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Flat Morphology on Power Lattices

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Abstract

Flat morphological operators, also called stack filters, are the natural extension of increasing set operators to grey-level images. The latter are usually modeled as functions \({E\rightarrow T}\), where T is a closed subset of \({\boldmath {\rm \bar R}}\) (for instance, \({\boldmath {\rm {\overline{Z}}}}\) or [a,b]).

We give here a general theory of flat morphological operators for functions defined on a space E of points and taking their values in an arbitrary complete lattice V of values. Several examples of such lattices have been considered in the litterature, and we illustrate our therory with them. Our approach relies on the usual techniques of thresholding and stacking. Some of the usual properties of flat operators for numerical functions extend unconditionally to this general framework. Others do not, unless the lattice V is completely distributive.

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Authors and Affiliations

  1. LSIIT UMR 7005 CNRS-ULP, Parc d’Innovation, Boulevard Sébastien Brant, BP 10413, 67412, ILLKIRCH Cedex, France

    Christian Ronse

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  1. Christian Ronse
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Correspondence to Christian Ronse.

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This paper is dedicated to Henk Heijmans, who made major contributions to the theory of Mathematical Morphology, until a health accident in March 2004 ended his scientific career.

Christian Ronse was born in 1954. He studied pure mathematics at the Université Libre de Bruxelles (Licence, 1976) and the University of Oxford (M.Sc., 1977; Ph.D., 1979), specializing in group theory. Between 1979 and 1991 he was Member of Scientific Staff at the Philips Research Laboratory Brussels, where he conducted research on combinatorics of switching circuits, feedback shift registers, discrete geometry, image processing, and mathematical morphology. During the academic year 1991–1992 he worked at the Université Bordeaux-1, where he obtained his Habilitation diploma. Since October 1992, he has been Professor of Computer Science at the Université Louis Pasteur, Strasbourg (promotion to First Class Professorship in 2001), where he contributed to the development of a research group on image analysis, and the teaching of image processing to students at various levels. His scientific interests include imaging theory, mathematical morphology, image segmentation and medical imaging.

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Ronse, C. Flat Morphology on Power Lattices. J Math Imaging Vis 26, 185–216 (2006). https://doi.org/10.1007/s10851-006-8304-1

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