Mesh Smoothing and Variational Subdivision

  • Jakob Andreas Bærentzen
  • Jens Gravesen
  • François Anton
  • Henrik Aanæs

Abstract

In this chapter, we cover how meshes are smoothed. This is an important topic in geometry processing since acquired meshes are always subject to noise. The basic principles of signal processing are discussed. Then we present the principles behind Laplacian smoothing and Taubin smoothing, which is based on Laplacian smoothing but suffers less from shrinkage. However, Taubin smoothing causes distortion if the mesh is not highly regular in its structure. Mean curvature flow is much better in this regard.

Using spectral smoothing, it is possible to create filters that more accurately manipulate features of a certain scale. Spectral smoothing works by observing that the Laplace–Beltrami operator, which is central to many smoothing schemes, can be written as a linear operator whose eigenvectors form a function space upon which the vertex positions can be projected—very much like the Fourier basis on a regular grid.

Smoothing generally smooths sharp edges and corners. Fortunately, there are several schemes which remove noise while preserving features such as corners and edges.

Finally, we can see smoothing as an energy minimization and the chapter concludes with a discussion of variational subdivision which is based on repeated refinement and smoothing of meshes.

Keywords

Attenuation Shrinkage Convolution 

References

  1. 1.
    Desbrun, M., Meyer, M., Schröder, P., Barr, A.H.: Implicit fairing of irregular meshes using diffusion and curvature flow. In: SIGGRAPH’99: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, pp. 317–324. ACM Press, New York (1999). doi: 10.1145/311535.311576 CrossRefGoogle Scholar
  2. 2.
    Taubin, G.: A signal processing approach to fair surface design. In: ACM SIGGRAPH’95 Proceedings (1995) Google Scholar
  3. 3.
    Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15–36 (1993) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Vallet, B., Lévy, B.: Spectral geometry processing with manifold harmonics. Comput. Graph. Forum 27(2), 251–260 (2008) CrossRefGoogle Scholar
  5. 5.
    Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. In: Hege, H.-C., Polthier, K. (eds.) Visualization and Mathematics III, pp. 35–57. Springer, Heidelberg (2003) CrossRefGoogle Scholar
  6. 6.
    Levy, B., Zhang, R.H.: Spectral geometry processing. In: ACM SIGGRAPH Course Notes (2010) Google Scholar
  7. 7.
    Shen, Y., Barner, K.E.: Fuzzy vector median-based surface smoothing. IEEE Trans. Vis. Comput. Graph. 10(3), 252–265 (2004). doi: 10.1109/TVCG.2004.1272725 CrossRefGoogle Scholar
  8. 8.
    Zheng, Y., Fu, H., Au, O.K.-C., Tai, C.-L.: Bilateral normal filtering for mesh denoising. IEEE Trans. Vis. Comput. Graph. 17(10), 1521–1530 (2011). doi: 10.1109/TVCG.2010.264 CrossRefGoogle Scholar
  9. 9.
    Kobbelt, L.P.: Discrete fairing and variational subdivision for freeform surface design. Vis. Comput. 16, 142–158 (2000) CrossRefGoogle Scholar
  10. 10.
    Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Dover, New York (2000) MATHGoogle Scholar
  11. 11.
    Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. John Hopkins, Baltimore (1996) MATHGoogle Scholar
  12. 12.
    Kobbelt, L., Campagna, S., Vorsatz, J., Seidel, H.-P.: Interactive multi-resolution modeling on arbitrary meshes. In: ACM SIGGRAPH’98 Proceedings, pp. 105–114 (1998) Google Scholar

Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  • Jakob Andreas Bærentzen
    • 1
  • Jens Gravesen
    • 2
  • François Anton
    • 1
  • Henrik Aanæs
    • 1
  1. 1.Department of Informatics and Mathematical ModellingTechnical University of DenmarkKongens LyngbyDenmark
  2. 2.Department of MathematicsTechnical University of DenmarkKongens LyngbyDenmark

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