Equivalence, Partial Order and Lattice of Neighborhood Sequences on the Triangular Grid

Abstract

In (digital) grids, neighbor relation is a crucial concept; digital distances are based on paths through neighbor points. Digital distances are significant, e.g., in digital image processing for giving an approximation of the Euclidean distance and allowing incremental algorithms on images. Neighborhood sequences (i.e., infinite sequences of the possible types of neighbors) are defining digital distances with a lower rotational dependency than the distances based only on a sole neighborhood. They allow one to change the used neighborhood condition in every step along a path. They are defined in various grids, and they can be periodic. Generalized neighborhood sequences do not need to be periodic. In this paper, the triangular grid is studied. An equivalence and two partial order relations on the set of generalized and periodic neighborhood sequences are shown on this grid. The first partial order, the faster relation, is based on distances defined by neighborhood sequences, and it does not provide a lattice but gives a relatively complex relation for neighborhood sequences with a short period. The other partial order, the relation componentwise dominate, defines a complete distributive lattice on the set of generalized neighborhood sequences. Finally, a relation of the above-mentioned relations is established. Important differences regarding the cases of the square and triangular grids are also highlighted.