Semi-discrete convex order and Laguerre tessellation fitting
Résumé
Laguerre tessellations provide a computationally tractable way to describe a large number of convex partitions of Euclidean space. 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 show that this problem can be reformulated as a Wasserstein projection onto the convex set of discrete measures dominated in convex order by an absolutely continuous measure. We provide a complete characterization of this set and exploit this to construct a regularized projection problem that can be solved efficiently and yields an approximation of the desired reconstruction. The same 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. Interestingly, our regularized problem can also be reinterpreted as a semi-discrete Wasserstein metric extrapolation problem.
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