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arxiv: 2502.00092 · v2 · submitted 2025-01-31 · 🧮 math.ST · cond-mat.dis-nn· math.MG· math.PR· stat.TH

Minkowski tensors for point clouds and voxelized data: robust, asymptotically unbiased estimators

Daniel Hug , Michael A. Klatt , Dominik Pabst This is my paper

Pith reviewed 2026-05-23 04:08 UTC · model grok-4.3

classification 🧮 math.ST cond-mat.dis-nnmath.MGmath.PRstat.TH
keywords Minkowski tensorspoint cloudsvoxelized dataasymptotically unbiased estimatorstensor valuationspositive reachrandom polytopes
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The pith

Improved estimators allow asymptotically unbiased computation of Minkowski tensors from point clouds and voxelized data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops better methods to calculate Minkowski tensors from discrete point clouds or voxelized representations of shapes. These tensors capture detailed geometric information about spatial structures. Previous estimators had unavoidable bias even at high resolutions, but the new approach reduces this bias significantly while improving robustness and speed. The authors extend the mathematical theory to cover finite unions of sets with positive reach, common in real-world materials and surfaces. They validate the method on simulated random polytopes and apply it to actual data from grains and rough surfaces, providing a Python package for use in any dimension.

Core claim

Minkowski tensors can be estimated from point clouds in an asymptotically unbiased manner by improving local estimators and generalizing the theory to finite unions of compact sets with positive reach. This allows precise estimates with relative errors of a few percent at practical resolutions for random spatial structures like beta polytopes, and works on real data such as metallic grains and nanorough surfaces.

What carries the argument

Asymptotically unbiased local estimators for Minkowski tensors (tensor valuations) from point clouds, extended to sets with positive reach.

If this is right

  • Estimates of interfacial tensors become asymptotically bias-free for finite unions of compact sets with positive reach.
  • Precise estimates with relative errors of a few percent are achievable for practically relevant resolutions in simulations of random polytopes.
  • The methods apply directly to real data of metallic grains and nanorough surfaces.
  • An open-source Python package enables computation in any dimension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These estimators could support more accurate analysis of complex materials where positive reach holds but data remains discrete.
  • Performance could be tested on sets that violate the positive reach condition to identify where bias reappears.
  • The efficiency improvements might allow integration into pipelines for extracting geometric features from voxelized data.

Load-bearing premise

The sets being estimated must be finite unions of compact sets with positive reach for the asymptotic unbiasedness to hold.

What would settle it

Observing that bias does not approach zero as resolution increases to infinity for sets with positive reach would falsify the asymptotic unbiasedness.

Figures

Figures reproduced from arXiv: 2502.00092 by Daniel Hug, Dominik Pabst, Michael A. Klatt.

Figure 1
Figure 1. Figure 1: Example data K0 (red points), which is a finite subset of an underlying set with positive reach (a), and the Voronoi diagram (b) of the data K0 and the set KR 0 (blue) of points with distance not bigger than some R > 0 from K0. The algorithm uses a random grid η (compare (4.6)) to estimate the Voronoi tensor V r,s R (K0) of K0, where only the points of η inside KR 0 are relevant for the estimation. Proof. … view at source ↗
Figure 2
Figure 2. Figure 2: , for three examples. The resolution of the grid is 1 µm. To exclude numerical artifacts, we apply a data cut and only consider grains with at least 500 voxel centers, which results in a data set with 2180 grains. For each grain, we took the average over 10 renditions. We varied Rn between 4 · R1 and 24 · R1. The results from [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Non-normalized histograms of the estimated volumes (a) and surface areas (b) of the metallic grains from Section 6.1. 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 [60]. 0.0 0.2 0.4 0.6 0.8 1.0 Ratio of the eig… view at source ↗
Figure 4
Figure 4. Figure 4: Non-normalized histograms of anisotropy indices for the metallic grains from Sec￾tion 6.1. More specifically, the plots show histograms of the ratios between the smallest and largest absolute eigenvalues of the interfacial tensor Φ0,2 1 (a) and the curvature tensor Φ0,2 2 (b). The underlying experimental data of the grains is from [60]. 6.2 Nanorough surfaces Next, we apply our algorithm to a distinctly di… view at source ↗
Figure 5
Figure 5. Figure 5: Examples of nanorough surfaces from [PITH_FULL_IMAGE:figures/full_fig_p031_5.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript presents an improved algorithm for estimating Minkowski tensors from point clouds and voxelized data that is claimed to be more robust and efficient than a recent predecessor. It generalizes the theoretical foundations for asymptotically bias-free estimation of interfacial tensors (among others) to finite unions of compact sets with positive reach. Explicit expressions for the expected Minkowski tensors of random beta polytopes are derived and compared to Monte Carlo simulations, yielding relative errors of a few percent at practical resolutions. The methods are applied to real data on metallic grains and nanorough surfaces, and an open-source Python package is provided.

Significance. If the asymptotic unbiasedness result holds under the stated positive-reach hypothesis, the work supplies a practical, reproducible tool for extracting tensor-valued morphological information from discrete spatial data. The explicit derivations for beta polytopes, direct comparison against independently computed expectations, and release of open code constitute verifiable strengths that support the central claims.

major comments (2)
  1. [Abstract / theoretical foundations] Abstract and the section on theoretical foundations: the generalization of asymptotically bias-free estimation is explicitly restricted to finite unions of compact sets with positive reach, yet the manuscript advertises applicability to nanorough surfaces. High-curvature or non-differentiable features on such surfaces typically yield zero reach, so the bias may fail to vanish; the paper provides no additional argument or numerical check that the estimators remain asymptotically unbiased when this hypothesis is violated.
  2. [real-data applications] Section on real-data applications: the nanorough-surface example is presented as a realistic test case covered by the new theory, but no verification is given that the digitized surfaces satisfy the positive-reach condition (e.g., via local curvature or tubular-neighborhood checks). This leaves the practical relevance of the bias-free guarantee for the advertised application unestablished.
minor comments (1)
  1. [simulation results] The abstract states relative errors of 'a few percent' for beta-polytope simulations; a table or figure reporting the precise errors per tensor component and resolution would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and indicate the revisions we are prepared to make.

read point-by-point responses

  1. Referee: [Abstract / theoretical foundations] Abstract and the section on theoretical foundations: the generalization of asymptotically bias-free estimation is explicitly restricted to finite unions of compact sets with positive reach, yet the manuscript advertises applicability to nanorough surfaces. High-curvature or non-differentiable features on such surfaces typically yield zero reach, so the bias may fail to vanish; the paper provides no additional argument or numerical check that the estimators remain asymptotically unbiased when this hypothesis is violated.

    Authors: We agree that the asymptotic unbiasedness result is proved only under the positive-reach hypothesis. The manuscript states this restriction explicitly and lists rough surfaces among potential application areas without claiming that every nanorough surface satisfies the condition. In the revised manuscript we will modify the abstract and the theoretical foundations section to state the hypothesis more prominently and to note that the bias-free guarantee does not extend to sets of zero reach. We will also emphasize that the beta-polytope experiments, which satisfy the hypothesis, serve as the rigorous numerical validation of the theory. revision: yes

  2. Referee: [real-data applications] Section on real-data applications: the nanorough-surface example is presented as a realistic test case covered by the new theory, but no verification is given that the digitized surfaces satisfy the positive-reach condition (e.g., via local curvature or tubular-neighborhood checks). This leaves the practical relevance of the bias-free guarantee for the advertised application unestablished.

    Authors: The referee correctly observes that no explicit verification of the positive-reach condition is supplied for the nanorough-surface data. In revision we will qualify the real-data section to present the nanorough-surface example as an illustration of practical use rather than as a case strictly covered by the theorem. If a simple geometric check on the voxelized representation can be performed without substantial additional work, we will include it; otherwise we will add an explicit caveat that the bias-free property is guaranteed only when the underlying set meets the positive-reach assumption. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior algorithm; central claims use independent integral-geometric derivations and beta-polytope benchmarks

full rationale

The paper improves a recently proposed algorithm and extends bias-free estimation to finite unions of positive-reach sets using integral-geometric foundations. It derives explicit expected Minkowski tensor expressions for beta polytopes independently of the estimators and validates via simulation comparisons with relative errors of a few percent. No quoted step reduces a claimed prediction or generalization to a fitted input, self-citation chain, or definitional equivalence. The positive-reach condition is stated as an explicit assumption without circular justification. This matches the default case of self-contained work against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard integral-geometric identities for sets of positive reach and on the existence of asymptotically unbiased local estimators; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Finite unions of compact sets with positive reach admit asymptotically bias-free estimation of interfacial Minkowski tensors via the improved local algorithm.
    Invoked to justify the generalization beyond the previously treated cases.

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