Minimal Non-simple and Minimal Non-cosimple Sets in Binary Images on Cell Complexes

  • T. Yung Kong
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4245)


The concepts of weak component and simple 1 are generalizations, to binary images on the n-cells of n-dimensional cell complexes, of the standard concepts of “26-component” and “26-simple” 1 in binary images on the 3-cells of a 3D cubical complex; the concepts of strong component and cosimple 1 are generalizations of the concepts of “6-component” and “6-simple” 1. Over the past 20 years, the problems of determining just which sets of 1’s can be minimal non-simple, just which sets can be minimal non-cosimple, and just which sets can be minimal non-simple (minimal non-cosimple) without being a weak (strong) foreground component have been solved for the 2D cubical and hexagonal, 3D cubical and face-centered-cubical, and 4D cubical complexes. This paper solves these problems in much greater generality, for a very large class of cell complexes of dimension ≤4.


Binary Image Cell Complex Algebraic Topology Cubical Complex Fundamental Lemma 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bertrand, G.: On P-simple points. C. R. Acad. Sci. Paris, Série I 321, 1077–1084 (1995)MATHMathSciNetGoogle Scholar
  2. 2.
    Bertrand, G.: A Boolean characterization of three-dimensional simple points. Pattern Recognition Letters 17, 115–124 (1996)CrossRefGoogle Scholar
  3. 3.
    Björner, A.: Topological methods. In: Graham, R.L., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, vol. II, pp. 1819–1872. MIT Press, Cambridge (1995)Google Scholar
  4. 4.
    Gau, C.J., Kong, T.Y.: Minimal nonsimple sets of voxels in binary images on a face-centered cubic grid. International Journal of Pattern Recognition and Artificial Intelligence 13, 485–502 (1999)CrossRefGoogle Scholar
  5. 5.
    Gau, C.J., Kong, T.Y.: Minimal nonsimple sets in 4D binary images. Graphical Models 65, 112–130 (2003)MATHCrossRefGoogle Scholar
  6. 6.
    Hall, R.W.: Tests for connectivity preservation for parallel reduction operators. Topology and Its Applications 46, 199–217 (1992)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Herman, G.T.: Geometry of Digital Spaces. Birkhäuser, Basel (1998)MATHGoogle Scholar
  8. 8.
    Hilditch, C.J.: Linear skeletons from square cupboards. In: Meltzer, B., Michie, D. (eds.) Machine Intelligence IV, pp. 403–420. Edinburgh University Press (1969)Google Scholar
  9. 9.
    Kong, T.Y.: On the problem of determining whether a parallel reduction operator for n-dimensional binary images always preserves topology. In: Melter, R.A., Wu, A.Y. (eds.) Vision Geometry II, Proceeding, Proc. SPIE, Boston, September 1993, vol. 2060, pp. 69–77 (1993)Google Scholar
  10. 10.
    Kong, T.Y.: On topology preservation in 2D and 3D thinning. International Journal of Pattern Recognition and Artificial Intelligence 9, 813–844 (1995)CrossRefGoogle Scholar
  11. 11.
    Kong, T.Y., Gau, C.-J.: Minimal non-simple sets in 4-dimensional binary images with (8,80)-adjacency. In: Klette, R., Žunić, J. (eds.) IWCIA 2004. LNCS, vol. 3322, pp. 318–333. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Kong, T.Y., Roscoe, A.W.: Characterizations of simply-connected finite polyhedra in 3-space. Bulletin of the London Mathematical Society 17, 575–578 (1985)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kong, T.Y., Saha, P.K., Rosenfeld, A.: Strongly normal sets of contractible tiles in N dimensions. Pattern Recognition (in press)Google Scholar
  14. 14.
    Kovalevsky, V.A.: Discrete topology and contour definition. Pattern Recognition Letters 2, 281–288 (1984)CrossRefGoogle Scholar
  15. 15.
    Ma, C.M.: On topology preservation in 3D thinning. CVGIP: Image Understanding 59, 328–339 (1994)CrossRefGoogle Scholar
  16. 16.
    Maunder, C.R.F.: Algebraic Topology. Dover Publications (1996)Google Scholar
  17. 17.
    Moise, E.E.: Geometric Topology in Dimensions 2 and 3. Springer, Heidelberg (1977)MATHGoogle Scholar
  18. 18.
    Ronse, C.: A topological characterization of thinning. Theoretical Computer Science 43, 31–41 (1986)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ronse, C.: Minimal test patterns for connectivity preservation in parallel thinning algorithms for binary digital images. Discrete Applied Mathematics 21, 67–79 (1988)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Rosenfeld, A.: Connectivity in digital pictures. Journal of the Association for Computing Machinery 17, 146–160 (1970)MATHMathSciNetGoogle Scholar
  21. 21.
    Saha, P.K., Chaudhuri, B.B.: Detection of 3D simple points for topology preserving transformation with applications to thinning. IEEE Transactions on Pattern Analysis and Machine Intelligence 16, 1028–1032 (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • T. Yung Kong
    • 1
  1. 1.Department of Computer ScienceQueens College, City University of New YorkFlushingU.S.A.

Personalised recommendations