Reconstructing masks from markers in non-distributive lattices
In mathematical morphology for sets (binary images), the geodesic reconstruction associates to a set called mask and a subset of it called marker, the union of all connected components of the mask intersected by the marker. It is obtained by iteration of a geodesic dilation applied to the marker inside the mask. This operation extends naturally to numerical functions (grey-level images), where it allows us to reconstruct flat zones. Considering that the family of images constitutes a complete lattice under some partial order relation, a general theory of geodesic dilations and reconstructions in a complete lattice was given in Ronse and Serra [Fundam Inf 46(4):349-395, 2001]. It relies on the assumption that the lattice is infinitely supremum distributive, and it fails for pictorial objects forming a non-distributive lattice. In this paper we give a more general theory of geodesic operations, that can be applied to non-distributive lattices; it is compatible with the previous theory when the lattice is infinitely supremum distributive. We study one particular form of geodesic operator, the impulsive geodesic dilation, which gives good results for images with values in a bundle lattice (e.g., the lattice of labels and the reference lattice). We also briefly discuss geodesy on the lattice of partitions.
non-distributive lattices mathematical morphology geodesic dilation and reconstruction