Local distances for distance transformations in two and three dimensions

A unified treatment is presented for determining the distance between two points in a two- or three-dimensional digitized space. Instead of a global Euclidean distance, a distance transformation based upon a sequence of optimal local distances is used. The optimal distance is derived in the context of minimizing the maximum error and the unbiased mean square error. Integer approximations for the local distances are developed for neighborhood sizes of three and five. Minimization is performed over circles and spheres to preserve the symmetries of the neighborhoods. In two dimensions the differences with previously published results are small. The results in three dimensions are new.