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Surface Thinning in 3D Cubical Complexes

  • John Chaussard
  • Michel Couprie
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5852)

Abstract

We introduce a parallel thinning algorithm with directional substeps based on the collapse operation, which is guaranteed to preserve topology and to provide a thin result. Then, we propose two variants of a surface-preserving thinning scheme, based on this parallel directional thinning algorithm. Finally, we propose a methodology to produce filtered surface skeletons, based on the above thinning methods and the recently introduced discrete λ-medial axis.

Keywords

Topology cubical complex thinning skeleton collapse 

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References

  1. 1.
    Attali, D., Boissonnat, J.-D., Edelsbrunner, H.: Stability and computation of the medial axis — a state-of-the-art report. In: Möller, T., Hamann, B., Russell, B. (eds.) Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration. LNCS, pp. 1–19. Springer, Heidelberg (to appear, 2009) Google Scholar
  2. 2.
    Attali, D., Lachaud, J.O.: Delaunay conforming iso-surface, skeleton extraction and noise removal. Computational Geometry: Theory and Applications 19, 175–189 (2001)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Attali, D., Montanvert, A.: Modelling noise for a better simplification of skeletons. In: Proc. International Conference on Image Processing (ICIP), vol. 3, pp. 13–16 (1996)Google Scholar
  4. 4.
    Bertrand, G.: On critical kernels. Comptes Rendus de l’Académie des Sciences, Série Math.  I(345), 363–367 (2007)Google Scholar
  5. 5.
    Bertrand, G., Couprie, M.: Two-dimensional parallel thinning algorithms based on critical kernels. Journal of Mathematical Imaging and Vision 31(1), 35–56 (2008)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bertrand, G., Couprie, M.: A new 3D parallel thinning scheme based on critical kernels. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 580–591. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Bertrand, G., Couprie, M.: On parallel thinning algorithms: minimal non-simple sets, P-simple points and critical kernels. Journal of Mathematical Imaging and Vision 35(1), 23–35 (2009)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Borgefors, G., Ragnemalm, I., Sanniti di Baja, G.: The Euclidean distance transform: finding the local maxima and reconstructing the shape. In: Proc. of the 7th Scandinavian Conference on Image Analysis, vol. 2, pp. 974–981 (1991)Google Scholar
  9. 9.
    Chaussard, J., Couprie, M., Talbot, H.: A discrete lambda-medial axis. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 421–433. Springer, Heidelberg (2009)Google Scholar
  10. 10.
    Chazal, F., Lieutier, A.: The lambda medial axis. Graphical Models 67(4), 304–331 (2005)zbMATHCrossRefGoogle Scholar
  11. 11.
    Couprie, M., Bertrand, G.: New characterizations of simple points in 2D, 3D and 4D discrete spaces. IEEE Trans. on Pattern Analysis and Machine Intelligence 31(4), 637–648 (2009)CrossRefGoogle Scholar
  12. 12.
    Davies, E.R., Plummer, A.P.N.: Thinning algorithms: a critique and a new methodology. Pattern Recognition 14, 53–63 (1981)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Ge, Y., Fitzpatrick, J.M.: On the generation of skeletons from discrete Euclidean distance maps. IEEE Trans. on Pattern Analysis and Machine Intelligence 18(11), 1055–1066 (1996)CrossRefGoogle Scholar
  14. 14.
    Hesselink, W.H., Roerdink, J.B.T.M.: Euclidean skeletons of digital image and volume data in linear time by the integer medial axis transform. IEEE Trans. on Pattern Analysis and Machine Intelligence 30(12), 2204–2217 (2008)CrossRefGoogle Scholar
  15. 15.
    Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Computer Vision, Graphics and Image Processing 48, 357–393 (1989)CrossRefGoogle Scholar
  16. 16.
    Kong, T.Y., Litherland, R., Rosenfeld, A.: Problems in the topology of binary digital images. In: Open problems in topology, pp. 376–385. Elsevier, Amsterdam (1990)Google Scholar
  17. 17.
    Kovalevsky, V.A.: Finite topology as applied to image analysis. Computer Vision, Graphics and Image Processing 46, 141–161 (1989)CrossRefGoogle Scholar
  18. 18.
    Liu, L.: 3d thinning on cell complexes for computing curve and surface skeletons. Master’s thesis, Washington University in Saint Louis (May 2009)Google Scholar
  19. 19.
    Malandain, G., Fernández-Vidal, S.: Euclidean skeletons. Image and Vision Computing 16, 317–327 (1998)CrossRefGoogle Scholar
  20. 20.
    Ogniewicz, R.L., Kübler, O.: Hierarchic Voronoi skeletons. Pattern Recognition 28(33), 343–359 (1995)CrossRefGoogle Scholar
  21. 21.
    Pudney, C.: Distance-ordered homotopic thinning: a skeletonization algorithm for 3D digital images. Computer Vision and Image Understanding 72(3), 404–413 (1998)CrossRefGoogle Scholar
  22. 22.
    Rémy, E., Thiel, E.: Exact medial axis with Euclidean distance. Image and Vision Computing 23(2), 167–175 (2005)CrossRefGoogle Scholar
  23. 23.
    Rosenfeld, A.: A characterization of parallel thinning algorithms. Information and Control 29, 286–291 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Serra, J.: Image analysis and mathematical morphology. Academic Press, London (1982)zbMATHGoogle Scholar
  25. 25.
    Siddiqi, K., Bouix, S., Tannenbaum, A., Zucker, S.: The Hamilton-Jacobi skeleton. In: International Conference on Computer Vision (ICCV), pp. 828–834 (1999)Google Scholar
  26. 26.
    Soille, P.: Morphological image analysis. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  27. 27.
    Talbot, H., Vincent, L.: Euclidean skeletons and conditional bisectors. In: Proc. VCIP 1992, SPIE, vol. 1818, pp. 862–876 (1992)Google Scholar
  28. 28.
    Vincent, L.: Efficient computation of various types of skeletons. In: Proc. Medical Imaging V, SPIE, vol. 1445, pp. 297–311 (1991)Google Scholar
  29. 29.
    Whitehead, J.H.C.: Simplicial spaces, nuclei and m-groups. Proceedings of the London Mathematical Society 45(2), 243–327 (1939)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • John Chaussard
    • 1
  • Michel Couprie
    • 1
  1. 1.Laboratoire d’Informatique Gaspard-Monge, Équipe A3SI, ESIEEUniversité Paris-EstParisFrance

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