Fitting regular point patterns with a hyperuniform perturbed lattice

Abstract

We introduce a flexible methodology for modelling regular spatial point patterns using hyperuniform perturbed lattices. We show that, under suitable mixing conditions on the displacement field, lattices perturbed by stationary random fields are hyperuniform in arbitrary dimension. In particular, Gaussian perturbations with absolutely summable covariances yield class-I hyperuniform point processes. We further derive an explicit formula for the $K$-function of Gaussian models, which enables efficient parameter estimation via the minimum contrast method. The proposed framework provides a computationally tractable alternative to classical Gibbs models for repulsive data. The methodology is illustrated on three-dimensional data describing grain centers in a polycrystalline nickel-titanium alloy.