Morphological decomposition of convex polytopes and its application in discrete image space

We present a new technique for the decomposition of convex structuring elements for morphological image processing. A unique feature of our approach is the use of linear integer programming technique to determine optimal decompositions for different parallel machine architectures. This technique is based on Shephard´s theorem for decomposing Euclidean convex polygons. We formulated the necessary and sufficient conditions to decompose a Euclidean convex polygon into a set of basis convex polygons. We used a set of linear equations to represent the relationships between the edges and the positions of the original convex polygon and those of the basis convex polygons. This is applied to a class of discrete convex polygons in the discrete space. Further, a cost function was used to represent the total processing time for performing dilations on different machine architectures. Then integer programming was used to solve the linear equations based on the cost function. Our technique is general and flexible, so that different cost functions could be used, thus achieving optimal decompositions for different parallel machine architectures