A Unified Topological Framework for Digital Imaging

  • Loïc Mazo
  • Nicolas Passat
  • Michel Couprie
  • Christian Ronse
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6607)

Abstract

In this article, a tractable modus operandi is proposed to model a (binary) digital image (i.e., an image defined on \(\mathbb Z^n\) and equipped with a standard pair of adjacencies) as an image defined in the space of cubical complexes (\(\mathbb F^n\)). In particular, it is shown that all the standard pairs of adjacencies in \(\mathbb Z^n\) can then be correctly modelled in \(\mathbb F^n\). Moreover, it is established that the digital fundamental group of a digital image in \(\mathbb Z^n\) is isomorphic to the fundamental group of its corresponding image in \(\mathbb F^n\), thus proving the topological correctness of the proposed approach. From these results, it becomes possible to establish links between topology-oriented methods developed either in classical digital spaces (\(\mathbb Z^n\)) or cubical complexes (\(\mathbb{F}^n\)).

Keywords

digital imaging digital topology cubical complexes homotopy fundamental group 

References

  1. 1.
    Aktouf, Z., Bertrand, G., Perroton, L.: A three-dimensional holes closing algorithm. Pattern Recognition Letters 23(5), 523–531 (2002)CrossRefMATHGoogle Scholar
  2. 2.
    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
  3. 3.
    Ayala, R., Domínguez, E., Francés, A.R., Quintero, A.: Digital homotopy with obstacles. Discrete Applied Mathematics 139(1-3), 5–30 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bertrand, G.: New notions for discrete topology. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 218–228. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Bertrand, G., Couprie, M.: A model for digital topology. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 229–241. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  6. 6.
    Bertrand, G., Malandain, G.: A new characterization of three-dimensional simple points. Pattern Recognition Letters 15(2), 169–175 (1994)CrossRefMATHGoogle Scholar
  7. 7.
    Faisan, S., Passat, N., Noblet, V., Chabrier, R., Meyer, C.: Topology-preserving warping of binary images according to one-to-one mappings. IEEE Transactions on Image Processing (to appear)Google Scholar
  8. 8.
    Gonzalez-Diaz, R., Jimenez, M.J., Medrano, B.: Well-composed cell complexes (November 2010), http://personal.us.es/majiro/ctic19.pdf, communication at CTIC 2010
  9. 9.
    Khalimsky, E., Kopperman, R., Meyer, P.R.: Computer graphics and connected topologies on finite ordered sets. Topology and its Applications 36(1), 1–17 (1990)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kong, T.Y.: A digital fundamental group. Computers and Graphics 13(2), 159–166 (1989)CrossRefGoogle Scholar
  11. 11.
    Kong, T.Y., Roscoe, A.W.: A theory of binary digital pictures. Computer Vision, Graphics and Image Processing 32(2), 221–243 (1985)CrossRefMATHGoogle Scholar
  12. 12.
    Kong, T.Y.: Topology-preserving deletion of 1’s from 2-, 3- and 4-dimensional binary images. In: Ahronovitz, E. (ed.) DGCI 1997. LNCS, vol. 1347, pp. 3–18. Springer, Heidelberg (1997)Google Scholar
  13. 13.
    Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Computer Vision, Graphics and Image Processing 48(3), 357–393 (1989)CrossRefGoogle Scholar
  14. 14.
    Kovalevsky, V.A.: Finite topology as applied to image analysis. Computer Vision, Graphics, and Image Processing 46(2), 141–161 (1989)CrossRefGoogle Scholar
  15. 15.
    Lachaud, J.O., Montanvert, A.: Continuous analogs of digital boundaries: A topological approach to iso-surfaces. Graphical Models and Image Processing 62, 129–164 (2000)CrossRefGoogle Scholar
  16. 16.
    Latecki, L.J.: 3d well-composed pictures. Graph. Models Image Process. 59(3), 164–172 (1997)CrossRefGoogle Scholar
  17. 17.
    Mazo, L., Passat, N., Couprie, M., Ronse, C.: Digital imaging: A unified topological framework. Tech. Rep. hal-00512270, Université Paris-Est (2010), http://hal.archives-ouvertes.fr/hal-00512270/fr/
  18. 18.
    Mazo, L., Passat, N., Couprie, M., Ronse, C.: Paths, homotopy and reduction in digital images. Acta Applicandae Mathematicae 113(2), 167–193 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Rosenfeld, A.: Connectivity in digital pictures. Journal of the Association for Computer Machinery 17(1), 146–160 (1970)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Loïc Mazo
    • 1
    • 2
  • Nicolas Passat
    • 1
  • Michel Couprie
    • 2
  • Christian Ronse
    • 1
  1. 1.LSIIT, UMR CNRS 7005Université de StrasbourgFrance
  2. 2.Laboratoire d’Informatique Gaspard-Monge, Équipe A3SIUniversité Paris-Est, ESIEEParisFrance

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