A New 3D Parallel Thinning Scheme Based on Critical Kernels

  • Gilles Bertrand
  • Michel Couprie
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4245)

Abstract

Critical kernels constitute a general framework settled in the category of abstract complexes for the study of parallel thinning in any dimension. We take advantage of the properties of this framework, and we derive a general methodology for designing parallel algorithms for skeletons of objects in 3D grids. In fact, this methodology does not need to handle the structure of abstract complexes, we show that only 3 masks defined in the classical cubic grid are sufficient to implement it. We illustrate our methodology by giving two new types of skeletons.

References

  1. 1.
    Rutovitz, D.: Pattern recognition. Journal of the Royal Statistical Society 129, 504–530 (1966)Google Scholar
  2. 2.
    Tsao, Y., Fu, K.: A parallel thinning algorithm for 3d pictures. CGIP 17(4), 315–331 (1981)Google Scholar
  3. 3.
    Saha, P., Chaudhuri, B., Dutta, D., Majumder, D.: A new shape-preserving parallel thinning algorithm for 3d digital images. PR 30(12), 1939–1955 (1997)Google Scholar
  4. 4.
    Bertrand, G.: A parallel thinning algorithm for medial surfaces. PRL 16, 979–986 (1995)Google Scholar
  5. 5.
    Gong, W., Bertrand, G.: A simple parallel 3d thinning algorithm. In: ICPR 1990, pp. 188–190 (1990)Google Scholar
  6. 6.
    Ma, C.M.: A 3d fully parallel thinning algorithm for generating medial faces. Pattern Recogn. Lett. 16(1), 83–87 (1995)CrossRefGoogle Scholar
  7. 7.
    Manzanera, A., Bernard, T., Prêteux, F., Longuet, B.: N-dimensional skeletonization: a unified mathematical framework. Journal of Electronic Imaging 11(1), 25–37 (2002)CrossRefGoogle Scholar
  8. 8.
    Ma, C.M., Sonka, M.: A 3d fully parallel thinning algorithm and its applications. Computer Vision and Image Understanding 64(3), 420–433 (1996)CrossRefGoogle Scholar
  9. 9.
    Palágyi, K., Kuba, A.: A 3d 6-subiteration thinning algorithm for extracting medial lines. Pattern Recognition Letters (19), 613–627 (1998)MATHCrossRefGoogle Scholar
  10. 10.
    Palágyi, K., Kuba, A.: A parallel 3d 12-subiteration thinning algorithm. Graphical Models and Image Processing (61), 199–221 (1999)CrossRefGoogle Scholar
  11. 11.
    Lohou, C., Bertrand, G.: A 3d 12-subiteration thinning algorithm based on p-simple points. Discrete Applied Mathematics 139, 171–195 (2004)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lohou, C., Bertrand, G.: A 3d 6-subiteration curve thinning algorithm based on p-simple points. Discrete Applied Mathematics 151, 198–228 (2005)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kong, T.Y.: On topology preservation in 2-d and 3-d thinning. International Journal on Pattern Recognition and Artificial Intelligence 9, 813–844 (1995)CrossRefGoogle Scholar
  14. 14.
    Bertrand, G.: On P-simple points. Comptes Rendus de l’Académie des Sciences, Série Math. I(321), 1077–1084 (1995)MathSciNetGoogle Scholar
  15. 15.
    Bertrand, G.: On critical kernels. Internal Report, Université de Marne-la-Vallée IGM2005-05 (2005) (also submitted for publication)Google Scholar
  16. 16.
    Bertrand, G., Couprie, M.: Two-dimensional parallel thinning algorithms based on critical kernels. Internal Report, Université de Marne-la-Vallée IGM2006-02 (2006) (also submitted for publication)Google Scholar
  17. 17.
    Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Comp. Vision, Graphics and Image Proc. 48, 357–393 (1989)CrossRefGoogle Scholar
  18. 18.
    Kovalevsky, V.: Finite topology as applied to image analysis. Computer Vision, Graphics and Image Processing 46, 141–161 (1989)CrossRefGoogle Scholar
  19. 19.
    Giblin, P.: Graphs, surfaces and homology. Chapman and Hall, Boca Raton (1981)MATHGoogle Scholar
  20. 20.
    Kong, T.Y.: Topology-preserving deletion of 1’s from 2-, 3- and 4-dimensional binary images. In: Ahronovitz, E. (ed.) DGCI 1997. LNCS, vol. 1347, pp. 3–18. Springer, Heidelberg (1997)Google Scholar
  21. 21.
    Bertrand, G.: Simple points, topological numbers and geodesic neighborhoods in cubic grids. Pattern Recognition Letters 15, 1003–1011 (1994)CrossRefGoogle Scholar
  22. 22.
    Bertrand, G., Malandain, G.: A new characterization of threedimensional simple points. Pattern Recognition Letters 15(2), 169–175 (1994)MATHCrossRefGoogle Scholar
  23. 23.
    Saha, P., Chaudhuri, B., Chanda, B., Dutta Majumder, D.: Topology preservation in 3d digital space. PR 27, 295–300 (1994)MathSciNetGoogle Scholar
  24. 24.
    Rosenfeld, A., Pfaltz, J.: Sequential operations in digital picture processing. Journal of the Association for Computer Machinery 13, 471–494 (1966)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Gilles Bertrand
    • 1
  • Michel Couprie
    • 1
  1. 1.Institut Gaspard-Monge, Laboratoire A2SI, Groupe ESIEENoisy-le-GrandFrance

Personalised recommendations