Idempotent Block Splitting on Partial Partitions, II: Non-isotone Operators

Our first paper introduced block splitting operators on the complete lattice of partial partitions, studied their algebraic properties and characterized block splitting openings (kernel operators) in terms of partial connections. In this second paper we study non-isotone idempotent block splitting operators on partial partitions. In particular we analyse the following two constructions:–  the residual combination of block splitting openings, where the (n + 1)-th opening is applied to the “residue” of the n-th one;–  a supremum of operators obtained by composition of a block splitting opening followed by a block selection operator guided by a predicate on sets.These operators belong to two families of idempotent operators generalizing openings that the author studied previously. In the same way as the openings analysed in the first paper, these two types of idempotent operators underlie recent image segmentation approaches due to Serra and Soille.