2D Topological Map Isomorphism for Multi-Label Simple Transformation Definition

  • Guillaume Damiand
  • Tristan Roussillon
  • Christine Solnon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8668)


A 2D topological map allows one to fully describe the topology of a labeled image. In this paper we introduce new tools for comparing the topology of two labeled images. First we define 2D topological map isomorphism. We show that isomorphic topological maps correspond to homeomorphic embeddings in the plane and we give a polynomial-time algorithm for deciding of topological map isomorphism. Then we use this notion to give a generic definition of multi-label simple transformation as a set of transformations of labels of pixels which does not modify the topology of the labeled image. We illustrate the interest of multi-label simple transformation by generating look-up tables of small transformations preserving the topology.


Combinatorial maps 2D topological maps isomorphism labeled image simple points simple sets 


  1. 1.
    Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley (1974)Google Scholar
  2. 2.
    Bazin, P.-L., Ellingsen, L.M., Pham, D.L.: Digital homeomorphisms in deformable registration. In: Karssemeijer, N., Lelieveldt, B. (eds.) IPMI 2007. LNCS, vol. 4584, pp. 211–222. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Damiand, G., Bertrand, Y., Fiorio, C.: Topological model for two-dimensional image representation: Definition and optimal extraction algorithm. Computer Vision and Image Understanding 93(2), 111–154 (2004)CrossRefGoogle Scholar
  4. 4.
    Damiand, G., Dupas, A., Lachaud, J.-O.: Combining topological maps, multi-label simple points, and minimum-length polygons for efficient digital partition model. In: Aggarwal, J.K., Barneva, R.P., Brimkov, V.E., Koroutchev, K.N., Korutcheva, E.R. (eds.) IWCIA 2011. LNCS, vol. 6636, pp. 56–69. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. 5.
    Damiand, G., Solnon, C., de la Higuera, C., Janodet, J.-C., Samuel, E.: Polynomial algorithms for subisomorphism of nd open combinatorial maps. Computer Vision and Image Understanding 115(7), 996–1010 (2011)CrossRefGoogle Scholar
  6. 6.
    Dupas, A., Damiand, G., Lachaud, J.-O.: Multi-label simple points definition for 3D images digital deformable model. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 156–167. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Gosselin, S., Damiand, G., Solnon, C.: Efficient search of combinatorial maps using signatures. Theoretical Computer Science 412(15), 1392–1405 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Khalimsky, E., Kopperman, R., Meyer, P.R.: Computer graphics and connected topologies on finite ordered sets. Topology and its Applications 36, 1–17 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Le Bodic, P., Locteau, H., Adam, S., Héroux, P., Lecourtier, Y., Knippel, A.: Symbol detection using region adjacency graphs and integer linear programming. In: Proc. of ICDAR, Barcelona, Spain, pp. 1320–1324. IEEE Computer Society (July 2009)Google Scholar
  10. 10.
    Lienhardt, P.: N-Dimensional generalized combinatorial maps and cellular quasi-manifolds. International Journal of Computational Geometry and Applications 4(3), 275–324 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Mazo, L.: A framework for label images. In: Ferri, M., Frosini, P., Landi, C., Cerri, A., Di Fabio, B. (eds.) CTIC 2012. LNCS, vol. 7309, pp. 1–10. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  12. 12.
    Rosenfeld, A.: Adjacency in digital pictures. Information and Control 26(1), 24–33 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Rosenfeld, A.: Digital Topology. The American Mathematical Monthly 86(8), 621–630 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Trémeau, A., Colantoni, P.: Regions adjacency graph applied to color image segmentation. IEEE Transactions on Image Processing 9, 735–744 (2000)CrossRefGoogle Scholar
  15. 15.
    Worboys, M.: The maptree: A fine-grained formal representation of space. In: Xiao, N., Kwan, M.-P., Goodchild, M.F., Shekhar, S. (eds.) GIScience 2012. LNCS, vol. 7478, pp. 298–310. Springer, Heidelberg (2012)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Guillaume Damiand
    • 1
  • Tristan Roussillon
    • 1
  • Christine Solnon
    • 1
  1. 1.Université de Lyon, CNRS, LIRIS, UMR5205, INSA-LyonFrance

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