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Viscous Lattices

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Abstract

Let E be an arbitrary space, and δ an extensive dilation of P(E) into itself, with an adjoint erosion ε. Then, the image δ[P(E)] of P(E) by δ is a complete lattice P where the sup is the union and the inf the opening of the intersection according to δ ε. The lattice L, named viscous, is not distributive, nor complemented. Any dilation α on P(E) admits the same expression in L. However, the erosion in L is the opening according to δ ε of the erosion in P(E). Given a connection C on P(E) the image of C under δ turns out to be a connection C ′ on L as soon as ε δ (C)⊂eq C. Moreover, the elementary connected openings γ x of C and γ′δ(x) are linked by the relation γ′δ(x) = δγ x ε. A comprehensive class of connection preverving closings ε δ is constructed. Two examples, binary and numerical (the latter comes from the heart imaging), prove the relevance of viscous lattices in interpolation and in segmentation problems.

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Authors and Affiliations

  1. Centre de Morphologie Mathématique, Ecole Nationale Superieure des Mines de Paris, 35, rue Saint-Honoré, 77305, Fontainebleau Cedex, France

    Jean Serra

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  1. Jean Serra
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Correspondence to Jean Serra.

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Jean Serra obtained the degree of Mining Engineer, in 1962 in Nancy, France, and in 1967 his Ph.D. for a work dealing with the estimation of the iron ore body of Lorraine by geostatistics. In cooperation with Georges Matheron, he laid the foundations of a new method, that he called Mathematical Morphology (1964). Its purpose was to describe quantitatively shapes and textures of natural phenomena, at micro and macro scales. In 1967, he founded with G. Matheron, the Centre de Morphologie Mathematique, at School of Mines of Paris, on the campus of Fontainebleau. Since this time, he has been working in this framework as a Directeur de Recherches. His main book is a two-volume treatise entitled “Image Analysis and Mathematical Morphology” (Ac. Press, 1982, 1988). He has been Vice President for Europe of the International Society for Stereology from 1979 to 1983. He founded the International Society for Mathematical Morphology in 1993, and was elected his first president. His achievements include several patents of devices for image processing, various awards and titles, such as the first AFCET award, in 1988, or Doctor Honoris Causa of the University of Barcelona (Spain) in 1993. He recently developed a new theory of segmentation, which is based on set connections (2001–2004), and currently works on colour image processing.

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Serra, J. Viscous Lattices. J Math Imaging Vis 22, 269–282 (2005). https://doi.org/10.1007/s10851-005-4894-2

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