Skip to main content

Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

  • Conference paper

Part of the Mathematics and Visualization book series (MATHVISUAL)

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.

Keywords

  • Gaussian Curvature
  • Voronoi Cell
  • Triangle Mesh
  • Discrete Operator
  • Voronoi Region

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-662-05105-4_2
  • Chapter length: 23 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   169.00
Price excludes VAT (USA)
  • ISBN: 978-3-662-05105-4
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   219.99
Price excludes VAT (USA)
Hardcover Book
USD   219.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. D. Aleksandrov and V. A. Zalgaller. Intrinsic Geometry of Surfaces. AMS, Rhode Island, USA, 1967.

    MATH  Google Scholar 

  2. U. Clarenz, U. Diewald, and M. Rumpf. Anisotropie Geometric Diffusion in Surface Processing. In IEEE Visualization, pages 397–405, 2000.

    Google Scholar 

  3. Qiang Du, Vance Faber, and Max Gunzburger. Centroidal Voronoi Tesselations: Applications and Algorithms. SIAM Review, 41 (4): 637–676, 1999.

    MathSciNet  MATH  CrossRef  Google Scholar 

  4. Ulrich Dierkes, Stefan Hildebrandt, Albrecht Küster, and Ortwin Wohlrab. Minimal Surfaces (I). Springer-Verlag, 1992.

    Google Scholar 

  5. 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.

    Google Scholar 

  6. 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.

    Google Scholar 

  7. G. Dziuk. An Algorithm for Evolutionary Surfaces. Numer. Math., 58, 1991.

    Google Scholar 

  8. J. Fu. Convergence of Curvatures in Secant Approximations. Journal of Differential Geometry, 37: 177–190, 1993.

    MathSciNet  MATH  Google Scholar 

  9. Michael Garland and Paul S. Heckbert. Surface Simplification Using Quadric Error Metrics. In SIGGRAPH 97 Conference Proceedings, pages 209–216, August 1997.

    CrossRef  Google Scholar 

  10. Alfred Gray. Modern Differential Geometry of Curves and Surfaces with Math-ematica. CRC Press, 1998.

    Google Scholar 

  11. Igor Guskov, Wim Sweldens, and Peter Schröder. Multiresolution Signal Processing for Meshes. In SIGGRAPH 99 Conference Proceedings, pages 325–334, 1999.

    CrossRef  Google Scholar 

  12. Bernd Hamann. Curvature Approximation for Triangulated Surfaces. In G. Farin et al., editor, Geometric Modelling, pages 139–153. Springer Verlag, 1993.

    Google Scholar 

  13. Paul S. Heckbert and Michael Garland. Optimal Triangulation and Quadric-Based Surface Simplification. Journal of Computational Geometry: Theory and Applications, November 1999.

    Google Scholar 

  14. 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.

    Google Scholar 

  15. 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.

    Google Scholar 

  16. Nelson Max. Weights for Computing Vertex Normals from Facet Normals. Journal of Graphics Tools, 4 (2): 1–6, 1999.

    CrossRef  Google Scholar 

  17. Mark Meyer. Differential Operators for Computer Graphics. Ph.D. Thesis, Caltech, December 2002.

    Google Scholar 

  18. J.M. Morvan. On Generalized Curvatures. Preprint, 2001.

    Google Scholar 

  19. Henry P. Moreton and Carlo H. Séquin. Functional Minimization for Fair Surface Design. In SIGGRAPH 92 Conference Proceedings, pages 167–176, July 1992.

    CrossRef  Google Scholar 

  20. 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.

    Google Scholar 

  21. 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.

    CrossRef  Google Scholar 

  22. Ulrich Pinkall and Konrad Polthier. Computing Discrete Minimal Surfaces and Their Conjugates. Experimental Mathematics, 2(1): 15 36, 1993.

    Google Scholar 

  23. T. Preußer and M. Rumpf. Anisotropic Nonlinear Diffusion in Flow Visualization. In IEEE Visualization, pages 323–332, 1999.

    Google Scholar 

  24. Konrad Polthier and Markus Schmies. Straightest Geodesics on Polyhedral Surfaces. In H.C. Hege and K. Polthier, editors, Mathematical Visualization. Springer Verlag, 1998.

    Google Scholar 

  25. 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.

    CrossRef  Google Scholar 

  26. B. Thibert and J.M. Morvan. Approximations of A Smooth Surface with a Triangulated Mesh. Preprint, 2002.

    Google Scholar 

  27. Grit Thürmer and Charles Wiithrich. Computing Vertex Normals from Polygonal Facets. Journal of Graphics Tools, 3 (1): 43–46, 1998.

    MATH  CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints 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