Similarity measures for convex polyhedra based on Minkowski addition

摘要

In this paper we introduce and investigate similarity measures for convex polyhedra based on Minkowski addition and inequalities for the mixed volume and volume related to the Brunn–Minkowski theory. All measures considered are invariant under translations; furthermore, some of them are also invariant under subgroups of the affine transformation group. For the case of rotation and scale invariance, we prove that to obtain the measures based on (mixed) volume, it is sufficient to compute certain functionals only for a finite number of critical rotations. The paper presents a theoretical framework for comparing convex shapes and contains a complexity analysis of the solution. Numerical implementations of the proposed approach are not discussed.