Curvature Estimation over Smooth Polygonal Meshes Using the Half Tube Formula

  • Ronen Lev
  • Emil Saucan
  • Gershon Elber
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4647)

Abstract

The interest, in recent years, in the geometric processing of polygonal meshes, has spawned a whole range of algorithms to estimate curvature properties over smooth polygonal meshes. Being a discrete approximation of a \(\mathcal{C}^2\) continuous surface, these methods attempt to estimate the curvature properties of the original surface. The best known methods are quite effective in estimating the total or Gaussian curvature but less so in estimating the mean curvature.

In this work, we present a scheme to accurately estimate the mean curvature of smooth polygonal meshes using a one sided tube formula for the volume above the surface. In the presented comparison, the proposed scheme yielded results whose accuracy is amongst the highest compared to similar techniques for estimating the mean curvature.

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References

  1. 1.
    Berger, M.: Geometry II. Universitext. Spinger, Berlin (1987)Google Scholar
  2. 2.
    Berger, M.B., Gostiaux, B.: Differential Geometry: Manifols, Curves and Surfaces. Springer, New York (1987)Google Scholar
  3. 3.
    Bobenko, A.I.: A conformal energy for simplicial surfaces. Combinatorial and Computational Geometry, MSRI Publications 52, 133–143 (2005)MathSciNetGoogle Scholar
  4. 4.
    Bobenko, A.I., Schröder, P.: Discrete Willmore Flow. In: Desbrun, M., Pottman, H. (eds.) Eurographics Symposium on Geometry Pocessing, pp. 101–110 (2005)Google Scholar
  5. 5.
    Borrelli, V., Cazals, F., Morvan, J.-M.: On the angular defect of triangulations and the pointwise approximation of Curvatures. Computer Aided Geometric Design 20, 319–341 (2003)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cheeger, J., Müller, W., Schrader, R.: On the Curvature of Piecewise Flat Spaces. Comm. Math. Phys. 92, 405–454 (1984)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cheeger, J., Müller, W., Schrader, R.: Kinematic and Tube Formulas for Piecewise Linear Spaces. Indiana Univ. Math. J. 35(4), 737–754 (1986)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cohen-Steiner, D., Morvan, J.M.: Restricted Delaunay triangulations and normal cycle. In: Proc. 19th Annu. ACM Sympos. Comput. Geom. pp. 237–246 (2003)Google Scholar
  9. 9.
    Cohen-Steiner, D., Morvan, J.M.: Approximation of Normal Cycles. Research Report 4723 INRIA (2003)Google Scholar
  10. 10.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs, NJ (1976)MATHGoogle Scholar
  11. 11.
    Dyn, N., Hormann, K., Kim, S.-J., Levin, D.: Optimizing 3d triangulations using discrete curvature analysis. In: Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods in CAGD, Oslo 2000, pp. 135–146 (2001)Google Scholar
  12. 12.
    Federer, H.: Curvature measures. Trans. Amer. Math. Soc. 93, 418–491 (1959)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Flaherty, F.J.: Curvature measures for piecewise linear manifolds. Bull. Amer. Math. Soc. 79(1) 100–102Google Scholar
  14. 14.
    Fu, J.H.G.: Convergence of Curvatures in Secant Approximation. J. Differential Geometry 37, 177–190 (1993)MATHGoogle Scholar
  15. 15.
    Gray, A.: Tubes. Addison-Wesley, Redwood City, CA (1990)MATHGoogle Scholar
  16. 16.
    Hamann, B.: Curvature approximation for triangulated surfaces. Computing Suppl. 8, 139–153 (1993)MathSciNetGoogle Scholar
  17. 17.
    Howland, J., Fu, J.H.G.: Tubular neighbourhoods in Euclidean spaces. Duke Math. J. 52(4), 1025–1045 (1985)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Hug, D., Last, G., Weil, W.: A local Steiner-type formula for general closed sets and applications. Mathematische Zeitschrift 246(1-2), 237–272 (2004)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kimmel, R., Malladi, R., Sochen, N.: Images as Embedded Maps and Minimal Surfaces: Movies, Color, Texture, and Volumetric Medical Images. International Journal of Computer Vision 39(2), 111–129 (2000)MATHCrossRefGoogle Scholar
  20. 20.
    Maltret, J.-L., Daniel, M.: Discrete curvatures and applications, a survey (preprint)Google Scholar
  21. 21.
    Martin, R.: Estimation of principal curvatures from range data. International Journal of Shape Modeling 4, 99–111 (1998)CrossRefGoogle Scholar
  22. 22.
    Meek, D., Walton, D.: On surface normal and gaussian curvature approximations given data sampled from a smooth surface. Computer Aided Geometric Design 17, 521–543 (2000)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    O’Neill, B.: Elementary Differential Geometry. Academic Press, NY (1966)MATHGoogle Scholar
  24. 24.
    Saucan, E., Appleboim, E., Barak, E., Elber, G., Lev, R., Zeevi, Y.Y.: Local versus Global in Quasiconformal Mapping for Medical Imaging (in preparation)Google Scholar
  25. 25.
    Stokely, E., Wu., S.Y.: Surface parameterization and curvature measurement of arbitrary 3d-objects: Five practicel methods. IEEE Transactions on Pattern Analysis and Machine Intelligence 14(8), 833–840 (1992)CrossRefGoogle Scholar
  26. 26.
    Sullivan, J.M.: Curvature Measures for Discrete Surfaces. In: Discrete Differential Geometry: An Applied Introduction. SIGGRAPH 2006, pp. 10–13 (2006)Google Scholar
  27. 27.
    Surazhsky, T., Magid, E., Soldea, O., Elber, G., Rivlin, E.: A Comparison of Gaussian and Mean Curvatures Estimation Methods on Triangular Meshes. In: Proceedings of the IEEE International Conference on Robotics and Automation Taipei, Taiwan 2003, pp. 1021–1026 (2003)Google Scholar
  28. 28.
    Taubin, G.: Estimating the tensor of curvature of a surface from a polyhedral approximation. In: ICCV, pp. 902–907 (1995)Google Scholar
  29. 29.
    Watanabe, K., Belyaev, A.G.: Detection of salient curvature features on polygonal surfaces. EUROGRAPHICS 2001 Computer Graphics Forum 20(3), 385–392 (2001)CrossRefGoogle Scholar
  30. 30.
    Willmore, T.J.: Riemannian Geometry. Clarendon Press, Oxford (1993)MATHGoogle Scholar
  31. 31.
    Yakoya, N., Levine, M.: Range image segmentation based on differential geometry: A hybrid approach. IEEE Transactions on Pattern Analysis and Machine Intelligence 11(6), 643–649 (1989)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Ronen Lev
    • 1
  • Emil Saucan
    • 2
  • Gershon Elber
    • 3
  1. 1.OptiTex Ltd., Petach-Tikva 49221Israel
  2. 2.Electrical Engineering Department, Technion, Haifa 32000Israel
  3. 3.Computer Science Department, Technion, Haifa 32000Israel

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