Digitally Continuous Multivalued Functions

  • Carmen Escribano
  • Antonio Giraldo
  • María Asunción Sastre
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4992)


We introduce in this paper a notion of continuity in digital spaces which extends the usual notion of digital continuity. Our approach uses multivalued maps. We show how the multivalued approach provides a better framework to define topological notions, like retractions, in a far more realistic way than by using just single-valued digitally continuous functions. In particular, we characterize the deletion of simple points, one of the most important processing operations in digital topology, as a particular kind of retraction.


Digital space continuous function simple point retraction 


  1. 1.
    Boxer, L.: Digitally continuous functions. Pattern Recognition Letters 15, 833–839 (1994)zbMATHCrossRefGoogle Scholar
  2. 2.
    Boxer, L.: A Classical Construction for the Digital Fundamental Group. Journal of Mathematical Imaging and Vision 10, 51–62 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Boxer, L.: Properties of Digital Homotopy. Journal of Mathematical Imaging and Vision 22, 19–26 (2005)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Boxer, L.: Homotopy properties of Sphere-Like Digital Images. Journal of Mathematical Imaging and Vision 24, 167–175 (2006)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Burguet, J., Malgouyres, R.: Strong thinning and polyhedric approximation of the surface of a voxel object. Discrete Applied Mathematics 125, 93–114 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fourey, S., Malgouyres, R.: Intersection number and topology preservation within digital surfaces. Theoretical Computer Science 283, 109–150 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hocking, J.G., Young, G.S.: Topology. Addison-Wesley, Reading (1961)zbMATHGoogle Scholar
  8. 8.
    Klette, R., Rosenfeld, A.: Digital Geometry. Elsevier, Amsterdam (2004)zbMATHGoogle Scholar
  9. 9.
    Kong, T.Y., Rosenfeld, A.: Digital Topology: Introduction and survey. Computer Vision, Graphics and Image Processing 48, 357–393 (1989)CrossRefGoogle Scholar
  10. 10.
    Kong, T.Y., Rosenfeld, A. (eds.): Topological algorithms for digital image processing. Elsevier, Amsterdam (1996)Google Scholar
  11. 11.
    Kovalevsky, V.: A new concept for digital geometry. In: Ying-Lie, O., et al. (eds.) Shape in Picture. Proc. of the NATO Advanced Research Workshop, Driebergen, The Netherlands (1992), Computer and Systems Sciences, vol. 126. Springer-Verlag (1994)Google Scholar
  12. 12.
    Malgouyres, R., Lenoir, A.: Topology preservation within digital surfaces. Comput. Graphics Image Process. Mach. Graphics Vision 7, 417–426 (1998)Google Scholar
  13. 13.
    Rosenfeld, A.: Continuous functions in digital pictures. Pattern Recognition Letters 4, 177–184 (1986)zbMATHCrossRefGoogle Scholar
  14. 14.
    Rosenfeld, A., Nakamurab, A.: Local deformations of digital curves. Pattern Recognition Letters 18, 613–620 (1997)CrossRefGoogle Scholar
  15. 15.
    Tsaur, R., Smyth, M.B.: Continuous multifunctions in discrete spaces with applications to fixed point theory. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry, Dagstuhl Seminar 2000. LNCS, vol. 2243, pp. 75–88. Springer, Heidelberg (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Carmen Escribano
    • 1
  • Antonio Giraldo
    • 1
  • María Asunción Sastre
    • 1
  1. 1.Departamento de Matemática Aplicada, Facultad de InformáticaUniversidad PolitécnicaMadridSpain

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