Minimal Simple Pairs in the 3-D Cubic Grid

  • N. Passat
  • M. CouprieEmail author
  • G. Bertrand


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. C. R. Acad. Sci. Sér. Math. I(321), 1077–1084 (1995) MathSciNetGoogle Scholar
  2. 2.
    Bertrand, G.: On critical kernels. C. R. Acad. Sci. Sér. Math. I(345), 363–367 (2007) MathSciNetGoogle Scholar
  3. 3.
    Bertrand, G., Couprie, M.: New 2D parallel thinning algorithms based on critical kernels. In: IWCIA. LNCS, vol. 4040, pp. 45–59. Springer, Berlin (2006) Google Scholar
  4. 4.
    Bertrand, G., Couprie, M.: A new 3D parallel thinning scheme based on critical kernels. In: DGCI. LNCS, vol. 4245, pp. 580–591. Springer, Berlin (2006) Google Scholar
  5. 5.
    Bertrand, G., Couprie, M.: Two-dimensional thinning algorithms based on critical kernels. J. Math. Imaging Vis. 31(1), 35–56 (2008) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bing, R.H.: Some aspects of the topology of 3-manifolds related to the Poincaré conjecture. In: Lectures on Modern Mathematics, vol. II, pp. 93–128. Wiley, New York (1964) Google Scholar
  7. 7.
    Chaussard, J.: Personal communication (2007) Google Scholar
  8. 8.
    Chillingworth, D.R.J.: Collapsing three-dimensional convex polyhedra. Proc. Camb. Philos. Soc. 63, 353–357 (1967) zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Couprie, M., Bertrand, G.: New characterizations of simple points in 2D, 3D and 4D discrete spaces. Technical Report IGM2007-07, Université de Marne-la-Vallée (2007) Google Scholar
  10. 10.
    Couprie, M., Coeurjolly, D., Zrour, R.: Discrete bisector function and Euclidean skeleton in 2D and 3D. Image Vis. Comput. 25(10), 1543–1556 (2007) CrossRefGoogle Scholar
  11. 11.
    Davies, E.R., Plummer, A.P.N.: Thinning algorithms: a critique and a new methodology. Pattern Recogn. 14(1–6), 53–63 (1981) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Dokládal, P., Lohou, C., Perroton, L., Bertrand, G.: Liver blood vessels extraction by a 3-D topological approach. In: MICCAI. LNCS, vol. 1679, pp. 98–105. Springer, Berlin (1999) Google Scholar
  13. 13.
    Fourey, S., Malgouyres, R.: A concise characterization of 3D simple points. Discrete Appl. Math. 125(1), 59–80 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Gau, C.-J., Kong, T.Y.: Minimal non-simple sets in 4D binary pictures. Graph. Models 65(1–3), 112–130 (2003) zbMATHCrossRefGoogle Scholar
  15. 15.
    Giblin, P.: Graphs, Surfaces and Homology. Chapman and Hall, London (1981) zbMATHGoogle Scholar
  16. 16.
    Yung Kong, T.: On topology preservation in 2-D and 3-D thinning. Int. J. Pattern Recogn. Artif. Intell. 9(5), 813–844 (1995) CrossRefGoogle Scholar
  17. 17.
    Yung Kong, T.: Topology-preserving deletion of 1’s from 2-, 3- and 4-dimensional binary images. In: DGCI. LNCS, vol. 1347, pp. 3–18. Springer, Berlin (1997) Google Scholar
  18. 18.
    Yung Kong, T., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vis. Graph. Image Process. 48(3), 357–393 (1989) CrossRefGoogle Scholar
  19. 19.
    Kovalevsky, V.A.: Finite topology as applied to image analysis. Comput. Vis. Graph. Image Process. 46(2), 141–161 (1989) CrossRefGoogle Scholar
  20. 20.
    Ma, C.M.: On topology preservation in 3D thinning. Comput. Vis. Graph. Image Process. 59(3), 328–339 (1994) Google Scholar
  21. 21.
    Morgenthaler, D.G.: Three-dimensional simple points: serial erosion, parallel thinning, and skeletonization. Technical Report TR-1005, University of Maryland (1981) Google Scholar
  22. 22.
    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: ISVC. LNCS, vol. 3804, pp. 60–67. Springer, Berlin (2005) Google Scholar
  23. 23.
    Ronse, C.: Minimal test patterns for connectivity preservation in parallel thinning algorithms for binary digital images. Discrete Appl. Math. 21(1), 67–79 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Rosenfeld, A.: Connectivity in digital pictures. J. Assoc. Comput. Mach. 17(1), 146–160 (1970) zbMATHMathSciNetGoogle Scholar
  25. 25.
    Ségonne, F.: Segmentation of medical images under topological constraints. Ph.D. thesis, MIT (2005) Google Scholar

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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.LSIIT, UMR 7005 CNRS/ULPStrasbourg 1 UniversityStrasbourgFrance
  2. 2.LABINFO-IGM, UMR CNRS 8049, A2SI-ESIEEUniversité Paris-EstMarne-le-ValléeFrance

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