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Using ideas from one-dimensional maximum entropy spectral estimation a two-dimensional spectral estimator is derived by extrapolating the two-dimensional sampled autocorrelation (or covariance) function. The method used maximizes the entropy of a set of random variables. The extrapolation (or prediction) process under this maximum entropy condition is shown to correspond to the most random extension or equivalently to the maximization of the mean-square prediction error when the optimum predictor is used. The two-dimensional extrapolation must he terminated by the investigator. The Fourier transform of the extrapolated autocorrelation function is the two-dimensional spectral estimator. Using this method one can apply windowing prior to calculating the spectral estimate. A specific algorithm for estimating the two-dimensional spectrum is presented, and its computational complexity is estimated. The algorithm has been programmed and computer examples are presented.