Flat Morphological Operators on Arbitrary Power Lattices
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 , , we make no assumption on the complete lattice V , and in contrast with Serra , 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 , we will consider the commutation of flat operators with anamorphoses (contrast functions) and thresholdings, duality by inversion, as well as related questions of continuity.
KeywordsComplete Lattice Mathematical Morphology Morphological Operator Contrast Function Complete Chain
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- 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
- 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
- G. Birkhoff: Lattice Theory (3rd edition), American Mathematical Society Colloquium Publications, Vol. 25, Providence, RI (1984).Google Scholar
- H. J. A. M. Heijmans: From binary to grey-level morphology. Unpublished (1991).Google Scholar
- H. J. A. M. Heijmans, R. Keshet: Inf-semilattice approach to self-dual morphology. J. Mathematical Imaging & Vision, to appear (2002).Google Scholar
- 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
- C. Ronse: Anamorphoses and flat morphological operators on power lattices. In preparation.Google Scholar
- C. Ronse, V. Agnus: Morphology on label images, and applications to video sequence processing. In preparation.Google Scholar
- J. Serra: Image Analysis and Mathematical Morphology, Vol. 2: Theoretical Advances. Academic Press, London, 1988.Google Scholar
- 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