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Using the ideas from one-dimensional (1-D) maximum entropy spectral estimation, we derive a 2-D spectral estimator by actually extrapolating the 2-D sampled autocorrelation function. The method used here is to maximize the entropy of a set of random variables. The extrapolation (prediction) process under the maximum entropy condition is shown to correspond to the most random extension or to the maximization of the mean square prediction error conditioned on using the optimum predictor. The 2-D extrapolation must be terminated by the investigator. The Fourier transform of the extrapolated autocorrelation function is our 2-D spectral estimator. A specific algorithm for estimating the 2-D spectrum is presented. The algorithm has been programmed and computer examples are presented.