Topological Properties of Thinning in 2-D Pseudomanifolds

  • Nicolas Passat
  • Michel Couprie
  • Loïc Mazo
  • Gilles Bertrand
Article

Abstract

Preserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2-D digital images (i.e. images defined on ℤ2) such procedures are usually based on the notion of simple point. In contrast to the situation in ℤn, n≥3, it was proved in the 80s that the exclusive use of simple points in ℤ2 was indeed sufficient to develop thinning procedures providing an output that is minimal with respect to the topological characteristics of the object. Based on the recently introduced notion of minimal simple set (generalising the notion of simple point), we establish new properties related to topology-preserving thinning in 2-D spaces which extend, in particular, this classical result to cubical complexes in 2-D pseudomanifolds.

Keywords

Topology preservation Simple points Simple sets Cubical complexes Collapse Confluence Pseudomanifolds 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Mangin, J.-F., Frouin, V., Bloch, I., Régis, J., López-Krahe, J.: From 3D magnetic resonance images to structural representations of the cortex topography using topology preserving deformations. J. Math. Imaging Vis. 5(4), 297–318 (1995) CrossRefGoogle Scholar
  2. 2.
    Faisan, S., Passat, N., Noblet, V., Chabrier, R., Meyer, C.: Topology preserving warping of binary images: application to atlas-based skull segmentation. In: MICCAI’08, Proceedings, Part I. Lecture Notes in Computer Science, vol. 5241, pp. 211–218. Springer, Berlin (2008) Google Scholar
  3. 3.
    Cornea, N., Silver, D., Yuan, X., Balasubramanian, R.: Computing hierarchical curve-skeletons of 3D objects. Vis. Comput. 21(11), 945–955 (2005) CrossRefGoogle Scholar
  4. 4.
    Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vis. Graph. Image Process. 48(3), 357–393 (1989) CrossRefGoogle Scholar
  5. 5.
    Couprie, M., Bertrand, G.: New characterizations of simple points in 2D, 3D and 4D discrete spaces. IEEE Trans. Pattern Anal. Mach. Intell. 31(4), 637–648 (2009) CrossRefGoogle Scholar
  6. 6.
    Davies, E.R., Plummer, A.P.: Thinning algorithms: a critique and a new methodology. Pattern Recognit. 14(16), 53–63 (1981) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Passat, N., Couprie, M., Bertrand, G.: Minimal simple pairs in the 3-D cubic grid. J. Math. Imaging Vis. 32(3), 239–249 (2008) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Rosenfeld, A.: Connectivity in digital pictures. J. Assoc. Comput. Mach. 17(1), 146–160 (1970) MathSciNetMATHGoogle Scholar
  9. 9.
    Rosenfeld, A.: Arcs and curves in digital pictures. J. Assoc. Comput. Mach. 20(1), 81–87 (1973) MathSciNetMATHGoogle Scholar
  10. 10.
    Ronse, C.: A topological characterization of thinning. Theor. Comput. Sci. 43(1), 31–41 (1986) CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Rosenfeld, A.: A characterization of parallel thinning algorithms. Inf. Control 29(3), 286–291 (1975) CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Kong, T.Y., Litherland, R., Rosenfeld, A.: Problems in the topology of binary digital images. In: van Mill, J., Reed, G. (eds.) Open Problems in Topology, pp. 377–385. Elsevier, Amsterdam (1990). Chap. 23 Google Scholar
  13. 13.
    Kovalesky, V.A.: Finite topology as applied to image analysis. Comput. Vis. Graph. Image Process. 46(2), 141–161 (1989) CrossRefGoogle Scholar
  14. 14.
    Kong, T.Y.: Topology-preserving deletion of 1’s from 2-, 3- and 4-dimensional binary images. In: DGCI’97, Proceedings. Lecture Notes in Computer Science, vol.  1347, pp. 3–18. Springer, Berlin (1997) Google Scholar
  15. 15.
    Bertrand, G.: On critical kernels. C. R. Acad. Sci., Ser. Math. 1(345), 363–367 (2007) MathSciNetGoogle Scholar
  16. 16.
    Bertrand, G., Couprie, M.: Two-dimensional parallel thinning algorithms based on critical kernels. J. Math. Imaging Vis. 31(1), 35–56 (2008) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Passat, N., Mazo, L.: An introduction to simple sets. Pattern Recognit. Lett. 30(15), 1366–1377 (2009) CrossRefGoogle Scholar
  18. 18.
    Giblin, P.: Graphs, Surfaces and Homology. Chapman and Hall, London (1981) MATHGoogle Scholar
  19. 19.
    Bertrand, G.: On topological watersheds. J. Math. Imaging Vis. 22(23), 217–230 (2005) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Zeeman, E.C.: Seminar on Combinatorial Topology. IHES (1963) Google Scholar
  21. 21.
    Kong, T.Y.: Minimal non-deletable sets and minimal non-codeletable sets in binary images. Theor. Comput. Sci. 406(12), 97–118 (2008) CrossRefMATHGoogle Scholar
  22. 22.
    Kong, T.Y.: On topology preservation in 2-D and 3-D thinning. Int. J. Pattern Recognit. Artif. Intell. 9(5), 813–844 (1995) CrossRefGoogle Scholar
  23. 23.
    Bing, R.H.: Some aspects of the topology of 3-manifolds related to the Poincaré conjecture. Lect. Mod. Math. II, 93–128 (1964) MathSciNetGoogle Scholar
  24. 24.
    Maunder, C.R.F.: Algebraic Topology. Dover, New York (1996) Google Scholar
  25. 25.
    Mazo, L., Passat, N.: On 2-dimensional simple sets in n-dimensional cubic grids. Discrete Comput. Geom. (in press). doi:10.1007/s00454-009-9195-x
  26. 26.
    Malgouyres, R., Lenoir, A.: Topology preservation within digital surfaces. Graph. Models 62, 71–84 (2000) CrossRefGoogle Scholar
  27. 27.
    Klette, R., Rosenfeld, A.: Digital Geometry. Morgan Kaufmann, San Mateo (2004) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Nicolas Passat
    • 1
  • Michel Couprie
    • 2
  • Loïc Mazo
    • 1
  • Gilles Bertrand
    • 2
  1. 1.LSIIT, UMR CNRS 7005Université de StrasbourgStrasbourgFrance
  2. 2.Laboratoire d’Informatique Gaspard-Monge, Équipe A3SI, ESIEE ParisUniversité Paris-EstMarne-la-ValléeFrance

Personalised recommendations