Advertisement

Discrete Bisector Function and Euclidean Skeleton

  • Michel Couprie
  • Rita Zrour
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3429)

Abstract

We propose a new definition and an exact algorithm for the discrete bisector function, which is an important tool for analyzing and filtering Euclidean skeletons. We also introduce a new thinning method which produces homotopic discrete Euclidean skeletons. Unlike previouly proposed approaches, this method is still valid in 3D.

References

  1. 1.
    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
  2. 2.
    Attali, D., Montanvert, A.: Modelling noise for a better simplification of skeletons. In: Procs. International Conference on Image Processing, vol. 3, pp. 13–16 (1996)Google Scholar
  3. 3.
    Bertrand, G.: Simple points, topological numbers and geodesic neighborhoods in cubic grids. Pattern Recognition Letters 15, 1003–1011 (1994)CrossRefGoogle Scholar
  4. 4.
    Bespamyatnikh, S.N.: An efficient algorithm for the three-dimensional diameter problem. In: Procs. ACM-SIAM symp. on discrete algorithms, pp. 137–146 (1998)Google Scholar
  5. 5.
    Blum, H.: An associative machine for dealing with the visual field and some of its biological implications. In: Biological prototypes and synthetic systems, vol. 1, pp. 244–260 (1961)Google Scholar
  6. 6.
    Borgefors, G., Ragnemalm, I., Sanniti di Baja, G.: The Euclidean distance transform: finding the local maxima and reconstructing the shape. In: Procs. of the 7th Scand. Conf. on image analysis, vol. 2, pp. 974–981 (1991)Google Scholar
  7. 7.
    Danielsson, P.E.: Euclidean distance mapping. Computer Graphics and Image Processing 14, 227–248 (1980)CrossRefGoogle Scholar
  8. 8.
    Chassery, J.M., Montanvert, A.: Géométrie discrète (1991)Google Scholar
  9. 9.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to algorithms. MIT Press, Cambridge (1990)zbMATHGoogle Scholar
  10. 10.
    Couprie, M., Zrour, R.: Discrete bisector function and Euclidean skeleton in 2D and 3D., report IGM2004-12 of the Institut Gaspard Monge (University of Marne-la-Vallée) (2004), http://www-igm.univ-mlv.fr/LabInfo/rapportsInternes/2004/12.pdf
  11. 11.
    Daragon, X., Couprie, M., Bertrand, G.: Discrete frontiers. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 236–245. Springer, Heidelberg (2003)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.
    Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Oxford University Press, Oxford (1938)Google Scholar
  15. 15.
    Yung Kong, T., Rosenfeld, A.: Digital topology: introduction and survey. Computer Vision, Graphics and Image Processing 48, 357–393 (1989)CrossRefGoogle Scholar
  16. 16.
    Lam, L., Lee, S.-W., Suen, C.Y.: Thinning methodologies - a comprehensive survey. IEEE PAMI 14(9), 869–885 (1992)Google Scholar
  17. 17.
    Luppe, M., da Fontoura Costa, L., Obac Roda, V.: Parallel implementation of exact dilations and multi-scale skeletonization. Real-Time Imaging 9, 163–169 (2003)CrossRefGoogle Scholar
  18. 18.
    Meyer, F.: Cytologie quantitative et morphologie mathématique, PhD thesis, École des mines de Paris (1979)Google Scholar
  19. 19.
    Malandain, G., Fernández-Vidal, S.: Euclidean Skeletons. Image and vision computing 16, 317–327 (1998)CrossRefGoogle Scholar
  20. 20.
    Rosenfeld, A., Kak, A.C.: Digital Image processing. Academic Press, London (1982)Google Scholar
  21. 21.
    Rémy, E., Thiel, E.: Exact Medial Axis with Euclidean Distance. to appear in Image and Vision Computing (2004)Google Scholar
  22. 22.
    Saito, T., Toriwaki, J.I.: New algorithms for Euclidean distance transformation of an n-dimensional digitized picture with applications. Pattern Recognition 27, 1551–1565 (1994)CrossRefGoogle Scholar
  23. 23.
    Hirata, T.: A unified linear-time algorithm for computing distance maps. Information Processing Letters 58(3), 129–133 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Shamos, M.I.: Computational geometry, PhD thesis, Yale University (1978)Google Scholar
  25. 25.
    Talbot, H., Vincent, L.: Euclidean skeletons and conditional bisectors. In: Proceedings of VCIP 1992, vol. 1818, pp. 862–876. SPIE, San Jose (1992)Google Scholar
  26. 26.
    Vincent, L.: Efficient Computation of Various Types of Skeletons. In: Proceedings of Medical Imaging, vol. 1445, pp. 297–311. SPIE, San Jose (1991)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Michel Couprie
    • 1
    • 2
  • Rita Zrour
    • 1
    • 2
  1. 1.Laboratoire A2SIGroupe ESIEENoisy-le-GrandFrance
  2. 2.IGMUnité Mixte de Recherche CNRS-UMLV-ESIEE UMR 8049 

Personalised recommendations