COMPUTER VISION, GRAPHICS, AND IMAGE PROCESSING 4,368-385 (1988) NOTE A Note on “Distance Transformations in Arbitrary Dimensions” P. P. DAS AND B. N. CHATTERJI Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology, Kharagpur-721302, India Received June 3,1987; revised March 18,1988 The upper limit of the absolute error between Euclidean and m-neighbor distance has been presented as a generalization of the results given in [l]. The conjecture regarding least upper Iimit has been proved under a restriction. A counterexample has been formulated to show that d18 in 3D may be greater than the Euclidean distance. The uniqueness of the counterexample has been established. 0 1988 Academic PW, hc. 1. INTRODUCTION In [l] Borgefors has presented the m-neighbor distance transformation (DT) for n-dimensional (n-D) digital pictures and has studied the upper limit (u.1.) of the absolute error between m-neighbor and Euclidean DT over an n-D hypercube of size (M + 1)“. Her study of error, however, is restricted to n = 1, 2, 3, and 4 only. In this note we present a generalization of such errors in arbitrary dimensions. The results, as obtained by Borgefors, are included in Table 1 and Table 2 for easy reference. We also show that her conjecture regarding the least u.1. holds under certain conditions. Though Borgefors has also discussed other classes of distances, in this note we are concerned only with the m-neighbor distances. That is, in all cases to follow, the “local distance” values are restricted to one or infinity. Recently Das, Chakrabarti, and Chatterji [2] have presented the functional form of m-neighbor distance, dz, in n-D. Here, we shall follow the notations there for the sake of convenient representations in n-D. We know that there are exactly n distinct DTs in n-D, each identified by the value of the neighborhood parameter m. In contrast to [l] we restrict m to take values 1,2,. . . , n. However, corresponding to each m we can detlne L(m) = g2’ . (I), where (7) denotes the binomial coeffi- cient, such that in n-D, an m-neighbor DT in our notation is the L(m)-neighbor DT defined by Borgefors. Hence in 2D, d, or a city block is df, d8 or a chessboard is di and in 3D, d, is d:, d,, is di, and dz6 is d:, and so on. Actually the parameter m characterizes the nature of the shortest path of which di is the length. It requires that on any such shortest path a pair of adjacent hypervoxels (hyper- cubes of unit side) should be m-neighbors; that is, they should ditfer in exactly m of their n coordinates by unity. Clearly the same requirement is satisfied by L(m)- neighbor DTs in [l]. Finally, we present a counterexample to show that in 3D, d,, may not be less than the Euclidean distance for all points. 368 0734-189X/88 $3.00 Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form resewed.