Universal Spaces for \((k, \overline{k})-\)Surfaces

  • J. C. Ciria
  • E. Domínguez
  • A. R. Francés
  • A. Quintero
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5810)


In the graph–theoretical approach to Digital Topology, the search for a definition of digital surfaces as subsets of voxels is still a work in progress since it was started in the early 1980’s. Despite the interest of the applications in which it is involved (ranging from visualization to image segmentation and graphics), there is not yet a well established general notion of digital surface that naturally extends to higher dimensions (see [5] for a proposal). The fact is that, after the first definition of surface, proposed by Morgenthaler [13] for \({\mathbb Z}^3\) with the usual adjacency pairs (26,6) and (6,26), each new contribution [10,4,9], either increasing the number of surfaces or extendeding the definition to other adjacencies, has still left out some objects considered as surfaces for practical purposes [12].

In this paper we find, for each adjacency pair \((k,{\overline{k}})\), \(k,{\overline{k}}\in\{6, 18, 26\}\) and \((k,{\overline{k}})\neq(6,6)\), a homogeneous \((k,{\overline{k}})\)-connected digital space whose set of digital surfaces is larger than any of those quoted above; moreover, it is the largest set of surfaces within that class of digital spaces as defined in [3]. This is an extension of a previous result for the (26,6)-adjacency in [7].


  1. 1.
    Ayala, R., Domínguez, E., Francés, A.R., Quintero, A.: Digital lighting functions. In: Ahronovitz, E. (ed.) DGCI 1997. LNCS, vol. 1347, pp. 139–150. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  2. 2.
    Ayala, R., Domínguez, E., Francés, A.R., Quintero, A.: A digital index theorem. Int. J. Pattern Recog. Art. Intell. 15(7), 1–22 (2001)zbMATHGoogle Scholar
  3. 3.
    Ayala, R., Domínguez, E., Francés, A.R., Quintero, A.: Weak lighting functions and strong 26 −surfaces. Theoretical Computer Science 283, 29–66 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bertrand, G., Malgouyres, R.: Some topological properties of surfaces in ℤ3. Jour. of Mathematical Imaging and Vision 11, 207–221 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brimkov, V.E., Klette, R.: Curves, hypersurfaces, and good pairs of adjacency relations. In: Klette, R., Žunić, J. (eds.) IWCIA 2004. LNCS, vol. 3322, pp. 276–290. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Ciria, J.C., Domínguez, E., Francés, A.R.: Separation theorems for simplicity 26 −surfaces. In: Braquelaire, A., Lachaud, J.-O., Vialard, A. (eds.) DGCI 2002. LNCS, vol. 2301, pp. 45–56. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Ciria, J.C., De Miguel, A., Domínguez, E., Francés, A.R., Quintero, A.: A maximum set of (26,6)-connected digital surfaces. In: Klette, R., Žunić, J. (eds.) IWCIA 2004. LNCS, vol. 3322, pp. 291–306. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. 8.
    Ciria, J.C., De Miguel, A., Domínguez, E., Francés, A.R., Quintero, A.: Local characterization of a maximum set of digital (26,6) −surfaces. Image and Vision Computing 25, 1685–1697 (2007)CrossRefzbMATHGoogle Scholar
  9. 9.
    Couprie, M., Bertrand, G.: Simplicity surfaces: a new definition of surfaces in \({\mathbb Z}^3\). In: SPIE Vision Geometry V, vol. 3454, pp. 40–51 (1998)Google Scholar
  10. 10.
    Kong, T.Y., Roscoe, A.W.: Continuous analogs fo axiomatized digital surfaces. Comput. Vision Graph. Image Process. 29, 60–86 (1985)CrossRefzbMATHGoogle Scholar
  11. 11.
    Kovalevesky, V.A.: Finite topology as applied to image analysis. Comput. Vis. Graph. Imag. Process. 46, 141–161 (1989)CrossRefGoogle Scholar
  12. 12.
    Malandain, G., Bertrand, G., Ayache, N.: Topological segmentation of discrete surfaces. Int. Jour. of Computer Vision 10(2), 183–197 (1993)CrossRefGoogle Scholar
  13. 13.
    Morgenthaler, D.G., Rosenfeld, A.: Surfaces in three–dimensional digital images. Inform. Control. 51, 227–247 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rourke, C.P., Sanderson, B.J.: Introduction to piecewise–linear topology. Ergebnisse der Math, vol. 69. Springer, Heidelberg (1972)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • J. C. Ciria
    • 1
  • E. Domínguez
    • 1
  • A. R. Francés
    • 1
  • A. Quintero
    • 2
  1. 1.Dpto. de Informática e Ingeniería de Sistemas, Facultad de CienciasUniversidad de ZaragozaZaragozaSpain
  2. 2.Dpto. de Geometría y Topología, Facultad de MatemáticasUniversidad de SevillaSevillaSpain

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