Consistent estimates of deformed isotropic Gaussian random fields on the plane

TL;DR

This paper establishes consistent estimation methods for recovering smooth deformations of isotropic Gaussian random fields on the plane, using kernel smoothing and quasiconformal theory, with convergence guarantees as data density increases.

Contribution

It introduces a novel estimation approach for deformations of Gaussian fields on the plane, proving almost sure uniform convergence under mild conditions.

Findings

Estimator converges to the true deformation up to rotation and translation.

Convergence holds uniformly on compact subsets as grid density increases.

Method combines kernel smoothing, Bergman projections, and quasiconformal theory.

Abstract

This paper proves fixed domain asymptotic results for estimating a smooth invertible transformation $f:\Bbb{R}^2\to\Bbb{R}^2$ when observing the deformed random field $Z\circ f$ on a dense grid in a bounded, simply connected domain $\Omega$, where $Z$ is assumed to be an isotropic Gaussian random field on $\Bbb{R}^2$. The estimate $\hat{f}$ is constructed on a simply connected domain $U$, such that $\overline{U}\subset\Omega$ and is defined using kernel smoothed quadratic variations, Bergman projections and results from quasiconformal theory. We show, under mild assumptions on the random field $Z$ and the deformation $f$, that $\hat{f}\to R_{\theta}f+c$ uniformly on compact subsets of $U$ with probability one as the grid spacing goes to zero, where $R_{\theta}$ is an unidentifiable rotation and $c$ is an unidentifiable translation.