This paper proves fixed domain asymptotic results for estimating a smooth invertible transformation f: ℝ2→ℝ2 when observing the deformed random field Z○f on a dense grid in a bounded, simply connected domain Ω, where Z is assumed to be an isotropic Gaussian random field on ℝ2. The estimate f̂ is constructed on a simply connected domain U, such that U̅⊂Ω 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 f̂→Rθf+c uniformly on compact subsets of U with probability one as the grid spacing goes to zero, where Rθ is an unidentifiable rotation and c is an unidentifiable translation.
"Consistent estimates of deformed isotropic Gaussian random fields on the plane." Ann. Statist. 37 (5A) 2324 - 2350, October 2009. https://doi.org/10.1214/08-AOS647