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Minkowski tensors for voxelized data: robust, asymptotically unbiased estimators

Daniel Hug, Michael A. Klatt, Dominik Pabst

TL;DR

This work tackles the challenge of unbiasedly estimating Minkowski tensors from voxelized data, where discrete orientations induce systematic bias. It introduces two robust estimators: a Voronoi-tensor estimator valid for finite unions with positive reach, and a least-squares reconstruction that avoids matrix inversion while recovering all $ ext{Φ}_k^{r,s}$; both are proven to be asymptotically unbiased under the stated geometric conditions. The authors provide a rigorous theory linking reach measures to boundary contributions, implement an open-source Python package, and validate the approach on convex and nonconvex test shapes, isotropic beta-polytopes with exact expectations, and real data from metallic grains and nanorough surfaces. The results show relative errors of only a few percent at practical resolutions and demonstrate applicability across dimensions and arbitrary discrete representations, enabling robust microstructure analysis in materials science and related fields.

Abstract

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.

Minkowski tensors for voxelized data: robust, asymptotically unbiased estimators

TL;DR

This work tackles the challenge of unbiasedly estimating Minkowski tensors from voxelized data, where discrete orientations induce systematic bias. It introduces two robust estimators: a Voronoi-tensor estimator valid for finite unions with positive reach, and a least-squares reconstruction that avoids matrix inversion while recovering all ; both are proven to be asymptotically unbiased under the stated geometric conditions. The authors provide a rigorous theory linking reach measures to boundary contributions, implement an open-source Python package, and validate the approach on convex and nonconvex test shapes, isotropic beta-polytopes with exact expectations, and real data from metallic grains and nanorough surfaces. The results show relative errors of only a few percent at practical resolutions and demonstrate applicability across dimensions and arbitrary discrete representations, enabling robust microstructure analysis in materials science and related fields.

Abstract

Minkowski tensors, also known as tensor valuations, provide robust -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.

Paper Structure

This paper contains 16 sections, 10 theorems, 110 equations, 5 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Minkowski tensors in integral geometry
  3. Asymptotics for estimators: a theoretical foundation
  4. Algorithms to estimate Minkowski tensors
  5. Description and analysis
  6. Implementation
  7. Choice of parameters
  8. Convex test cases
  9. Nonconvex test cases
  10. Isotropic random polytopes
  11. Exact expectations
  12. Simulation study
  13. Experimental data
  14. Metallic grains
  15. Nanorough surfaces
  16. ...and 1 more sections

Key Result

Lemma 3.2

If $A$ is a finite union of compact sets of positive reach, then the total variation measures of the reach measures of $A$ are concentrated on $\mathop{\mathrm{Nor}}\nolimits(A)$ and satisfy

Figures (5)

  • Figure 1: Example data $K_0$ (red points), which is a finite subset of an underlying set with positive reach (a), and the Voronoi diagram (b) of the data $K_0$ and the set $K_0^R$ (blue) of points with distance not bigger than some $R>0$ from $K_0$. The algorithm uses a random grid $\eta$ (compare (\ref{['VoroM_Estimator']})) to estimate the Voronoi tensor $\mathcal{V}_R^{r,s}(K_0)$ of $K_0$, where only the points of $\eta$ inside $K_0^R$ are relevant for the estimation.
  • Figure 2: Examples of three distinct metallic grains that are represented by voxel centers. Here, each voxel center is indicated by a ball with a diameter equal to the lattice spacing, i.e., 1 $\mu$m. The underlying experimental data of the grains is from stinville_multi-modal_2022.
  • Figure 3: Non-normalized histograms of the estimated volumes (a) and surface areas (b) of the metallic grains from Section \ref{['sec:grains']}. Besides a large number of small cells, we observe a few exceptionally large cells. For better visualization, only a range of values is shown, i.e., the most extreme cases are excluded here. The underlying experimental data of the grains is from stinville_multi-modal_2022.
  • Figure 4: Non-normalized histograms of anisotropy indices for the metallic grains from Section \ref{['sec:grains']}. More specifically, the plots show histograms of the ratios between the smallest and largest absolute eigenvalues of the interfacial tensor $\Phi_1^{0,2}$ (a) and the curvature tensor $\Phi_2^{0,2}$ (b). The underlying experimental data of the grains is from stinville_multi-modal_2022.
  • Figure 5: Examples of nanorough surfaces from Table \ref{['tab:nanorough']} with three different degrees of roughness, i.e., an RMS value of (a) 7 nm, (b) 24 nm, and (c) 35 nm. The data is taken from spengler_strength_2019; the figures have been created by Jens Uwe Neurohr.

Theorems & Definitions (24)

  • Remark 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Corollary 4.1
  • proof
  • Lemma 4.2
  • ...and 14 more