Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

A semi-Lagrangian scheme with radial basis approximation for surface reconstruction

  • Published:
Computing and Visualization in Science

Abstract

We propose a Semi-Lagrangian scheme coupled with Radial Basis Function interpolation for approximating a curvature-related level set model, which has been proposed by Zhao et al. (Comput Vis Image Underst 80:295–319, 2000) to reconstruct unknown surfaces from sparse data sets. The main advantages of the proposed scheme are the possibility to solve the level set method on unstructured grids, as well as to concentrate the reconstruction points in the neighbourhood of the data set, with a consequent reduction of the computational effort. Moreover, the scheme is explicit. Numerical tests show the accuracy and robustness of our approach to reconstruct curves and surfaces from relatively sparse data sets.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Similar content being viewed by others

References

  1. Behrens, J., Iske, A.: Grid-free adaptive semi-Lagrangian advection using radial basis functions. Comput. Math. Appl. 43, 319–327 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Carlini, E., Falcone, M., Ferretti, R.: Convergence of a large time-step scheme for mean curvature motion. Interf. Free Bound. 12, 409–441 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carlini, E., Ferretti, R.: A Semi-Lagrangian scheme for area-preserving flows. In: Proceedings of the conference “ICPR2012”, IEEExplore (2012)

  4. Carlini, E., Ferretti, R.: A semi-Lagrangian approximation for the AMSS model of image processing. Appl. Numer. Math. 73, 16–32 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carr, J.C., Beatson, R.K., Cherrie, J.B., Mitchell, T.J., Fright, W.R., McCallum, B.C., Evans, T.R.: Reconstruction and representation of 3d objects with radial basis functions. In: Proceedings of ACM SIGGRAPH, pp. 67–76 (2001)

  6. Carr, J.C., Fright, W.R., Beatson, R.K.: Surface interpolation with radial basis functions for medical imaging. IEEE Trans. Med. Imag. 16, 96–107 (1997)

    Article  Google Scholar 

  7. Chirokov, A.: Interpolation and approximation using radial base function (RBF). http://www.mathworks.com/

  8. Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Falcone, M., Ferretti, R.: Consistency of a Large Time-Step Scheme for Mean Curvature Motion, Numerical Mathematics and Advanced Applications—ENUMATH 2001. Springer, Milano (2003)

    MATH  Google Scholar 

  10. Falcone, M., Ferretti, R.: Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. SIAM, Philadelphia (2013)

    Book  MATH  Google Scholar 

  11. Franchini, E., Morigi, S., Sgallari, F.: Implicit shape reconstruction of unorganized points using PDE-based deformable 3D manifolds. Numer. Math. Theory Methods Appl. 3, 405–430 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Goldstein, T., Bresson, X., Osher, S.: Geometric applications of the Split Bregman method: segmentation and surface reconstruction. J. Sci. Comput. 45, 272–293 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, Y., Lee, D., Lee, C., Lee, J., Lee, S., Kim, J., Kim, J., Ahn, S.: Surface embedding narrow volume reconstruction from unorganized points. Comput. Vis. Image Underst. 121, 100–107 (2014)

    Article  Google Scholar 

  14. Liang, J., Park, F., Zhao, H.: Robust and efficient implicit surface reconstruction for point clouds based on convexified image segmentation. J. Sci. Comput. 54, 577–602 (2013)

    Article  MathSciNet  Google Scholar 

  15. Liu, G.R.: Mesh Free Methods: Moving Beyond the Finite Element Method. CRC Press, Boca Raton (2002)

    Book  Google Scholar 

  16. Savchenko, V.V., Pasko, A.A., Okunev, O.G., Kunii, T.L.: Function representation of solids reconstructed from scattered surface points and contours. Comput. Gr. Forum 14, 181–188 (1995)

    Article  Google Scholar 

  17. Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  18. Ye, J., Yanowsky, I., Dong, B., Gandlin, R., Brandt, A., Osher, S.: Multigrid narrow band surface reconstruction via level set functions. In: Proceedings of the 8th International Symposium on Visual Computing—ISVC 2012 (Rethymnon, Crete), pp. 61–70. Springer, Berlin (2012)

  19. Tsai, Y.R., Osher, S.: Total variation and level set methods in image science. Acta Numer. 14, 509–573 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhao, H., Osher, S., Merriman, B., Kang, M.: Implicit and non-parametric shape reconstruction from unorganized points using variational level set method. Comput. Vis. Image Underst. 80, 295–319 (2000)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica “G. Castelnuovo”, Sapienza Università di Roma, P.le A. Moro, 5, 00185, Rome, Italy

    E. Carlini

  2. Dipartimento di Matematica e Fisica, Università Roma Tre, L.go S. Leonardo Murialdo, 1, 00146, Rome, Italy

    R. Ferretti

Authors
  1. E. Carlini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Ferretti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Ferretti.

Additional information

This research has been partially funded by Sapienza Università di Roma, Università Roma Tre and INdAM–GNCS.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carlini, E., Ferretti, R. A semi-Lagrangian scheme with radial basis approximation for surface reconstruction. Comput. Visual Sci. 18, 103–112 (2017). https://doi.org/10.1007/s00791-016-0274-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00791-016-0274-2

Keywords

Search

Navigation