Similarity and Symmetry Measures for Convex Shapes Using Minkowski Addition

This paper is devoted to similarity and symmetry measures for convex shapes whose deenition is based on Minkowski addition and the Brunn-Minkowski inequality. This means in particular that these measures are region-based, in contrast to most of the literature, where one considers contour-based measures. All measures considered in this paper are invariant under translations; furthermore, they can be chosen to be invariant under rotations, multiplications, reeections, or the class of aane transformations. It is shown that the mixed volume of a convex polygon and a rotation of another convex polygon over an angle is a piecewise concave function of. This and other results of a similar nature form the basis for the development of eecient algorithms for the computation of the given measures. Various results obtained in this paper are illustrated by experimental data. Although the paper deals exclusively with the 2-dimensional case, many of the theoretical results carry over almost directly to higher-dimensional spaces.