Semi-discrete convex order and Laguerre tessellation fitting

Abstract

Laguerre tessellations offer an efficient way to parameterize a large class of convex partitions of Euclidean space using only a set of points and scalar weights. For this reason, they have become popular in computational geometry, imaging and numerical analysis, both as a modeling and a discretization tool. In this paper we study the problem of reconstructing a Laguerre tessellation with prescribed cell volumes from the barycenters of its cells. We establish a geometric interpretation of this problem in terms of the set of discrete measures dominated in convex order by an absolutely continuous measure. In particular, we show that the reconstruction problem can be solved approximately by computing a Wasserstein projection onto this set. More generally, our method can also be applied to fit a Laguerre tessellation to an arbitrary set of barycenters. We give a concrete application of this in materials science, of fitting a Laguerre tessellation to an electron backscatter diffraction (EBSD) image of a steel.