Properties of a Natural Ordering Relation for Octagonal Neighborhood Sequences

Abstract

13] and the book [16]. In this paper we give a brief survey of the literature on so-called octagonal neighborhood Neighborhood sequences play an important role in sevsequences which are derived from the above notions in a dieral branches of discrete geometry and image processing. rect way. In particular, we focus on results about the strucThe literature of such sequences is very wide. In this patural behavior of such sequences under a nice natural partial per we give a survey on results on a natural partial orderordering relation, which can also be important in several aping relationfor generalized nD octagonal neighborhood seplications. First we briefly summarize the most important quences. As this ordering does not have nice properties for results and directions of investigations of this area. each subset ofsuch neighborhood sequences, we also invesDas, Chakrabarti and Chatterji (see [5]) considered arbitigate another relation and provide several properties for trary periodic sequences of cityblock and chessboard moit. We put special emphasize on neighborhood sequences tions, called periodic neighborhood sequences, and also which generate metrics on Zn. In certain applications it their corresponding generalizations in Zn. Moreover, they can be useful to compare any two neighborhood sequencesestablished a formula for calculating the distance d(p, q; A) however; none of these partial orderings is a total ordering. of any two points p, q C Zn, determined by such a neighFor this purpose, we investigate a norm-like concept, called borhood sequence A. They introduced a natural partial velocity, which fits very well to the natural ordering relaordering relation for periodic neighborhood sequences: if tion. We also define a metric for neighborhood sequences, for two periodic neighborhood sequences A, B we have and investigate its properties. d(p, q; A) < d(p, q; B) for all p, q C Zn then A is "faster" than B. Das [1] investigated the lattice properties of the set of periodic neighborhood sequences and some of its subsets