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The two-dimensional (2-D) doubly stochastic Gaussian (DSG) model was introduced by one of the authors to provide a complete model for spatial filters which adapt to the local structure in an image signal. Here we present the optimal estimator and 2-D fixed-lag smoother for this DSG model extending earlier work of Ackerson and Fu. As the optimal estimator has an exponentially growing state space, we investigate suboptimal estimators using both a tree and a decision-directed method. Experimental results are presented.