Kalman estimation for a class of rational isotropic random fields

This paper considers the application of Kalman estimation theory to the problem of estimating two-dimensional isotropic random fields, whose equations are expressed in terms of the Laplacian, given some noisy observations on a finite disk. It is shown that this problem is equivalent to that of solving a countably infinite number of one-dimensional estimation problems. Markovian models for the one-dimensional processes are developed and the associated Kalman filters are shown to be asymptotically stable. The desired field estimate is then obtained by combining the smoothed estimates resulting from each of the one-dimensional problems weighted in an appropriate fashion.