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Fitting regular point patterns with a hyperuniform perturbed lattice

Daniela Flimmel

TL;DR

The paper addresses fitting regular spatial point patterns that exhibit suppressed large-scale fluctuations by introducing hyperuniform perturbed lattices as ground models. It develops finite-moment and strong-m mixing conditions under which perturbed lattices are hyperuniform in general dimension, and provides an explicit $K$-function formula for Gaussian perturbations to enable fast minimal-contrast fitting. The NiTi 3D-XRD grain-center data analysis demonstrates hyperuniform behavior with an estimated exponent $\\hat{\\alpha}=0.81$ and shows that dependent Gaussian perturbations better capture the observed structure than independent perturbations or cloaked variants. The work delivers a computationally efficient, physically interpretable baseline for repulsive spatial data and informs model choice for materials microstructure analyses, with avenues for marked perturbations and tessellation-based extensions.

Abstract

We introduce a new methodology for modeling regular spatial data using hyperuniform point processes. We show that, under some mixing conditions on the perturbations, perturbed lattices in general dimension are hyperuniform. Due to their inherent repulsive structure, they serve as an effective baseline model for data sets in which points exhibit repulsiveness. Specifically, we derive an explicit formula for the $K$-function of lattices perturbed by a Gaussian random field, which proves particularly useful in conjunction with the minimal contrast method. We apply this approach to a data set representing the grain centers of a polycrystalline metallic material composed of nickel and titanium.

Fitting regular point patterns with a hyperuniform perturbed lattice

TL;DR

The paper addresses fitting regular spatial point patterns that exhibit suppressed large-scale fluctuations by introducing hyperuniform perturbed lattices as ground models. It develops finite-moment and strong-m mixing conditions under which perturbed lattices are hyperuniform in general dimension, and provides an explicit -function formula for Gaussian perturbations to enable fast minimal-contrast fitting. The NiTi 3D-XRD grain-center data analysis demonstrates hyperuniform behavior with an estimated exponent and shows that dependent Gaussian perturbations better capture the observed structure than independent perturbations or cloaked variants. The work delivers a computationally efficient, physically interpretable baseline for repulsive spatial data and informs model choice for materials microstructure analyses, with avenues for marked perturbations and tessellation-based extensions.

Abstract

We introduce a new methodology for modeling regular spatial data using hyperuniform point processes. We show that, under some mixing conditions on the perturbations, perturbed lattices in general dimension are hyperuniform. Due to their inherent repulsive structure, they serve as an effective baseline model for data sets in which points exhibit repulsiveness. Specifically, we derive an explicit formula for the -function of lattices perturbed by a Gaussian random field, which proves particularly useful in conjunction with the minimal contrast method. We apply this approach to a data set representing the grain centers of a polycrystalline metallic material composed of nickel and titanium.

Paper Structure

This paper contains 16 sections, 7 theorems, 72 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. General notation
  4. Point processes and second order measures
  5. Hyperuniformity of a point process
  6. Perturbed lattices
  7. Strong mixing of the perturbation field
  8. Gaussian perturbations
  9. Real data analysis
  10. Description of the data
  11. Relation to the existing literature
  12. Hyperuniformity in the NiTi data
  13. Models description
  14. Model fitting
  15. Conclusion
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\mathbf{p}=\{p_i, i \in \mathcal{L}\}$ be a strictly stationary random field such that there exist $s, r>0$ with $1/s+1/r=1$, $\mathbb{E} |p_0|^r<\infty$ and Then the perturbed lattice $\Xi= \mathcal{L}+\mathbf{p}+U$ is hyperuniform.

Figures (9)

  • Figure 1: Theoretical values of the $K$-function and the centered $L$-function of the lattice perturbed by independent Gaussian random variables with standard deviation $0.25$.
  • Figure 2: Positions of grain centers in NiTi alloy extracted using 3D-XRD.
  • Figure 3: Estimated (a) centered L-function, (b) pair-correlation function and (c) G-function. The estimated values are plotted in solid lines, resp. red dashed line, while being compared to a theoretical values of homogeneous Poisson point process (green dashed line).
  • Figure 4: Estimated pair correlation function of the rescalled data set. After $r=1.5$, it becomes negligible.
  • Figure 5: Histograms presenting the number of points appearing in 64 distant boxes of the size of $2.7^3$.
  • ...and 4 more figures

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Definition 1: m-th factorial moment measure
  • Definition 2: $m$-th factorial cumulant measure
  • Definition 3: Hyperuniform point process
  • Definition 4: Structure factor
  • Definition 5: Perturbed lattice
  • Lemma 1
  • ...and 10 more