Fuzzy morphology and fuzzy distances: New definitions and links in both euclidean and geodesic cases

  • Isabelle Bloch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1566)


The aim of this paper is to establish links between fuzzy morphology and fuzzy distances. We show how distances can be derived from fuzzy mathematical morphology, in particular fuzzy dilation, leading to interesting definitions involving membership information and spatial information. Conversely, we propose a way to define fuzzy morphological operations from fuzzy distances. We deal with both the Euclidean and the geodesic cases. In particular in the geodesic case, we propose a definition of fuzzy balls, from which fuzzy geodesic dilation and erosion are derived, based on fuzzy geodesic distances. These new operations enhance the set of fuzzy morphological operators, leading to transformations of a fuzzy set conditionally to another fuzzy set. The proposed definitions are valid in any dimension, in both Euclidean and geodesic cases.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B. De Baets. Idempotent Closing and Opening Operations in Fuzzy Mathematical Morphology. In ISUMA-NAFIPS’95, pages 228–233, College Park, MD, September 1995.Google Scholar
  2. 2.
    I. Bloch. Distances in Fuzzy Sets for Image Processing derived from Fuzzy Mathematical Morphology. In Information Processing and Management of Uncertainty in Knowledge-Based Systems, pages 1307–1312, Granada, Spain, July 1996.Google Scholar
  3. 3.
    I. Bloch. Fuzzy Geodesic Distance in Images. In A. Ralescu and T. Martin, editors, Lecture Notes in Artificial Intelligence: Fuzzy Logic in Artificial Intelligence, towards Intelligent Systems, pages 153–166. Springer Verlag, 1996.Google Scholar
  4. 4.
    I. Bloch. Fuzzy relative position between objects in images: a morphological approach. In IEEE Int. Conf. on Image Processing ICIP’96, volume II, pages 987–990, Lausanne, September 1996.Google Scholar
  5. 5.
    I. Bloch. On Fuzzy Distances and their Use in Image Processing under Imprecision. Technical report, ENST Paris 97D012, 1997.Google Scholar
  6. 6.
    I. Bloch. Fuzzy Geodesic Mathematical Morphology from Fuzzy Geodesic Distance. In ISMM’98, Amsterdam, 1998.Google Scholar
  7. 7.
    I. Bloch. On Links between Fuzzy Morphology and Fuzzy Distances: Euclidean and Geodesic Cases. In IPMU’98, Paris, 1998.Google Scholar
  8. 8.
    I. Bloch and H. Maître. Constructing a fuzzy mathematical morphology: alternative ways. In Second IEEE International Conference on Fuzzy Systems, FUZZ IEEE 93, pages 1303–1308, San Fransisco, California, March 1993.Google Scholar
  9. 9.
    I. Bloch and H. Maître. Fuzzy Mathematical Morphologies: A Comparative Study. Pattern Recognition, 28(9):1341–1387, 1995.MathSciNetCrossRefGoogle Scholar
  10. 10.
    I. Bloch, H. Maître, and M. Anvari. Fuzzy Adjacency between Image Objects. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5(6), 1997.Google Scholar
  11. 11.
    V. di Gesu, M. C. Maccarone, and M. Tripiciano. Mathematical Morphology based on Fuzzy Operators. In R. Lowen and M. Roubens, editors, Fuzzy Logic, pages 477–486. Kluwer Academic, 1993.Google Scholar
  12. 12.
    D. Dubois and H. Prade. Fuzzy Sets and Systems: Theory and Applications. Academic Press, New-York, 1980.Google Scholar
  13. 13.
    J. Serra (Ed.). Image Analysis and Mathematical Morphology, Part II: Theoretical Advances. Academic Press, London. 1988.Google Scholar
  14. 14.
    C. Lantuejoul and F. Maisonneuve. Geodesic Methods in Image Analysis. Pattern Recognition, 17(2):177–187, 1984.MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    A. T. Popov. Morphological Operations on Fuzzy Sets. In IEE Image Processing and its Applications, pages 837–840, Edinburgh, UK, July 1995.Google Scholar
  16. 16.
    A. Rosenfeld. The Fuzzy Geometry of Image Subsets. Pattern Recognition Letters, 2:311–317, 1984.CrossRefGoogle Scholar
  17. 17.
    A. Rosenfeld. Distances between Fuzzy Sets. Pattern Recognition Letters 3:229–233, 1985.MATHCrossRefGoogle Scholar
  18. 18.
    M. Schmitt and J. Mattioli. Morphologie mathématique. Masson, Paris, 1994.Google Scholar
  19. 19.
    J. Serra. Image Analysis and Mathematical Morphology. Academic Press, London, 1982.MATHGoogle Scholar
  20. 20.
    D. Sinha and E. Dougherty. Fuzzy Mathematical Morphology. Journal of Visual Communication and Image Representation, 3(3):286–302, 1992.CrossRefGoogle Scholar
  21. 21.
    D. Sinha and E. R. Dougherty. Fuzzification of Set Inclusion: Theory and Applications. Fuzzy Sets and Systems, 55:15–42, 1993.MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    D. Sinha, P. Sinha, E. R. Dougherty, and S. Batman. Design and Analysis of Fuzzy Morphological Algorithms for Image Processing. IEEE Trans. on Fuzzy Systems, 5(4):570–584, 1997.CrossRefGoogle Scholar
  23. 23.
    L. A. Zadeh. The Concept of a Linguistic Variable and its Application to Approximate Reasoning. Information Sciences, 8:199–249, 1975.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Isabelle Bloch
    • 1
  1. 1.École Nationale Supérieure des Télécommunications, département ImagesCNRS URA820ParisFrance

Personalised recommendations