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Removal and Contraction for n-Dimensional Generalized Maps

  • Guillaume Damiand
  • Pascal Lienhardt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2886)

Abstract

Removal and contraction are basic operations for several methods conceived in order to handle irregular image pyramids, for multi-level image analysis for instance. Such methods are often based upon graph-like representations which do not maintain all topological information, even for 2-dimensional images. We study the definitions of removal and contraction operations in the generalized maps framework. These combinatorial structures enable us to unambiguously represent the topology of a well-known class of subdivisions of n-dimensional (discrete) spaces. The results of this study make a basis for a further work about irregular pyramids of n-dimensional images.

Keywords

Removal contraction irregular pyramids generalized maps 

References

  1. 1.
    Burt, P., Hong, T.H., Rosenfeld, A.: Segmentation and estimation of image region properties through cooperative hierarchical computation. IEEE Transactions on Systems, Man and Cybernetics 11, 802–809 (1981)CrossRefGoogle Scholar
  2. 2.
    Meer, P.: Stochastic image pyramids. Computer Vision, Graphics and Image Processing 45, 269–294 (1989)CrossRefGoogle Scholar
  3. 3.
    Montanvert, A., Meer, P., Rosenfeld, A.: Hierarchical image analysis using irregular tesselations. IEEE Transactions on Pattern Analysis and Machine Intelligence 13, 307–316 (1991)CrossRefGoogle Scholar
  4. 4.
    Jolion, J., Montanvert, A.: The adaptive pyramid: a framework for 2d image analysis. Computer Vision, Graphics and Image Processing 55, 339–348 (1992)zbMATHGoogle Scholar
  5. 5.
    Kropatsch, W.: Abstraction pyramids on discrete representations. In: Braquelaire, A., Lachaud, J.-O., Vialard, A. (eds.) DGCI 2002. LNCS, vol. 2301, pp. 1–21. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Kropatsch, W.: Building irregular pyramids by dual graph contraction. Technical report PRIP-TR-35, Dept. for Pattern Recognition and Image Processing, Institute for Automation, Technical Univerity of Vienna, Austria (1994)Google Scholar
  7. 7.
    Kropatsch, W.: Building irregular pyramids by dual-graph contraction. Vision, Image and Signal Processing 142, 366–374 (1995)CrossRefGoogle Scholar
  8. 8.
    Baumgart, B.: A polyhedron representation for computer vision. In: Samelson, K. (ed.) ECI 1976. LNCS, vol. 44, pp. 589–596. Springer, Heidelberg (1976)Google Scholar
  9. 9.
    Guibas, L., Stolfi, J.: Primitives for the manipulation of general subdivisions and the computation of voronoí diagrams. ACM Transactions on Graphics 4, 74–123 (1985)zbMATHCrossRefGoogle Scholar
  10. 10.
    Dobkin, D., Laszlo, M.: Primitives for the manipulation of three-dimensional subdivisions. Algorithmica 4, 3–32 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Brisson, E.: Representing geometric structures in d dimensions: topology and order. Discrete Comput. Geom. 9, 387–426 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lienhardt, P.: N-dimensional generalized combinatorial maps and cellular quasimanifolds. International Journal of Computational Geometry and Applications 4, 275–324 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    de Floriani, L., Mesmoudi, M., Morando, F., Puppo, E.: Non-manifold decomposition in arbitrary dimensions. In: Braquelaire, A., Lachaud, J.-O., Vialard, A. (eds.) DGCI 2002. LNCS, vol. 2301, pp. 69–80. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  14. 14.
    Edmonds, J.: A combinatorial representation for polyhedral surfaces. Notices of the American Mathematical Society 7 (1960)Google Scholar
  15. 15.
    Jacques, A.: Constellations et graphes topologiques. In: Combinatorial Theory and Applications, vol. 2, pp. 657–673 (1970)Google Scholar
  16. 16.
    Cori, R.: Un code pour les graphes planaires et ses applications. PhD thesis, Universitè de Paris VII (1973)Google Scholar
  17. 17.
    Elter, H., Lienhardt, P.: Cellular complexes as structured semi-simplicial sets. International Journal of Shape Modeling 1, 191–217 (1994)zbMATHCrossRefGoogle Scholar
  18. 18.
    Brun, L.: Segmentation d’images couleur à base topologique. PhD thesis, Universit é de Bordeaux I (1996)Google Scholar
  19. 19.
    Fiorio, C.: Approche interpixel en analyse d’images: une topologie et des algorithmes de segmentation. PhD thesis, Université Montpellier II (1995)Google Scholar
  20. 20.
    Braquelaire, J., Desbarats, P., Domenger, J., Wüthrich, C.: A topological structuring for aggregates of 3d discrete objects. In: Workshop on Graph based representations, Austria, IAPR-TC15, pp. 193–202 (1999)Google Scholar
  21. 21.
    Bertrand, Y., Damiand, G., Fiorio, C.: Topological encoding of 3d segmented images. In: Nyström, I., Sanniti di Baja, G., Borgefors, G. (eds.) DGCI 2000. LNCS, vol. 1953, pp. 311–324. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  22. 22.
    Bertrand, Y., Damiand, G., Fiorio, C.: Topological map: minimal encoding of 3d segmented images. In: Workshop on Graph based representations, Ischia, Italy, IAPR-TC15, pp. 64–73 (2001)Google Scholar
  23. 23.
    Braquelaire, J., Desbarats, P., Domenger, J.: 3d split and merge with 3-maps. In: Workshop on Graph based representations, Ischia, Italy, IAPR-TC15, pp. 32–43 (2001)Google Scholar
  24. 24.
    Damiand, G.: Définition et étude d’un modèle topologique minimal de représentation d’images 2d et 3d. PhD thesis, Université de Montpellier II (2001)Google Scholar
  25. 25.
    Brun, L., Kropatsch, W.: Dual contraction of combinatorial maps. In: Workshop on Graph based representations, Austria, IAPR-TC15, pp. 145–154 (1999)Google Scholar
  26. 26.
    Brun, L., Kropatsch, W.: Pyramids with combinatorial maps. Technical report 57, Institute of Computer Aided Automation, Vienna University of Technology, Austria (1999), URL: http://www.prip.tuwien.ac.at/
  27. 27.
    Brun, L., Kropatsch, W.: The construction of pyramids with combinatorial maps. Technical report 63, Institute of Computer Aided Automation, Vienna University of Technology, Austria (2000), URL: http://www.prip.tuwien.ac.at/
  28. 28.
    Brun, L., Kropatsch, W.: Contraction kernels and combinatorial maps. In: Workshop on Graph based representations, Ischia, Italy, IAPR-TC15, pp. 12–21 (2001)Google Scholar
  29. 29.
    Lienhardt, P.: Topological models for boundary representation: a comparison with n-dimensional generalized maps. Commputer Aided Design 23, 59–82 (1991)zbMATHGoogle Scholar
  30. 30.
    Elter, H.: Etude de structures combinatoires pour la représentation de complexes cellulaires. PhD thesis, Université Louis-Pasteur, Strasbourg (1994)Google Scholar
  31. 31.
    Damiand, G., Lienhardt, P.: Removal and contraction for n-dimensional generalized maps. Technical Report 2003-01, Laboratoire IRCOM-SIC (2003), URL: http://damiands.free.fr/
  32. 32.
    Brun, L., Kropatsch, W.: Receptive fields within the combinatorial pyramid framework. In: Braquelaire, A., Lachaud, J.-O., Vialard, A. (eds.) DGCI 2002. LNCS, vol. 2301, pp. 92–101. Springer, Heidelberg (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Guillaume Damiand
    • 1
  • Pascal Lienhardt
    • 1
  1. 1.IRCOM-SICUMR-CNRS 6615Futuroscope Chasseneuil CedexFrance

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