Flat morphological operators on arbitrary power lattices

• Authors: Christian Ronse
• Venue: Proceedings of the 11th international conference on Theoretical foundations of computer vision
• Year: 2002

We give here the basis for a general theory of flat morphological operators for functions defined on a space E and taking their values in an arbitrary complete lattice V . Contrarily to Heijmans [4, 6], we make no assumption on the complete lattice V, and in contrast with Serra [18], we rely exclusively on the usual construction of flat operators by thresholding and stacking. Some known properties of flat operators for numerical functions (V = Z or R) extend to this general framework: flat dilations and erosions, flat extension of a union of operators or of a composition of an operator by a dilation. Others don't, unless V is completely distributive: flat extension of an intersection or of a composition of operators; for these we give counterexamples with V being the non-distributive lattice of labels. In another paper [15], we will consider the commutation of flat operators with anamorphoses (contrast functions) and thresholdings, duality by inversion, as well as related questions of continuity.