A comparison among distances based on neighborhood sequences in regular grids

Abstract

The theory of neighborhood sequences is applicable in many image-processing algorithms. The theory is well developed for the square grid. Recently there are some results for the hexagonal grid as well. In this paper, we are considering all the three regular grids in the plane. We show that there are some very essential differences occurring. On the triangular plane the distance has metric properties. The distances on the square and the hexagonal case may not meet the triangular inequality. There are non-symmetric distances on the hexagonal case. In addition, contrary to the other two grids, the distance can depend on the order of the initial elements of the neighborhood sequence. Moreover in the hexagonal grid it is possible that circles with different radii are the same (using different neighborhood sequences). On the square grid the circles with the same radius are in a well ordered set, but in the hexagonal case there can be non-comparable circles. © Springer-Verlag Berlin Heidelberg 2005.