Weighted Distances and Digital Disks on the Khalimsky Grid

Abstract

In this paper, after providing an appropriate coordinate system, we investigate the weighted distances on the Khalimsky grid. There are two types of natural neighborhood relations, and one semi-neighborhood is also defined. Weighted distances are defined for both cases, i.e., allowing or not the semi-neighborhood. We give formulae for computing the weighted distance of any point-pair on the Khalimsky grid in these cases. Digital disks based on the weighted distances are also investigated. In some cases, these disks may not be convex; moreover, they may contain holes. Sometimes, if semi-neighborhood is allowed, they are not connected, i.e., they contain islands. The conditions of concavities, holes and islands are characterized.