Digital Steiner sets and Matheron semi-groups

摘要

There are various ways to define digital convexity in Zn. The proposed approach focuses on structuring elements (and not the sets under study), whose digital versions should allow to construct hierarchies of operators satisfying Matheron semi-groups law γλγμ=γmax(λ,μ), where λ is a size factor. In Rn the convenient class is the Steiner one. Its elements are Minkowski sums of segments. We prove that it admits a digital equivalent when the segments of Zn are Bezout. The conditions under which the Steiner sets are convex in Zn, and are connected, are established. The approach is then extended to structuring elements that vary according to the law of perspective, and also to anamorphoses, so that the digital Steiner class and its properties can extend to digital spaces as a sphere or a torus.