A fast algorithm for linear estimation of two- dimensional isotropic random fields

The problem considered involves estimating a two-dimensional isotropic random field given noisy observations of this field over a disk of finite radius. By expanding the field and observations in Fourier series, and exploiting the covariance structure of the resulting Fourier coefficient processes, recursions are obtained for efficiently constructing the linear least-squares estimate of the field as the radius of the observation disk increases. These recursions are similar to the Levinson equations of one-dimensional linear prediction. In the spectral domain they take the form of Schrödinger equations, which are used to give an inverse spectral interpretation of our estimation procedure.