Local digital algorithms applied to Boolean models

TL;DR

This paper studies how to estimate geometric features of Boolean models using local digital algorithms, revealing limitations and providing new unbiased estimators in specific cases.

Contribution

It demonstrates the non-existence of asymptotically unbiased estimators for certain measures in higher dimensions and derives new estimators for 3D isotropic models.

Findings

Unbiased estimators for surface area and mean curvature exist in 3D isotropic models.

No asymptotically unbiased estimators for surface area in dimensions ≥2.

Unbiased estimator for Euler characteristic in ball-based Boolean models.

Abstract

We investigate the estimation of specific intrinsic volumes of stationary Boolean models by local digital algorithms; that is, by weighted sums of $n \times\ldots \times n$ configuration counts. We show that asymptotically unbiased estimators for the specific surface area or integrated mean curvature do not exist if the dimension is at least two or three, respectively. For 3-dimensional stationary, isotropic Boolean models, we derive asymptotically unbiased estimators for the specific surface area and integrated mean curvature. For a Boolean model with balls as grains we even obtain an asymptotically unbiased estimator for the specific Euler characteristic.