Similarity between Hypergraphs Based on Mathematical Morphology

  • Isabelle Bloch
  • Alain Bretto
  • Aurélie Leborgne
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7883)


In the framework of structural representations for applications in image understanding, we establish links between similarities, hypergraph theory and mathematical morphology. We propose new similarity measures and pseudo-metrics on lattices of hypergraphs based on morphological operators. New forms of these operators on hypergraphs are introduced as well. Some examples based on various dilations and openings on hypergraphs illustrate the relevance of our approach.


Hypergraphs similarity pseudo-metric valuation mathematical morphology on hypergraphs 


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  1. 1.
    Birkhoff, G.: Lattice theory, 3rd edn., vol. 25. American Mathematical Society (1979)Google Scholar
  2. 2.
    Bloch, I., Bretto, A.: Mathematical Morphology on Hypergraphs: Preliminary Definitions and Results. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds.) DGCI 2011. LNCS, vol. 6607, pp. 429–440. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Bloch, I., Bretto, A.: Mathematical morphology on hypergraphs, application to similarity and positive kernel. Computer Vision and Image Understanding 117(4), 342–354 (2013)CrossRefGoogle Scholar
  4. 4.
    Bloch, I., Heijmans, H., Ronse, C.: Mathematical Morphology. In: Aiello, M., Pratt-Hartman, I., van Benthem, J. (eds.) Handbook of Spatial Logics, ch. 13, pp. 857–947. Springer (2007)Google Scholar
  5. 5.
    Bretto, A.: Hypergraph Theory: an Introduction. Mathematical Engineering. Springer (2013)Google Scholar
  6. 6.
    Chen, Y., Garcia, E.K., Gupta, M.R., Rahimi, A., Cazzanti, L.: Similarity-based classification: Concepts and algorithms. Journal of Machine Learning Research 10, 747–776 (2009)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Choi, S.S., Cha, S.H., Tappert, C.C.: A survey of binary similarity and distance measures. Journal of Systemics, Cybernetics and Informatics 8(1), 43–48 (2010)Google Scholar
  8. 8.
    Cousty, J., Najman, L., Dias, F., Serra, J.: Morphological filtering on graphs. Computer Vision and Image Understanding 117, 370–385 (2013)CrossRefGoogle Scholar
  9. 9.
    Dias, F., Cousty, J., Najman, L.: Some Morphological Operators on Simplicial Complex Spaces. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds.) DGCI 2011. LNCS, vol. 6607, pp. 441–452. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Duchenne, O., Bach, F., Kweon, I., Ponce, J.: A tensor-based algorithm for high-order graph matching. IEEE Transactions on Pattern Analysis and Machine Intelligence 33(12), 2383–2395 (2011)CrossRefGoogle Scholar
  11. 11.
    Gao, Y., Wang, M., Tao, D., Ji, R., Dai, Q.: 3-D object retrieval and recognition with hypergraph analysis. IEEE Transactions on Image Processing 21(9), 4290–4303 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Heijmans, H.J.A.M.: Morphological Image Operators. Academic Press, Boston (1994)zbMATHGoogle Scholar
  13. 13.
    Heijmans, H.J.A.M., Ronse, C.: The Algebraic Basis of Mathematical Morphology – Part I: Dilations and Erosions. Computer Vision, Graphics and Image Processing 50(3), 245–295 (1990)zbMATHCrossRefGoogle Scholar
  14. 14.
    Jouili, S., Tabbone, S.: Hypergraph-based image retrieval for graph-based representation. Pattern Recognition 45, 4054–4068 (2012)CrossRefGoogle Scholar
  15. 15.
    Liang, Z., Chi, Z., Fu, H., Feng, D.: Salient object detection using content-sensitive hypergraph representation and partitioning. Pattern Recognition 45, 3886–3901 (2012)CrossRefGoogle Scholar
  16. 16.
    Loménie, N., Stamon, G.: Morphological mesh filtering and α-objects. Pattern Recognition Letters 29(10), 1571–1579 (2008)CrossRefGoogle Scholar
  17. 17.
    Meyer, F., Stawiaski, J.: Morphology on Graphs and Minimum Spanning Trees. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.) ISMM 2009. LNCS, vol. 5720, pp. 161–170. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  18. 18.
    Ronse, C., Heijmans, H.J.A.M.: The Algebraic Basis of Mathematical Morphology – Part II: Openings and Closings. Computer Vision, Graphics and Image Processing 54(1), 74–97 (1991)zbMATHGoogle Scholar
  19. 19.
    Serra, J. (ed.): Image Analysis and Mathematical Morphology, Part II: Theoretical Advances. Academic Press, London (1988)Google Scholar
  20. 20.
    Simovici, D.: Betweenness, metrics and entropies in lattices. In: 38th IEEE International Symposium on Multiple Valued Logic, ISMVL 2008. pp. 26–31 (2008)Google Scholar
  21. 21.
    Ta, V.-T., Elmoataz, A., Lézoray, O.: Partial Difference Equations over Graphs: Morphological Processing of Arbitrary Discrete Data. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part III. LNCS, vol. 5304, pp. 668–680. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  22. 22.
    Vincent, L.: Graphs and Mathematical Morphology. Signal Processing 16(4), 365–388 (1989)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Voloshin, V.I.: Introduction to Graph and Hypergraph Theory. Nova Science Publishers (2009)Google Scholar
  24. 24.
    Zhang, Z., Hancock, E.: Hypergraph based information-theoretic feature selection. Pattern Recognition Letters 33, 1991–1999 (2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Isabelle Bloch
    • 1
  • Alain Bretto
    • 2
  • Aurélie Leborgne
    • 3
  1. 1.Institut Mines-Telecom, Telecom ParisTech, CNRS LTCIParisFrance
  2. 2.GREYC CNRS-UMR 6072CaenFrance
  3. 3.INSA-Lyon, CNRS LIRISUniversité de LyonFrance

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