Viscous lattices
Let E be an arbitrary space, and sigma an extensive dilation of P(E) into itself, with an adjoint erosion epsilon. Then, the image sigma[P(E)] of P(E) by delta is a complete lattice L where the sup is the union and the inf the opening of the intersection according to delta epsilon. The lattice L, named viscous, is not distributive, nor complemented. Any dilation alpha on P(E) admits the same expression in L. However, the erosion in L is the opening according to delta epsilon of the erosion in P(E). Given a connection C on P(E) the image of C under 8 turns out to be a connection C' on L as soon as epsilon delta(C) subset of C. Moreover, the elementary connected openings gamma(x) of C and gamma(delta)'((x)) are linked by the relation gamma(delta)'((x)) = delta gamma(x)epsilon. A comprehensive class of connection preverving closings epsilon delta 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.
lattices connections mathematical morphology image interpolation