Properties of a Natural Ordering Relation for Octagonal Neighborhood Sequences

Neighborhood sequences play an important role in several branches of discrete geometry and image processing. The literature of such sequences is very wide. In this paper we give a survey on results on a natural partial ordering relation for generalized nD octagonal neighborhood sequences. As this ordering does not have nice properties for each subset of such neighborhood sequences, we also investigate another relation and provide several properties for it. We put special emphasize on neighborhood sequences which generate metrics on Zⁿ. In certain applications it can be useful to compare any two neighborhood sequences -however, none of these partial orderings is a total ordering. For this purpose, we investigate a norm-like concept, called velocity, which fits very well to the natural ordering relation. We also define a metric for neighborhood sequences, and investigate its properties.