wuthrich, C., Computing vertex normals from polygonal facets, J. Graphics Tools, 3, 1, 43-46 (1998) · Zbl 0927.68103
[22] Vollmer, J.; Mencl, R.; Müller, H., Geometric modeling: improved Laplacian smoothing of noisy surface meshes, Comput. Graphics Forum, 18, 3, 131-138 (1999)
[23] Xu, G., Surface fairing and featuring by mean curvature motions, J. Comput. Appl. Math., 163, 1, 295-309 (2004) · Zbl 1040.65015
[24] S. Yoshizawa, A.G. Belyaev, Fair triangle mesh generation with discrete elastica, GMP’02, pp. 119-123.; S. Yoshizawa, A.G. Belyaev, Fair triangle mesh generation with discrete elastica, GMP’02, pp. 119-123.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.