Minkowski tensors, also known as tensor valuations, provide robust $n$-point information for a wide range of random spatial structures. Local estimators for voxelized data, however, are unavoidably biased even in the limit of infinitely high resolution. Here, we substantially improve a recently proposed, asymptotically unbiased algorithm to estimate Minkowski tensors for voxelized data. Our improved algorithm is more robust and efficient. Moreover we generalize the theoretical foundations for an asymptotically bias-free estimation of the interfacial tensors to the case of finite unions of compact sets with positive reach, which is relevant for many applications like rough surfaces or composite materials. As a realistic test case, we consider, among others, random (beta) polytopes. We first derive explicit expressions of the expected Minkowski tensors, which we then compare to our simulation results. We obtain precise estimates with relative errors of a few percent for practically relevant resolutions. Finally, we apply our methods to real data of metallic grains and nanorough surfaces, and we provide an open-source python package, which works in any dimension.

翻译：闵可夫斯基张量（亦称张量赋值）为各类随机空间结构提供了稳健的n点信息。然而，对于体素化数据，局部估计器即使在无限高分辨率的极限情况下也不可避免地存在偏差。本文对近期提出的体素化数据闵可夫斯基张量渐近无偏估计算法进行了实质性改进。改进后的算法具有更强的稳健性与计算效率。此外，我们将界面张量渐近无偏估计的理论基础推广至具有正接触距离的紧集有限并集情形，这对粗糙表面或复合材料等众多应用场景具有重要意义。作为实际测试案例，我们重点考察了随机（β）多胞体。首先推导了期望闵可夫斯基张量的显式表达式，进而与仿真结果进行对比。对于实际应用中的分辨率，我们获得了相对误差仅为百分之几的精确估计。最后，我们将该方法应用于金属晶粒和纳米粗糙表面的真实数据，并提供了可在任意维度运行的开源Python软件包。