On approximating Euclidean metrics by digital distances in 2D and 3D

In this paper a geometric approach is suggested to ®nd the closest approximation to Euclidean metric based on geometric measures of the digital circles in 2D and the digital spheres in 3D for the generalized octagonal distances. First we show that the vertices of the digital circles (spheres) for octagonal distances can be suitably approximated as a function of the number of neighborhood types used in the sequence. Then we use these approximate vertex formulae to compute the geometric features in an approximate way. Finally we minimize the errors of these measurements with respect to respective Euclidean discs to identify the best distances. We have veri®ed our results by experimenting with analytical error measures suggested earlier. We have also compared the performances of the good octagonal distances with good weighted distances. It has been found that the best octagonal distance in 2D (f1; 1; 2g) performs equally good with respect to the best one for the weighted distances (h3; 4i). In fact in 3D, the octagonal distance f1; 1; 3g has an edge over the other good weighted distances. Ó 2000 Elsevier Science B.V. All rights reserved.