Minimal Simple Pairs in the Cubic Grid

  • Nicolas Passat
  • Michel Couprie
  • Gilles Bertrand
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4992)


Preserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. This paper constitutes an introduction to the study of non-trivial simple sets in the framework of cubical 3-D complexes. A simple set has the property that the homotopy type of the object in which it lies is not changed when the set is removed. The main contribution of this paper is a characterisation of the non-trivial simple sets composed of exactly two voxels, such sets being called minimal simple pairs.


Cubical complexes topology preservation collapse thinning 3-D space 


  1. 1.
    Bertrand, G.: On P-simple points. Comptes Rendus de l’Académie des Sciences, Série Math. I(321), 1077–1084 (1995)Google Scholar
  2. 2.
    Bertrand, G.: On critical kernels. Comptes Rendus de l’Académie des Sciences, Série Math. I(345), 363–367 (2007)MathSciNetGoogle Scholar
  3. 3.
    Bertrand, G., Couprie, M.: New 2D Parallel Thinning Algorithms Based on Critical Kernels. In: Reulke, R., Eckardt, U., Flach, B., Knauer, U., Polthier, K. (eds.) IWCIA 2006. LNCS, vol. 4040, pp. 45–59. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    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
  5. 5.
    Bertrand, G., Couprie, M.: Two-dimensional thinning algorithms based on critical kernels. Journal of Mathematical Imaging and Vision (to appear, 2008)Google Scholar
  6. 6.
    Bing, R.H.: Some aspects of the topology of 3-manifolds related to the Poincaré conjecture. Lectures on Modern Mathematics II, 93–128 (1964)MathSciNetGoogle Scholar
  7. 7.
    Cohen, M.M.: A course in simple-homotopy theory. Springer, Heidelberg (1973)zbMATHGoogle Scholar
  8. 8.
    Couprie, M., Coeurjolly, D., Zrour, R.: Discrete bisector function and Euclidean skeleton in 2D and 3D. Image and Vision Computing 25(10), 1543–1556 (2007)CrossRefGoogle Scholar
  9. 9.
    Davies, E.R., Plummer, A.P.N.: Thinning algorithms: a critique and a new methodology. Pattern Recognition 14(1–6), 53–63 (1981)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Dokládal, P., Lohou, C., Perroton, L., Bertrand, G.: Liver Blood Vessels Extraction by a 3-D Topological Approach. In: Taylor, C., Colchester, A. (eds.) MICCAI 1999. LNCS, vol. 1679, pp. 98–105. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. 11.
    Fourey, S., Malgouyres, R.: A concise characterization of 3D simple points. Discrete Applied Mathematics 125(1), 59–80 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gau, C.-J., Kong, T.Y.: Minimal non-simple sets in 4D binary pictures. Graphical Models 65(1–3), 112–130 (2003)zbMATHCrossRefGoogle Scholar
  13. 13.
    Giblin, P.: Graphs, surfaces and homology. Chapman and Hall, Boca Raton (1981)zbMATHGoogle Scholar
  14. 14.
    Kong, T.Y.: On topology preservation in 2-D and 3-D thinning. International Journal on Pattern Recognition and Artificial Intelligence 9(5), 813–844 (1995)CrossRefGoogle Scholar
  15. 15.
    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
  16. 16.
    Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Computer Vision, Graphics and Image Processing 48(3), 357–393 (1989)CrossRefGoogle Scholar
  17. 17.
    Kovalevsky, V.A.: Finite topology as applied to image analysis. Computer Vision, Graphics and Image Processing 46(2), 141–161 (1989)CrossRefGoogle Scholar
  18. 18.
    Passat, N., Couprie, M., Bertrand, G.: Minimal simple pairs in the 3-D cubic grid. Technical Report IGM2007-04, Université de Marne-la-Vallée (2007),
  19. 19.
    Passat, N., Ronse, C., Baruthio, J., Armspach, J.-P., Bosc, M., Foucher, J.: Using multimodal MR data for segmentation and topology recovery of the cerebral superficial venous tree. In: Bebis, G., Boyle, R., Koracin, D., Parvin, B. (eds.) ISVC 2005. LNCS, vol. 3804, pp. 60–67. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  20. 20.
    Ronse, C.: Minimal test patterns for connectivity preservation in parallel thinning algorithms for binary digital images. Discrete Applied Mathematics 21(1), 67–79 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Rosenfeld, A.: Connectivity in digital pictures. Journal of the Association for Computer Machinery 17(1), 146–160 (1970)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Ségonne, F.: Segmentation of Medical Images under Topological Constraints. PhD thesis, MIT (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Nicolas Passat
    • 1
  • Michel Couprie
    • 2
  • Gilles Bertrand
    • 2
  1. 1.LSIIT, UMR 7005 CNRS/ULPStrasbourg 1 UniversityFrance
  2. 2.LABINFO-IGM, UMR CNRS 8049Université Paris-EstFrance

Personalised recommendations