On two-dimensional Markov spectral estimation

We give systematic methods for two-dimensional (2-D) spectral estimation from raw data using Gaussian Markov random field (MRF) models. The MRF models considered in this paper characterize the statistical dependence of the observation y(s) on its neighbors in all directions. Due to the modeling assumption, the 2-D spectrum is an explicit function of the parameters of the MRF model. Thus, the spectral estimation problem reduces to that of estimating the appropriate structure and the parameters of the MRF model. A tractable algorithm for maximum likelihood (ML) estimation of parameters is obtained by using doubly periodic boundary conditions for the given data. Asymptotically consistent decision rules are used to choose the appropriate neighbor set of the MRF model. The MRF spectral estimate obtained using the doubly periodic boundary conditions has the property that its inverse discrete Fourier transforms are in perfect agreement with the periodic sample correlation values of the given observations, over a symmetric window identical to the structure of the MRF model. Using this result, and the fact that the 2-D maximum entropy spectral (MES) estimate has a structure similar to the MRF power spectrum, we show that the MRF spectral estimate developed in this paper is a good approximation to the MES estimate. This result provides a reasonable basis (i.e., maximum likelihood) on which to justify the use of the MES estimate with correlation estimates obtained from a finite piece of data samples.