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

TL;DR

This paper extends weighted distance transforms to modules and demonstrates their computation on various point lattices, including FCC and BCC grids, ensuring correctness of the chamfer algorithm.

Contribution

It formalizes weighted distances on modules and proves the correctness of the chamfer algorithm on general point lattices.

Findings

Correctness of the chamfer algorithm on general point lattices.

Application of weighted distances to FCC and BCC grids.

Formalization of weighted distances on modules.

Abstract

This paper presents the generalization of weighted distances to modules and their computation through the chamfer algorithm on general point lattices. The first part is dedicated to formalization of definitions and properties (distance, metric, norm) of weighted distances on modules. It resumes tools found in literature to express the weighted distance of any point of a module and to compute optimal weights in the general case to get rotation invariant distances. The second part of this paper proves that, for any point lattice, the sequential two-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.