Inverting Laguerre tessellations: Recovering tessellations from the volumes and centroids of their cells using optimal transport

In this paper we study an inverse problem in convex geometry, inspired by a problem in materials science. Firstly, we consider the question of whether a Laguerre tessellation (a partition by convex polytopes) can be recovered from only the volumes and centroids of its cells. We show that this problem has a unique solution and give a constructive way of computing it using optimal transport theory and convex optimisation. Secondly, we consider the problem of fitting a Laguerre tessellation to synthetic volume and centroid data. Given some target volumes and centroids, we seek a Laguerre tessellation such that the difference between the volumes and centroids of its cells and the target volumes and centroids is minimised. For an appropriate objective function and suitable data, we prove that local minimisers of this problem can be constructed using convex optimisation. We also illustrate our results numerically. There is great interest in the computational materials science community in fitting Laguerre tessellations to electron backscatter diffraction (EBSD) and x-ray diffraction images of polycrystalline materials. As an application of our results we fit a 2D Laguerre tessellation to an EBSD image of steel.