On Two-Dimensional Spectral Realization

Reconstruction of a spectral density function from a finite set of covariances can be performed by maximizing an entropy functional. The method of the maximum entropy on the mean is used for computing a discrete version of this spectral density and allows one to give a new interpretation of these reconstruction methods. In fact, we show that the choice of the entropy is directly related to a prior distribution. In particular, we consider processes on Z2. Steepest descent procedures permit the numerical computation of discrete realizations for a wide class of entropies. To ensure the nonnegativity of the solution related to the Burg entropy, we present a new algorithm based on a fixed-point method and the Yule-Walker equations to compute this solution. Then, the solution of the dual problem is obtained as the limit of the trajectory of an ordinary differential equation.