We investigate the estimation of specific intrinsic volumes of stationary Boolean models by local digital algorithms; that is, by weighted sums of $n \times\ldots \times n$ configuration counts. We show that asymptotically unbiased estimators for the specific surface area or integrated mean curvature do not exist if the dimension is at least two or three, respectively. For 3-dimensional stationary, isotropic Boolean models, we derive asymptotically unbiased estimators for the specific surface area and integrated mean curvature. For a Boolean model with balls as grains we even obtain an asymptotically unbiased estimator for the specific Euler characteristic.

中文翻译：

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应用于布尔模型的局部数字算法

我们研究了通过局部数字算法对固定布尔模型的特定固有体积的估计；也就是说，通过 $n \times\ldots \times n$ 配置计数的加权和。我们表明，如果维度分别至少为两个或三个，则不存在比表面积或积分平均曲率的渐近无偏估计量。对于 3 维平稳、各向同性布尔模型，我们推导出比表面积和积分平均曲率的渐近无偏估计量。对于以球为颗粒的布尔模型，我们甚至可以获得特定欧拉特征的渐近无偏估计量。