Connectivity on Complete Lattices

Abstract

Classically, connectivity is a topological notion for sets, often introduced by means of arcs. A nontopological axiomatics has been proposed by Matheron and Serra. The present paper extends it to complete sup-generated lattices. A connection turns out to be characterized by a family of openings labelled by the sup-generators, which partition each element of the lattice into maximal terms, of zero infima. When combined with partition closings, these openings generate strong sequential alternating filters. Starting from a first connection several others may be designed by acting on some dilations or symmetrical operators. When applying this theory to function lattices, one interprets the so-called connected operators in terms of actual connections, as well as the watershed mappings. But the theory encompasses the numerical functions and extends, among others, to multivariate lattices.

DOI: 10.1023/A:1008324520475

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@article{Serra1996ConnectivityOC, title={Connectivity on Complete Lattices}, author={Jean Paul Fr{\'e}d{\'e}ric Serra}, journal={Journal of Mathematical Imaging and Vision}, year={1996}, volume={9}, pages={231-251} }