On the perimeter estimation of pixelated excursion sets of 2D anisotropic random fields

TL;DR

This paper introduces a consistent estimator for the perimeter of excursion sets of 2D random fields from digital images, applicable without Gaussianity or isotropy assumptions, and proves its asymptotic normality for affine, strongly mixing fields.

Contribution

The paper proposes a novel perimeter estimator for excursion sets of 2D random fields that is consistent, does not require Gaussianity or isotropy, and establishes its asymptotic distribution.

Findings

Estimator is consistent as pixel size decreases.

No normalization or Gaussianity/isotropy assumptions needed.

Asymptotic normality proven for affine, strongly mixing fields.

Abstract

We are interested in creating statistical methods to provide informative summaries of random fields through the geometry of their excursion sets. To this end, we introduce an estimator for the length of the perimeter of excursion sets of random fields on $\mathbb{R}^2$ observed over regular square tilings. The proposed estimator acts on the empirically accessible binary digital images of the excursion regions and computes the length of a piecewise linear approximation of the excursion boundary. The estimator is shown to be consistent as the pixel size decreases, without the need of any normalization constant, and with neither assumption of Gaussianity nor isotropy imposed on the underlying random field. In this general framework, even when the domain grows to cover $\mathbb{R}^2$, the estimation error is shown to be of smaller order than the side length of the domain. For affine, strongly mixing random fields, this translates to a multivariate Central Limit Theorem for our estimator when multiple levels are considered simultaneously. Finally, we conduct several numerical studies to investigate statistical properties of the proposed estimator in the finite-sample data setting.