Flat Morphological Operators on Arbitrary Power Lattices

  • Christian Ronse
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2616)


We give here the basis for a general theory of flat morphological operators for functions defined on a space E and taking their values in an arbitrary complete lattice V . Contrarily to Heijmans [4], [6], we make no assumption on the complete lattice V , and in contrast with Serra [18], we rely exclusively on the usual construction of flat operators by thresholding and stacking.

Some known properties of flat operators for numerical functions (V = \( \overline Z \) or \( \overline R \) ) extend to this general framework: flat dilations and erosions, flat extension of a union of operators or of a composition of an operator by a dilation. Others don’t, unless V is completely distributive: flat extension of an intersection or of a composition of operators; for these we give counterexamples with V being the non-distributive lattice of labels. In another paper [15], we will consider the commutation of flat operators with anamorphoses (contrast functions) and thresholdings, duality by inversion, as well as related questions of continuity.


Complete Lattice Mathematical Morphology Morphological Operator Contrast Function Complete Chain 
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  1. [1]
    V. Agnus, C. Ronse, F. Heitz: Segmentation spatiotemporelle morphologique de séquences d’images. In RFIA’2000: 12ème Congr`es Francophone “Reconnaissance des Formes et Intelligence Artificielle”, Paris, France (2000), Vol. 1, pp. 619–627.Google Scholar
  2. [2]
    V. Agnus, C. Ronse, F. Heitz: Spatio-temporal segmentation using morphological tools. Proc. 15th International Conference on Pattern Recognition, Barcelona, Spain (2000), Vol. 3, pp. 885–888.Google Scholar
  3. [3]
    G. Birkhoff: Lattice Theory (3rd edition), American Mathematical Society Colloquium Publications, Vol. 25, Providence, RI (1984).Google Scholar
  4. [4]
    H. J. A. M. Heijmans: Theoretical aspects of gray-level morphology. IEEE Trans. Pattern Analysis & Machine Intelligence, Vol. 13 (1991), pp. 568–582.CrossRefGoogle Scholar
  5. [5]
    H. J. A. M. Heijmans: From binary to grey-level morphology. Unpublished (1991).Google Scholar
  6. [6]
    H. J. A. M. Heijmans: Morphological Image Operators, Acad. Press, Boston, MA (1994).zbMATHGoogle Scholar
  7. [7]
    H. J. A. M. Heijmans, R. Keshet: Inf-semilattice approach to self-dual morphology. J. Mathematical Imaging & Vision, to appear (2002).Google Scholar
  8. [8]
    H. J. A. M. Heijmans, C. Ronse: The algebraic basis of mathematical morphology I: dilations and erosions. Computer Vision, Graphics & Image Processing, Vol. 50, no. 3 (1990), pp. 245–295.zbMATHCrossRefGoogle Scholar
  9. [9]
    R. Kresch: Extensions of morphological operations to complete semilattices and its applications to image and video processing. In H. Heijmans & J. Roerdink, editors, International Symposium on Mathematical Morphology 1998. Mathematical morphology and its applications to image and signal processing IV, pp. 35–42, Kluwer Academic Publishers, June 1998.Google Scholar
  10. [10]
    R. Keshet (Kresch): Mathematical Morphology on complete semilattices and its applications to image processing. Fundamenta Informaticae, Vol. 41 (2000), pp. 33–56.MathSciNetGoogle Scholar
  11. [11]
    P. Maragos, R. W. Schafer: Morphological filters-Part II: Their relations to median, order-statistics, and stack filters. IEEE Trans. Acoustics, Speech and Signal Processing, Vol. 35 (1987), pp. 1170–1184.CrossRefMathSciNetGoogle Scholar
  12. [12]
    G. N. Raney: Completely distributive complete lattices. Proceedings of the American Mathematical Society, Vol. 3 (1952), pp. 677–680.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    C. Ronse: Order-configuration functions: mathematical characterizations and applications to digital signal and image processing. Information Sciences, Vol. 50, no. 3 (1990), pp. 275–327.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    C. Ronse: Why mathematical morphology needs complete lattices. Signal Processing, Vol. 21, no. 2 (1990), pp. 129–154.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    C. Ronse: Anamorphoses and flat morphological operators on power lattices. In preparation.Google Scholar
  16. [16]
    C. Ronse, V. Agnus: Morphology on label images, and applications to video sequence processing. In preparation.Google Scholar
  17. [17]
    J. Serra: Image Analysis and Mathematical Morphology, Vol. 2: Theoretical Advances. Academic Press, London, 1988.Google Scholar
  18. [18]
    J. Serra: Anamorphoses and function lattices (multivalued morphology). In E. R. Dougherty, editor, Mathematical Morphology in Image Processing, pp. 483–523, Marcel Dekker, New York, 1993.Google Scholar
  19. [19]
    P. D. Wendt, E. J. Coyle, N. C. Callagher: Stack Filters. IEEE Trans. Acoustics, Speech and Signal Processing, Vol. 34(1986), pp. 898–911.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Christian Ronse
    • 1
  1. 1.LSIIT UMR 7005 CNRS-ULPIllkirchFrance

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