# Flat Morphological Operators on Arbitrary Power Lattices

## Abstract

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.

## Keywords

Complete Lattice Mathematical Morphology Morphological Operator Contrast Function Complete Chain## Preview

Unable to display preview. Download preview PDF.

## References

- [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]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]G. Birkhoff:
*Lattice Theory*(3rd edition), American Mathematical Society Colloquium Publications, Vol. 25, Providence, RI (1984).Google Scholar - [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]H. J. A. M. Heijmans: From binary to grey-level morphology.
*Unpublished*(1991).Google Scholar - [6]H. J. A. M. Heijmans:
*Morphological Image Operators*, Acad. Press, Boston, MA (1994).zbMATHGoogle Scholar - [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]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]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]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]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]G. N. Raney: Completely distributive complete lattices.
*Proceedings of the American Mathematical Society*, Vol. 3 (1952), pp. 677–680.zbMATHCrossRefMathSciNetGoogle Scholar - [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]C. Ronse: Why mathematical morphology needs complete lattices.
*Signal Processing*, Vol. 21, no. 2 (1990), pp. 129–154.zbMATHCrossRefMathSciNetGoogle Scholar - [15]C. Ronse: Anamorphoses and flat morphological operators on power lattices.
*In preparation*.Google Scholar - [16]C. Ronse, V. Agnus: Morphology on label images, and applications to video sequence processing.
*In preparation*.Google Scholar - [17]J. Serra:
*Image Analysis and Mathematical Morphology*,*Vol. 2*:*Theoretical Advances*. Academic Press, London, 1988.Google Scholar - [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]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