Metrically independent sets in the digital plane

Abstract

In computer vision a variety of metrics are used to determine the distance between lattice points. The three which are encountered most often are city block distance, chessboard distance and Euclidean distance. A set S of lattice points will be said to be ( r, s)-metrically independent if the congruence of S and T under the metric r implies congruence under the metric s for every digital set T. Necessary and sufficient conditions are obtained for sets to be metrically independent with respect to the three given distances. Conditions on the interpoint distances are also determined which permit a set to be embedded in the digital plane with these metrics.