Discrete & Computational Geometry

, Volume 43, Issue 4, pp 893–913 | Cite as

On 2-dimensional Simple Sets in n-dimensional Cubic Grids

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Abstract

Preserving topological properties of objects during reduction procedures is an important issue in the field of discrete image analysis. Such procedures are generally based on the notion of simple point, the exclusive use of which may result in the appearance of “topological artifacts.” This limitation leads to consider a more general category of objects, the simple sets, which also enable topology-preserving image reduction. A study of two-dimensional simple sets in two-dimensional spaces has been proposed recently. This article is devoted to the study of two-dimensional simple sets in spaces of higher dimension (i.e., n-dimensional spaces, n≥3). In particular, several properties of minimal simple sets (i.e., which do not strictly include any other simple sets) are proposed, leading to a characterisation theorem. It is also proved that the removal of a two-dimensional simple set from an object can be performed by only considering the minimal ones, thus authorising the development of efficient thinning algorithms.

Keywords

Digital topology Thinning Topology preservation Simple sets Cubical complexes n-dimensional spaces 

References

  1. 1.
    Duda, O., Hart, P.E., Munson, J.H.: Graphical data processing research study and experimental investigation. Tech. Rep. AD650926, Stanford Research Institute (1967) Google Scholar
  2. 2.
    Davies, E.R., Plummer, A.P.N.: Thinning algorithms: A critique and a new methodology. Pattern Recognit. 14(1–6), 53–63 (1981) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Kong, T.Y., Rosenfeld, A.: Digital topology: Introduction and survey. Comput. Vis. Graph. Image Process. 48(3), 357–393 (1989) CrossRefGoogle Scholar
  4. 4.
    Rosenfeld, A.: Connectivity in digital pictures. J. Assoc. Comput. Mach. 17(1), 146–160 (1970) MATHMathSciNetGoogle Scholar
  5. 5.
    Bertrand, G., Malandain, G.: A new characterization of three-dimensional simple points. Pattern Recognit. Lett. 15(2), 169–175 (1994) MATHCrossRefGoogle Scholar
  6. 6.
    Kong, T.Y.: Topology-preserving deletion of 1’s from 2-, 3- and 4-dimensional binary images. In: Ahronovitz, E., Fiorio, C. (eds.) Discrete Geometry for Computer Imagery—DGCI’97, 7th International Workshop, Proceedings. Lecture Notes in Computer Science, vol. 1347, pp. 3–18. Springer, Berlin (1997) Google Scholar
  7. 7.
    Fourey, S., Malgouyres, R.: A concise characterization of 3D simple points. Discrete Appl. Math. 125(1), 59–80 (2003) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    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
  9. 9.
    Bertrand, G.: On P-simple points. C.R. Acad. Sci. Sér. Math. 1(321), 1077–1084 (1995) MathSciNetGoogle Scholar
  10. 10.
    Ma, C.M.: On topology preservation in 3D thinning. Comput. Vis. Graph. Image Process. 59(3), 328–339 (1994) Google Scholar
  11. 11.
    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
  12. 12.
    Ronse, C.: A topological characterization of thinning. Theor. Comput. Sci. 43(1), 31–41 (1986) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    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
  14. 14.
    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
  15. 15.
    Passat, N., Mazo, L.: An introduction to simple sets. Tech. Rep. LSIIT-2008-1, Université Strasbourg 1 (2008, submitted) Google Scholar
  16. 16.
    Passat, N., Couprie, M., Mazo, L., Bertrand, G.: Topological properties of thinning in 2-D pseudomanifolds. Tech. Rep. IGM2008-5, Université Marne-la-Vallée (2008, submitted) Google Scholar
  17. 17.
    Kovalesky, V.A.: Finite topology as applied to image analysis. Comput. Vis. Graph. Image Process. 46(2), 141–161 (1989) CrossRefGoogle Scholar
  18. 18.
    Bertrand, G.: On critical kernels. C.R. Acad. Sci. Sér. Math. 1(345), 363–367 (2007) MathSciNetGoogle Scholar
  19. 19.
    Zeeman, E.C.: On the dunce hat. Topology 2, 341–358 (1964) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Bing, R.H.: Some aspects of the topology of 3-manifolds related to the Poincaré conjecture. In: Lectures on Modern Mathematics II, pp. 93–128 (1964) Google Scholar
  21. 21.
    González-Díaz, R., Real, P.: On the cohomology of 3D digital images. Discrete Appl. Math. 147(2–3), 245–263 (2005) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Peltier, S., Alayrangues, S., Fuchs, L., Lachaud, J.O.: Computation of homology groups and generators. Comput. Graph. 30(1), 62–69 (2006) CrossRefGoogle Scholar
  23. 23.
    Mrozek, M., Pilarczyk, P., Żelazna, N.: Homology algorithm based on acyclic subspace. Comput. Math. Appl. 55(11), 2395–2412 (2008) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Zeeman, E.C.: Seminar on Combinatorial Topology. IHES (1963) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Université de Strasbourg, LSIIT, UMR CNRS 7005Parc d’InnovationIllkirch CedexFrance

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