Lattice-theoretical fixpoint theorems in morphological image filtering

In mathematical morphology, Matheron has described the complete lattice structure of several classes of increasing (isotone) operators, such as filters. These results can be interpreted in terms of fixpoints of certain types of transformations of the lattice of increasing operators. Moreover, Heijmans and Serra have given conditions for the construction of such operators by convergence of an iteration of increasing operators. We recall some known lattice-theoretical fixpoint ideas originating from Tarski's 1955 paper. A slight elaboration on the underlying methodology leads to the aforementioned results and some others as a consequence of the general theory.

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