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Fuzzy Bipolar Mathematical Morphology: A General Algebraic Setting

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Mathematical Morphology and Its Applications to Image and Signal Processing (ISMM 2011)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 6671))

Abstract

Bipolar information is an important component in information processing, to handle both positive information (e.g. preferences) and negative information (e.g. constraints) in an asymmetric way. In this paper, a general algebraic framework is proposed to handle such information using mathematical morphology operators, leading to results that apply to any partial ordering.

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References

  1. Dubois, D., Kaci, S., Prade, H.: Bipolarity in Reasoning and Decision, an Introduction. In: International Conference on Information Processing and Management of Uncertainty, IPMU 2004, Perugia, Italy, pp. 959–966 (2004)

    Google Scholar 

  2. Bloch, I.: Dilation and erosion of spatial bipolar fuzzy sets. In: Masulli, F., Mitra, S., Pasi, G. (eds.) WILF 2007. LNCS (LNAI), vol. 4578, pp. 385–393. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  3. Bloch, I.: Bipolar fuzzy mathematical morphology for spatial reasoning. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.) ISMM 2009. LNCS, vol. 5720, pp. 24–34. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  4. Bloch, I.: Bipolar Fuzzy Spatial Information: Geometry, Morphology, Spatial Reasoning. In: Jeansoulin, R., Papini, O., Prade, H., Schockaert, S. (eds.) Methods for Handling Imperfect Spatial Information, pp. 75–102. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  5. Bloch, I.: Lattices of fuzzy sets and bipolar fuzzy sets, and mathematical morphology. Information Sciences 181, 2002–2015 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Nachtegael, M., Sussner, P., Mélange, T., Kerre, E.: Some Aspects of Interval-Valued and Intuitionistic Fuzzy Mathematical Morphology. In: IPCV 2008 (2008)

    Google Scholar 

  7. Mélange, T., Nachtegael, M., Sussner, P., Kerre, E.: Basic Properties of the Interval-Valued Fuzzy Morphological Operators. In: IEEE World Congress on Computational Intelligence, WCCI 2010, Barcelona, Spain, pp. 822–829 (2010)

    Google Scholar 

  8. Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)

    MATH  Google Scholar 

  9. Ronse, C.: Why Mathematical Morphology Needs Complete Lattices. Signal Processing 21(2), 129–154 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Keshet, R.: Mathematical Morphology on Complete Semilattices and its Applications to Image Processing. Fundamenta Informaticae 41, 33–56 (2000)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  12. Heijmans, H.: Morphological Image Operators. Academic Press, Boston (1994)

    MATH  Google Scholar 

  13. Grabisch, M., Greco, S., Pirlot, M.: Bipolar and bivariate models in multicriteria decision analysis: Descriptive and constructive approaches. International Journal of Intelligent Systems 23(9), 930–969 (2008)

    Article  MATH  Google Scholar 

  14. Öztürk, M., Tsoukias, A.: Bipolar preference modeling and aggregation in decision support. International Journal of Intelligent Systems 23(9), 970–984 (2008)

    Article  MATH  Google Scholar 

  15. Konieczny, S., Marquis, P., Besnard, P.: Bipolarity in bilattice logics. International Journal of Intelligent Systems 23(10), 1046–1061 (2008)

    Article  MATH  Google Scholar 

  16. Dubois, D., Prade, H.: An Overview of the Asymmetric Bipolar Representation of Positive and Negative Information in Possibility Theory. Fuzzy Sets and Systems 160, 1355–1366 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Aptoula, E., Lefèvre, S.: A Comparative Study in Multivariate Mathematical Morphology. Pattern Recognition 40, 2914–2929 (2007)

    Article  MATH  Google Scholar 

  18. Bouyssou, D., Dubois, D., Pirlot, M., Prade, H.: Concepts and Methods of Decision-Making. In: ISTE. Wiley, Chichester (2009)

    Google Scholar 

  19. Bloch, I.: Mathematical morphology on bipolar fuzzy sets: general algebraic framework. Technical Report 2010D024, Télécom ParisTech (November 2010)

    Google Scholar 

  20. Atanassov, K.T.: Intuitionistic Fuzzy Sets. Fuzzy Sets and Systems 20, 87–96 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zadeh, L.A.: The Concept of a Linguistic Variable and its Application to Approximate Reasoning. Information Sciences 8, 199–249 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  22. Neumaier, A.: Clouds, fuzzy sets, and probability intervals. Reliable Computing 10(4), 249–272 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dubois, D., Gottwald, S., Hajek, P., Kacprzyk, J., Prade, H.: Terminology Difficulties in Fuzzy Set Theory – The Case of “Intuitionistic Fuzzy Sets”. Fuzzy Sets and Systems 156, 485–491 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Deschrijver, G., Cornelis, C., Kerre, E.: On the Representation of Intuitionistic Fuzzy t-Norms and t-Conorms. IEEE Transactions on Fuzzy Systems 12(1), 45–61 (2004)

    Article  MATH  Google Scholar 

  25. Bloch, I., Maître, H.: Fuzzy Mathematical Morphologies: A Comparative Study. Pattern Recognition 28(9), 1341–1387 (1995)

    Article  MathSciNet  Google Scholar 

  26. Bloch, I.: Duality vs. Adjunction for Fuzzy Mathematical Morphology and General Form of Fuzzy Erosions and Dilations. Fuzzy Sets and Systems 160, 1858–1867 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Bloch, I. (2011). Fuzzy Bipolar Mathematical Morphology: A General Algebraic Setting. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds) Mathematical Morphology and Its Applications to Image and Signal Processing. ISMM 2011. Lecture Notes in Computer Science, vol 6671. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21569-8_2

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  • DOI: https://doi.org/10.1007/978-3-642-21569-8_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-21568-1

  • Online ISBN: 978-3-642-21569-8

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