An efficient maximum entropy technique for 2-D isotropic random fields

In this paper, we present a new linear MEM algorithm for 2-D isotropic random fields. Our procedure differs from previous 2-D MEM algorithms by the fact that we take maximal advantage of the symmetries implied by isotropy, which is the natural generalization to several dimensions of the l-D notion of stationarity. Unlike general 2-D covariances, isotropic covariance functions which are positive definite on a disk are known to be extendible. Here, we develop a computationally efficient procedure for computing the MEM isotropic spectral estimate corresponding to an isotropic covariance function which is given over a finite disk of radius 2R. We show that the isotropic MEM problem has a linear solution and that it is equivalent to the problem of constructing the optimal linear filter for estimating the underlying isotropic field at a point on the boundary of a disk of radius R given noisy measurements of the field inside the disk. Our procedure is based on Fourier series expansions in both the space and wave-number domains of the inverse of the MEM spectral estimate. Furthermore, our method is guaranteed to yield a valid isotropic spectral estimate and it is computationally efficient since it requires only O(BRL2 ) operations where L is the number of points used to discretize the interval [0, R], and where B is the bandwidth in the wave-number plane of the spectrum that we want to estimate. We also present examples to illustrate the behavior of our algorithm and its high resolution property.