While it is well-known how to compute the cells of a Laguerre tessellation for a given set of weighted generator points, it is not obvious how to invert a Laguerre tessellation. That is, given that one observes a Laguerre tessellation, how can one retrieve the weighted generators corresponding to the observed cells. In this paper, we consider inversion of a class of random Laguerre tessellations known as Poisson-Laguerre tessellations. The weighted generators of observed cells of a Poisson-Laguerre tessellation are of interest because knowledge of these weighted generators is useful for statistical inference of Poisson-Laguerre tessellations. For general Laguerre tessellations we provide a characterization of all configurations of weighted generator points which yield the same Laguerre tessellation. For Poisson-Laguerre tessellations we propose a method for consistent inversion, meaning that as one observes the tessellation through increasing observation windows, a closer approximation of the original weighted generators can be obtained. In a simulation study we examine both performance of the inversion procedure, as well as the use of the obtained approximated weighted generators for nonparametrically estimating the weight distribution function corresponding to a Poisson-Laguerre tessellation.

翻译：摘要：对于给定的一组加权生成点，如何计算拉盖尔剖分的胞元已是众所周知的问题，但拉盖尔剖分的反演却并非显而易见。也就是说，当观测到一个拉盖尔剖分时，如何恢复出与观测胞元对应的加权生成点？本文考虑一类随机拉盖尔剖分——泊松-拉盖尔剖分的反演问题。泊松-拉盖尔剖分中观测胞元的加权生成点具有研究价值，因为这些加权生成点的知识有助于对泊松-拉盖尔剖分进行统计推断。对于一般拉盖尔剖分，我们给出了所有能生成相同拉盖尔剖分的加权生成点构型的刻画。针对泊松-拉盖尔剖分，我们提出了一种一致反演方法，即随着观测窗口的扩大，可以渐近地逼近原始加权生成点。通过仿真研究，我们评估了反演过程的性能，并检验了所获近似加权生成点在非参数估计泊松-拉盖尔剖分对应的权重分布函数中的作用。