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.
Categories: math.ST stat.TH