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Mathematical Morphology Operators over Concept Lattices

  • Jamal Atif
  • Isabelle Bloch
  • Felix Distel
  • Céline Hudelot
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7880)

Abstract

Although mathematical morphology and formal concept analysis are two lattice-based data analysis theories, they are still developed in two disconnected research communities. The aim of this paper is to contribute to fill this gap, beyond the classical relationship between the Galois connections defined by the derivation operators and the adjunctions underlying the algebraic mathematical morphology framework. In particular we define mathematical morphology operators over concept lattices, based on distances, valuations, or neighborhood relations in concept lattices. Their properties are also discussed. These operators provide new tools for reasoning over concept lattices.

Keywords

Complete Lattice Concept Lattice Mathematical Morphology Formal Context Neighborhood Relation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Atif, J., Hudelot, C., Bloch, I.: Abduction in description logics using formal concept analysis and mathematical morphology: Application to image interpretation. In: 8th International Conference on Concept Lattices and Their Applications (CLA 2011), Nancy, Paris, pp. 405–408 (October 2011)Google Scholar
  2. 2.
    Baader, F.: Computing a minimal representation of the subsumption lattice of all conjunctions of concepts defined in a terminology. In: 1st International KRUSE Symposium on Knowledge Retrieval, Use and Storage for Efficiency, pp. 168–178 (1995)Google Scholar
  3. 3.
    Birkhoff, G.: Lattice theory, 3rd edn., vol. 25. American Mathematical Society (1979)Google Scholar
  4. 4.
    Bloch, I.: On Links between Mathematical Morphology and Rough Sets. Pattern Recognition 33(9), 1487–1496 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    Bloch, I., Maître, H.: Fuzzy Mathematical Morphologies: A Comparative Study. Pattern Recognition 28(9), 1341–1387 (1995)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bloch, I., Pino-Pérez, R., Uzcategui, C.: A Unified Treatment of Knowledge Dynamics. In: International Conference on the Principles of Knowledge Representation and Reasoning, KR 2004, Canada, pp. 329–337 (2004)Google Scholar
  8. 8.
    Ferré, S., Ridoux, O.: A logical generalization of formal concept analysis. In: Ganter, B., Mineau, G.W. (eds.) ICCS 2000. LNCS (LNAI), vol. 1867, pp. 371–384. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  9. 9.
    Ganter, B., Wille, R., Franzke, C.: Formal concept analysis: Mathematical foundations. Springer-Verlag New York, Inc. (1997)Google Scholar
  10. 10.
    Heijmans, H.J.A.M.: Morphological Image Operators. Academic Press, Boston (1994)zbMATHGoogle Scholar
  11. 11.
    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, 245–295 (1990)zbMATHCrossRefGoogle Scholar
  12. 12.
    Leclerc, B.: Lattice valuations, medians and majorities. Discrete Mathematics 111(1), 345–356 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Monjardet, B.: Metrics on partially ordered sets–a survey. Discrete Mathematics 35(1), 173–184 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Orum, C., Joslyn, C.: Valuations and metrics on partially ordered sets, arXiv preprint arXiv:0903.2679 (2009)Google Scholar
  15. 15.
    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, 74–97 (1991)zbMATHGoogle Scholar
  16. 16.
    Ronse, C.: Adjunctions on the lattices of partitions and of partial partitions. Applicable Algebra in Engineering, Communication and Computing 21(5), 343–396 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, New-York (1982)zbMATHGoogle Scholar
  18. 18.
    Serra, J. (ed.): Image Analysis and Mathematical Morphology, Part II: Theoretical Advances. Academic Press, London (1988)Google Scholar
  19. 19.
    Simovici, D.: Betweenness, metrics and entropies in lattices. In: 38th IEEE International Symposium on Multiple Valued Logic, ISMVL, pp. 26–31 (2008)Google Scholar
  20. 20.
    Stern, M.: Semimodular lattices: Theory and applications. Cambridge University Press (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Jamal Atif
    • 1
  • Isabelle Bloch
    • 2
  • Felix Distel
    • 3
  • Céline Hudelot
    • 4
  1. 1.LRI - TAOUniversité Paris SudOrsayFrance
  2. 2.Telecom ParisTech - CNRS LTCIInstitut Mines TelecomParisFrance
  3. 3.Fakultät Informatik - Institut für theoretische InformatikTechnische Universität DresdenDresdenGermany
  4. 4.MAS LaboratoryEcole Centrale de ParisFrance

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