Internal models and recursive estimation for 2-D isotropic random fields

Efficient recursive smoothing algorithms are developed for isotropic random fields that can be obtained by passing white noise through rational filters. The estimation problem is shown to be equivalent to a countably infinite set of 1-D separable two-point boundary value smoothing problems. The 1-D smoothing problems are solved using a Markovianization approach followed by a standard 1-D smoothing algorithm. The desired field estimate is then obtained as properly weighted sum of the 1-D smoothed estimates. The 1-D two-point boundary value problems are also shown to have the same asymptotic properties and yield a stable spectral factorization of the power spectrum of the isotropic random fields