Inverting Laguerre Tessellations

A Laguerre tessellation is a generalization of a Voronoi tessellation where the proximity between points is measured via a power distance rather than the Euclidean distance. Laguerre tessellations have found significant applications in materials science, providing improved modeling of (poly)crystalline microstructures and grain growth. There exist efficient algorithms to construct Laguerre tessellations from given sets of weighted generator points, similar to methods used for Voronoi tessellations. The purpose of this paper is to provide theory and methodology for the inverse construction; that is, to recover the weighted generator points from a given Laguerre tessellation. We show that, unlike the Voronoi case, the inverse problem is in general non-unique: different weighted generator points can create the same tessellation. To recover pertinent generator points we formulate the inversion problem as a multimodal optimization problem and apply the cross-entropy method to solve it.