Skip to main content
SpringerLink
Log in

Global diffusion method for smoothing triangular mesh

  • Original Article
  • Published:
The Visual Computer Aims and scope Submit manuscript

Abstract

In this paper, we propose a new smoothing method based on physical principles. The smoothing process is modeled using the heat transfer process. We start from the global equation of heat conservation and we decompose it into basic laws. The numerical scheme is derived directly from the discretization of the basic heat transfer laws using computation algebraic topological tools, thus providing a physical and topological explanation for each step of the discretization process. In such a way, the geometry, topology and physics are concurring together in a unified framework to define and simulate the diffusion process to reduce random noise on the surface of the object. Experimental results show a good performance in improvement of the proposed approach compared to existing smoothing methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Clarenz, U., Diewald, U., Rumpf, M.: Processing textured surfaces via anisotropic geometric diffusion. IEEE Trans. Image Process. 13(2), 248–261 (2004)

    Article  Google Scholar 

  2. Bajaj, C.L., Xu, G.: Anisotropic diffusion of surfaces and functions on surfaces. ACM Trans. Graph. 22(1), 4–32 (2003)

    Article  Google Scholar 

  3. Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. In: Proc. VisMath., pp. 35–57 (2002)

  4. Hildebrandt, K., Polthier, K.: Anisotropic filtering of non-linear surface features. Comput. Graph. Forum 23(3), 391–400 (2004)

    Article  Google Scholar 

  5. Ohtake, Y., Belyaev, A.G., Bogaevski, I.: Mesh regularization and adaptive smoothing. Comput. Aides Geom. D. 33(11), 789–800 (2001)

    Article  Google Scholar 

  6. Yoshizawa, S., Belyaev, A.G., Seidel, H.-P.: Smoothing by example: mesh denoising by averaging with similarity-based weights. In: Proc. SMI, pp. 9 (2006)

  7. Zhang, Y., Ben, A.: Hamza, vertex-based diffusion for 3D mesh denoising. IEEE Trans. Image Process. 16(4), 1036–1045 (2007)

    Article  MathSciNet  Google Scholar 

  8. Tarmissi, K., Ben Hamza, A.: Multivariate kernel diffusion for surface denoising. Signal Image Video Process. 5(2), 191–201 (2011)

  9. Mattiussi, C.: A reference discretization strategy for the numerical solution of physical field problems. Adv. Imag. Elect. Phys. 121, 143–279 (2002)

    Google Scholar 

  10. Tonti, E.: Finite formulation of the electromagnetic field. IEEE Trans. Magn. 38(2), 333–336 (2002)

    Article  Google Scholar 

  11. Incropera, F.P.: Fundamentals of Heat and Mass Transfer. Wiley, New York (2006)

    Google Scholar 

  12. Burke, W.L.: Applied Differential Geometry. Cambridge University Press, Cambridge (1985)

    Book  MATH  Google Scholar 

  13. Chard, J.A., Shapiro, V.: A multivector data structure for differential forms and equations. Math. Comput. Simul. 54(1–3), 33–64 (2000)

    Article  MathSciNet  Google Scholar 

  14. Bettini, P., Trevisan, F.: 3-D magnetostatic with the finite formulation. IEEE Trans. Magn. 39(3), 1135–1138 (2003)

    Google Scholar 

  15. Desbrun, M., Kanso, E., Tong, Y.: Discrete differential forms for computational modeling. In: ACM SIGGRAPH 2005 Courses, p. 7 (2005)

  16. Elcott, S., Tong, Y., Kanso, E., Schröder, P., Desbrun, M.: Stable, circulation-preserving, simplicial fluids. ACM Trans. Graph. 26(1), 4 (2007)

    Article  Google Scholar 

  17. Gu, X.D., Yau, S.-T.: Computational conformal geometry. In: Advanced Lectures in Mathematics. High Education Press and International Press, Somerville (2008)

  18. van de Weygaert, R., Platen, E., Vegter, G., Eldering, B., Kruithof, N.: Alpha shape topology of the cosmic web. In: International Symposium on Voronoi Diagrams in Science and Engineering, pp. 224–234 (2010)

  19. Gross, P.W., Kotiuga, P.R.: Data structures for geometric and topological aspects of finite element algorithms. Progress Electromagn. Res. 32, 151–169 (2001)

    Article  Google Scholar 

  20. Croom, F.H.: Basic Concepts of Algebraic Topology. Springer, Berlin (1978)

    Book  MATH  Google Scholar 

  21. Fortier, Auclair: M., Ziou, D.: A global approach for solving evolutive heat transfer for image denoising and inpainting. IEEE Trans. Image Process 15(9), 2558–2574 (2006)

  22. Huang, C.-H., Yan, J.-Y.: An inverse problem in predicting temperature dependent heat capacity per unit volume without internal measurements. Int. J. Numer. Meth. Eng. 39(4), 605–618 (1996)

    Google Scholar 

  23. Voitovich, T.V., Vandewalle, S.: Exact integration formulas for the finite volume element method on simplicial meshes. Numer. Methods Partial Differ. Equ. 1–27 (2006)

  24. Weickert, J.: Theoretical foundations of anisotropic diffusion in image processing. In: Theoretical Foundations of Computer Vision, pp. 221–236 (1994)

  25. Rusinkiewicz, S.: Estimating curvatures and their derivatives on triangle meshes. In: Symposium on 3D Data Processing, Visualization, and Transmission (2004)

  26. Delfinado, C.J.A., Edelsbrunner, H.: An incremental algorithm for Betti numbers of simplicial complexes. In: 9th Annual ACM Conference on Computational Geometry, pp. 232–239 (1993)

  27. Branin Jr, F.H.: The algebraic-topological basis for network analogies and for vector calculus. In: Symposium on Generalized, Networks, pp. 453–491 (1966)

  28. Kaczynski, T., Mrozek, M., Slusarek, M.: Homology computation by reduction of chain complexes. Comput. Math. Appl. 35, 59–70 (1998)

    Google Scholar 

  29. Ziou, D., Allili, M.: Generating cubical complexes from image data and computation of the euler number. Pattern Recognit. 35(12), 2833–2839 (2002)

    Article  MATH  Google Scholar 

  30. Belyaev, A., Ohtake, Y.: A comparison of mesh smoothing methods. In: Israel-Korea Bi-National Conference on Geometric Modeling and Computer Graphics, pp. 83–87 (2003)

  31. Sun, X., Rosin, P.L., Martin, R.R., Langbein, F.C.: The 3D model acquisition pipeline. Graph. Models 71(2), 34–48 (2009)

    Article  Google Scholar 

  32. Bernardini, F., Rushmeier, H.E.: The 3D model acquisition pipeline. Comput. Graph. Forum 21(2), 149–172 (2002)

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank anonymous reviewers for their constructive comments. They would also like to thank Dr. Xian fang Sun from Cardiff University for providing the scanned metal plate object. The Kitten, Gargoyle objects are courtesy from AIM@SHAPE repository.

Author information

Authors and Affiliations

  1. Faculté Polydisciplinaire Ouarzazate, Ouarzazate, Morocco

    Ahmed Fouad El Ouafdi

  2. Faculté des sciences, Universit de Sherbrooke, Sherbrooke, Canada

    Djemel Ziou

Authors
  1. Ahmed Fouad El Ouafdi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Djemel Ziou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmed Fouad El Ouafdi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

El Ouafdi, A.F., Ziou, D. Global diffusion method for smoothing triangular mesh. Vis Comput 31, 295–310 (2015). https://doi.org/10.1007/s00371-014-0922-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00371-014-0922-9

Keywords

Search

Navigation