Connectivity on Complete Lattices

  • Authors:

  • Jean Serra

  • Affiliations:

  • Centre de Morphologie Mathématique, Ecole des Mines de Paris, 35, rue Saint-Honoré, 77305 Fontainebleau, France. E-mail: serra@cmm.ensmp.fr

  • Venue:
  • Journal of Mathematical Imaging and Vision
  • Year:
  • 1998

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Abstract

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