论文标题
表面积和游览集的体积在基于点云的多物质镶嵌上观察到
Surface area and volume of excursion sets observed on point cloud based polytopic tessellations
论文作者
论文摘要
$ C^2 $平滑随机场的游览集在其各种几何措施中带有相关信息。从计算角度来看，人们永远无法访问对游览集的连续观察，而是在空间中离散点的观察。据报道，对于维度2和3中的特定定期晶格，即使在观察域中晶格变强，偏移的表面积的通常估计仍然存在偏差。在目前的工作中，在平稳性和各向同性的关键假设下，我们证明了这种限制偏见是观察点位置不变的。确实，我们确定了偏见的明确公式，表明它仅取决于空间尺寸$ d $。这使我们能够为游览集的表面积定义一个无偏的估计器，该估计值是由$ \ mathbb {r}^d $中的多层的一般特性近似的，其中包括Poisson-Voronoi Tessellations。我们还为表面积建立了一个联合中央限制定理，并在高管晶格上观察到的游览集的体积估计值。
The excursion set of a $C^2$ smooth random field carries relevant information in its various geometric measures. From a computational viewpoint, one never has access to the continuous observation of the excursion set, but rather to observations at discrete points in space. It has been reported that for specific regular lattices of points in dimensions 2 and 3, the usual estimate of the surface area of the excursions remains biased even when the lattice becomes dense in the domain of observation. In the present work, under the key assumptions of stationarity and isotropy, we demonstrate that this limiting bias is invariant to the locations of the observation points. Indeed, we identify an explicit formula for the bias, showing that it only depends on the spatial dimension $d$. This enables us to define an unbiased estimator for the surface area of excursion sets that are approximated by general tessellations of polytopes in $\mathbb{R}^d$, including Poisson-Voronoi tessellations. We also establish a joint central limit theorem for the surface area and volume estimates of excursion sets observed over hypercubic lattices.
