Summary
This paper proposes a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures on arbitrary triangle meshes. We present a consistent derivation of these first and second order differential properties using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method, and compare them to existing formulations. Building upon previous work in discrete geometry, these operators are closely related to the continuous case, guaranteeing an appropriate extension from the continuous to the discrete setting: they respect most intrinsic properties of the continuous differential operators. We show that these estimates are optimal in accuracy under mild smoothness conditions, and demonstrate their numerical quality. We also present applications of these operators, such as mesh smoothing, enhancement, and quality checking, and show results of denoising in higher dimensions, such as for tensor images.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. D. Aleksandrov and V. A. Zalgaller. Intrinsic Geometry of Surfaces. AMS, Rhode Island, USA, 1967.
U. Clarenz, U. Diewald, and M. Rumpf. Anisotropie Geometric Diffusion in Surface Processing. In IEEE Visualization, pages 397–405, 2000.
Qiang Du, Vance Faber, and Max Gunzburger. Centroidal Voronoi Tesselations: Applications and Algorithms. SIAM Review, 41 (4): 637–676, 1999.
Ulrich Dierkes, Stefan Hildebrandt, Albrecht Küster, and Ortwin Wohlrab. Minimal Surfaces (I). Springer-Verlag, 1992.
Mathieu Desbrun, Mark Meyer, Peter Schröder, and Alan H. Barr. Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow. In SIG-GRAPH 99 Conference Proceedings, pages 317–324, 1999.
Mathieu Desbrun, Mark Meyer, Peter Schröder, and Alan H. Barr. Anisotropie Feature-Preserving Denoising of Height Fields and Images. In Graphics Interface’2000 Conference Proceedings, pages 145–152, 2000.
G. Dziuk. An Algorithm for Evolutionary Surfaces. Numer. Math., 58, 1991.
J. Fu. Convergence of Curvatures in Secant Approximations. Journal of Differential Geometry, 37: 177–190, 1993.
Michael Garland and Paul S. Heckbert. Surface Simplification Using Quadric Error Metrics. In SIGGRAPH 97 Conference Proceedings, pages 209–216, August 1997.
Alfred Gray. Modern Differential Geometry of Curves and Surfaces with Math-ematica. CRC Press, 1998.
Igor Guskov, Wim Sweldens, and Peter Schröder. Multiresolution Signal Processing for Meshes. In SIGGRAPH 99 Conference Proceedings, pages 325–334, 1999.
Bernd Hamann. Curvature Approximation for Triangulated Surfaces. In G. Farin et al., editor, Geometric Modelling, pages 139–153. Springer Verlag, 1993.
Paul S. Heckbert and Michael Garland. Optimal Triangulation and Quadric-Based Surface Simplification. Journal of Computational Geometry: Theory and Applications, November 1999.
J. M. Hyman and M. Shashkov. Natural Discretizations for the Divergence, Gradient and Curl on Logically Rectangular Grids. Applied Numerical Mathematics, 25: 413–442, 1997.
J. M. Hyman, M. Shashkov, and S. Steinberg. The numerical solution of diffusion problems in strongly heterogenous non-isotropic materials. Journal of Computational Physics, 132:130–148, 1997.
Nelson Max. Weights for Computing Vertex Normals from Facet Normals. Journal of Graphics Tools, 4 (2): 1–6, 1999.
Mark Meyer. Differential Operators for Computer Graphics. Ph.D. Thesis, Caltech, December 2002.
J.M. Morvan. On Generalized Curvatures. Preprint, 2001.
Henry P. Moreton and Carlo H. Séquin. Functional Minimization for Fair Surface Design. In SIGGRAPH 92 Conference Proceedings, pages 167–176, July 1992.
R. Malladi and J.A. Sethian. Irrrage Processing: Flows under Min/Max Curvature and Mean Curvature. Graphical Models and Image Processing 58(2):127141, March 1996.
P. Perona and J. Malik. Scale-space and Edge Detection Using Anisotropic Diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (7): 629–639, July 1990.
Ulrich Pinkall and Konrad Polthier. Computing Discrete Minimal Surfaces and Their Conjugates. Experimental Mathematics, 2(1): 15 36, 1993.
T. Preußer and M. Rumpf. Anisotropic Nonlinear Diffusion in Flow Visualization. In IEEE Visualization, pages 323–332, 1999.
Konrad Polthier and Markus Schmies. Straightest Geodesics on Polyhedral Surfaces. In H.C. Hege and K. Polthier, editors, Mathematical Visualization. Springer Verlag, 1998.
Gabriel Taubin. Estimating the Tensor of Curvature of a Surface from a Polyhedral Approximation. In Proc. 5th Intl. Conf. on Computer Vision (ICCV’95), pages 902–907, June 1995.
B. Thibert and J.M. Morvan. Approximations of A Smooth Surface with a Triangulated Mesh. Preprint, 2002.
Grit Thürmer and Charles Wiithrich. Computing Vertex Normals from Polygonal Facets. Journal of Graphics Tools, 3 (1): 43–46, 1998.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Meyer, M., Desbrun, M., Schröder, P., Barr, A.H. (2003). Discrete Differential-Geometry Operators for Triangulated 2-Manifolds. In: Hege, HC., Polthier, K. (eds) Visualization and Mathematics III. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05105-4_2
Download citation
DOI: https://doi.org/10.1007/978-3-662-05105-4_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05682-6
Online ISBN: 978-3-662-05105-4
eBook Packages: Springer Book Archive