The surface area of a set which is only observed as a binary pixel image is often estimated by a weighted sum of pixel configurations counts. In this paper we examine these estimators in a design based setting—we assume that the observed set is shifted uniformly randomly. Bounds for the difference between the essential supremum and the essential infimum of such an estimator are derived, which imply that the variance is in \(O(t^2)\) as the lattice distance t tends to zero. In particular, it is asymptotically neglectable compared to the bias. A simulation study shows that the theoretically derived convergence order is optimal in general, but further improvements are possible in special cases.

仅作为二进制像素图像观察到的集合的表面积通常通过像素配置计数的加权总和来估计。在本文中，我们在基于设计的设置中检查这些估计量——我们假设观察到的集合是随机均匀移动的。推导出了这种估计量的基本上界和基本下界之间的差异的界限，这意味着方差在\(O(t^2)\) 中，因为格距t趋于零。特别是，与偏差相比，它是渐近可忽略的。模拟研究表明，理论上推导出的收敛阶次通常是最优的，但在特殊情况下可能会进一步改进。