Minkowski Tensors for Point Clouds and Voxelized Data: Robust, Asymptotically Unbiased Estimators

Minkowski tensors, also known as tensor valuations, provide robust n-point information for a wide range of random spatial structures. Local estimators for point clouds, e.g., representing 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 from point clouds. Our improved algorithm is more robust and efficient. Moreover we generalize the theoretical foundations for an asymptotically bias-free estimation of the interfacial tensors, among others, 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 of random spatial structures, we consider 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.