Consistent estimates of deformed isotropic gaussian random fields on the plane

Abstract

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 U ? 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 f Rθ f + 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. © Institute of Mathematical Statistics, 2009.