Weighted distance transforms generalized to modules and their computation on point lattices

This paper presents the generalization of weighted distances to modules and their computation through the chamfer algorithm on general point-lattices. A first part is devoted to the definitions and properties (distance, metric, norm) of weighted distances on modules, with a presentation of weight optimization in the general case, to get rotation invariant distances. A general formula for the weighted distance on any module is presented. The second part of this paper proves that, for any pointlattice, the sequential 2-scan chamfer algorithm produces correct distance maps. Finally, the definitions and computation of weighted distances are applied to the face-centered cubic (FCC) and body-centered cubic (BCC) grids.