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We are interested in creating statistical methods to provide informativesummaries of random fields through the geometry of their excursion sets. Tothis end, we introduce an estimator for the length of the perimeter ofexcursion sets of random fields on $\mathbb{R}^2$ observed over regular squaretilings. The proposed estimator acts on the empirically accessible binarydigital images of the excursion regions and computes the length of a piecewiselinear approximation of the excursion boundary. The estimator is shown to beconsistent as the pixel size decreases, without the need of any normalizationconstant, and with neither assumption of Gaussianity nor isotropy imposed onthe underlying random field. In this general framework, even when the domaingrows to cover $\mathbb{R}^2$, the estimation error is shown to be of smallerorder than the side length of the domain. For affine, strongly mixing randomfields, this translates to a multivariate Central Limit Theorem for ourestimator when multiple levels are considered simultaneously. Finally, weconduct several numerical studies to investigate statistical properties of theproposed estimator in the finite-sample data setting.

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