Distance transforms for three-dimensional grids with non-cubic voxels

Distance transforms on the face-centered cubic (fcc) grid and the body-centered cubic (bcc) grid are examined. Since the voxels on the fcc and bcc grids are better approximations of a Euclidean ball than the cube, the distance transforms (DTs) on these grids can be less rotation dependent than those in Z3, which is a desirable feature. Optimal (according to the error function) weights are calculated and integer approximations of these weights are found. Also, the two-dimensional city block distance is generalized to the fcc and bcc grids by considering a unit distance between gridpoints whose corresponding voxels share a face. A method to compute the DTs is presented. The results are evaluated both theoretically and by actually computing some DTs.