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This paper proves fixed domain asymptotic results for estimating a smoothinvertible transformation $f:\Bbb{R}^2\to\Bbb{R}^2$ when observing the deformedrandom 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 smoothedquadratic variations, Bergman projections and results from quasiconformaltheory. We show, under mild assumptions on the random field $Z$ and thedeformation $f$, that $\hat{f}\to R_{\theta}f+c$ uniformly on compact subsetsof $U$ with probability one as the grid spacing goes to zero, where$R_{\theta}$ is an unidentifiable rotation and $c$ is an unidentifiabletranslation.

Consistent estimates of deformed isotropic Gaussian random fields on the plane