Mathematical morphology and poset geometry

The aim of this paper is to characterize morphological convex geometries (resp., antimatroids). We define these two structures by using closure operators, and kernel operators. We show that these convex geometries are equivalent to poset geometries. 2000 Mathematics Subject Classification. 37F20, 06A07. 1. Introduction. Convex geometries are particular closure operators. These objects link the theory of convex sets to combinatorial theory. More precisely poset geometries are basic structures, because any convex geometry can be generated from them (6). Mathematical morphology, introduced by Serra (7) is a very important tool in im- age processing and pattern recognition. The framework of mathematical morphology consists in erosions and dilations (resp., Galois functions) which result in morpholog- ical closures and kernels. These particular operators are well adapted to algorithmic computation (5). Theorem 4.2 proves that poset geometries and morphological geometries defined below are equivalent. The convexity is very important in mathematical morphology (5). So this article tries to demonstrate the relation between combinatorial convexity, mathematical morphology, and image processing.