Abstract

We propose a new variational model for surface fairing. We extend nonlocal smoothing techniques for image regularization to surface smoothing or fairing, with surfaces represented by triangular meshes. Our method is able to smooth the surfaces and preserve features due to geometric similarities using a mean curvature based local geometric descriptor. We present an efficient two step approach that first smoothes the mean curvature normal map, and then corrects the surface to fit the smoothed normal field. This leads to a fast implementation of a feature preserving fourth order geometric flow. We demonstrate the efficacy of the model with several surface fairing examples.

Keywords

Triangular Mesh Curvature Vector Image Denoising Signed Distance Function Image Regularization 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Clarenz, U., Diewald, U., Dziuk, G., Rumpf, M., Rusu, R.: A finite element method for surface restoration with smooth boundary conditions. Computer Aided Geometric Design 21(5), 427–445 (2004)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Desbrun, M., Meyer, M., Schroeder, P., Barr, A.: Implicit fairing of Irregular meshes using diffusion and curvature flow. In: Computer Graphics (SIGGRAPH 1999 Proceedings), pp. 317–324 (1999)Google Scholar
  3. 3.
    Meyer, M., Desbrun, M., Schroeder, P., Barr, A.: Discrete Differential Geometry Operators for Triangulated 2-Manifolds. In: Proc. VisMath 2002, Berlin-Dahlem, Germany, pp. 237–242 (2002)Google Scholar
  4. 4.
    Schneider, R., Kobbelt, L.: Geometric fairing of irregular meshes for free-form surface design. Computer Aided Geometric Design 18(4), 359–379 (2001)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Tasdizen, T., Whitaker, R., Burchard, P., Osher, S.: Geometric surface processing via normal maps. ACM Transactions on Graphics (TOG) 22/4, 1012–1033 (2003)CrossRefGoogle Scholar
  6. 6.
    Buades, A., Coll, B., Morel, J.: A Non-Local Algorithm for Image Denoising. In: Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2005), vol. 2, pp. 60–65. IEEE Computer Society, Washington, DC (2005)Google Scholar
  7. 7.
    Gilboa, G., Osher, S.: Nonlocal linear image regularization and supervised segmentation. Multiscale Modeling and Simulation 6(2), 595–630 (2007)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Dong, B., Ye, J., Osher, S., Dinov, I.: Level Set Based Nonlocal Surface Restoration. Multiscale Modeling and Simulation 7(2), 589–598 (2008)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Modeling and Simulation (SIAM Interdisciplinary Journal) 4(2), 490–530 (2005)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Jung, M., Bresson, X., Vese, L.: Nonlocal Mumford-Shah Regularizers for Color Image Restoration. IEEE Trans. Image Process (2010)Google Scholar
  11. 11.
    Xu, G., Pan, Q., Bajaj, C.L.: Discrete surface modelling using partial differential equations. Computer Aided Geometric Design 23(2), 125–145 (2006)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Tasdizen, T., Whitaker, R., Burchard, P., Osher, S.: Geometric surface processing via normal maps. ACM Transactions on Graphics 22(4), 1012–1033 (2003)CrossRefGoogle Scholar
  13. 13.
    Ohtake, Y., Belyaeva, A., Bogaevski, I.: Mesh regularization and adaptive smoothing. Computer-Aided Design 33/11, 789–800 (2001)CrossRefGoogle Scholar
  14. 14.
    Morigi, S.: Geometric Surface Evolution with Tangential Contribution. Journal of Computational and Applied Mathematics 233, 1277–1287 (2010)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Deschaud, J.E., Goulette, F.: Point cloud non local denoising using local surface descriptor similarity. In: Paparoditis, N., Pierrot-Deseilligny, M., Mallet, C., Tournaire, O. (eds.) IAPRS, vol. XXXVIII, Part 3A - Saint-Mandé, France (2010)Google Scholar
  16. 16.
    Yoshizawa, S., Belyaev, A., Seidel, H.P.: Smoothing by Example: Mesh Denoising by Averaging with Similarity-based Weights. In: Proc. IEEE International Conference on Shape Modeling and Applications (SMI), Matsushima, Japan, June 14-16, pp. 38–44 (2006)Google Scholar
  17. 17.
    Xu, G.: Convergent Discrete Laplace-Beltrami Operators over Triangular Surfaces. In: Proceedings of the Geometric Modeling and Processing 2004, GMP 2004 (2004)Google Scholar
  18. 18.
    Lysaker, M., Osher, S., Tai, X.C.: Noise Removal Using Smoothed Normals and Surface Fitting. IEEE Transaction on Image Processing 13(10), 1345–1457 (2004)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Serena Morigi
    • 1
  • Marco Rucci
    • 1
  • Fiorella Sgallari
    • 1
  1. 1.Department of Mathematics-CIRAMUniversity of BolognaBolognaItaly

Personalised recommendations