Two-dimensional (2-D) spectrum estimation from raw data is of interest in signal and image processing. A parametric technique for spectrum estimation using 2-D noncausal autoregressive (NCAR) models is given. The NCAR models characterize the statistical dependency of the observation at location s on its neighbors in all directions. This modeling assumption reduces the spectrum estimation problem to two subproblems: the choice of appropriate structure of the NCAR model and the estimation of parameters in NCAR models. By assuming that the true structure of the NCAR model is known, we first analyze the existence and uniqueness of Gaussian maximum likelihood (GML) estimates of NCAR model parameters. Due to the noncausal nature of the models, the computation of GML estimates is burdensome. By assuming specific boundary conditions, computationally tractable expressions are obtained for the likelihood function. Expressions for the asymptotic covariance matrix of the GML estimates as well as the simultaneous confidence bands for the estimated spectrum using GML estimates are derived. Finally, the usefulness of the method is illustrated by computer simulation results.

The problem of deriving simultaneous confidence bands for a class of 2-D spectral estimates is considered. Specifically, we consider the class of spectral estimates obtained by fitting 2-D noncausal autoregressive and Gaussian Markov random field models to the given lattice data. In such parametric methods the spectrum is a function of estimates of model parameters. Using the asymptotic normality of the parameter estimates and Scheffe's projection theorem, simultaneous confidence bands for the estimated spectrum are derived. Supporting computer simulation results are also provided.

A stochastic model-based approach is presented for estimation of the two-dimensional maximum entropy power spectrum (MEPS) from given finite uniform array data. The method consists of fitting an appropriate two-dimensional noncausal Gaussian-Markov random field (GMRF) model to the given data using the maximum likelihood (ML) technique for parameter estimation. The nonlinear criterion function used for ML estimation is similar in structure to the function arising in the deterministic approach of Lang and McClellan. The model-based approach provides new insights into the two-dimensional MEPS estimation problem. For example, using the asymptotic normality of ML estimates, we derive simultaneous confidence bands for the estimated MEPS. It turns out that when the true correlations are generated by a noncausal GMRF model, the two-dimensional MEPS can be obtained by solving linear equations. This approach also suggests techniques for realizing two-dimensional GMRF models from the given correlation data. Several numerical examples are given to illustrate the usefulness of the approach.

Most of the parametric techniques of the spectrum estimation assume a specific distribution for the observations. Even a small number of outliers in the observations, violating the distribution assumption can yield poor spectral estimates. We present an algorithm which yields 2-D spectral estimates, robust to additive outliers. The algorithm is iterative in nature and involves fitting a 2-D noncausal spatial autoregressive (SAR) to the given data.

A stochastic model based approach for 2-D MEPS analysis of a given finite uniform array data is presented. The method consists of fitting an appropriate 2-D noncausal Gaussian Markov Random Field (GMRF) model to the given data using maximum likelihood (ML) method for parameter estimation. The nonlinear criterion function used for ML estimation is similar in structure to the function arising in the algebraic approach of Lang and McClellan [1]. An example is given to illustrate the usefulness of our approach.

Two-dimensional spectral estimation from raw data is of interest in signal and image processing. In this paper, a parametric technique using non causal spatial autoregressive models for spectral estimation is given. The spatial autoregressive models characterize the statistical dependency of the observation at location s on its neighbors in all directions. Once an appropriate model is fitted, the spectrum is a function of the model parameters. By assuming specific boundary conditions maximum likelihood estimates of model parameters are obtained. The usefulness of the method developed here is illustrated by resolving two closely spaced sinusoids on the plane.