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This paper considers a detection theory approach to the restoration of digitized images. The images are modeled as second-order Markov meshes. This model is not only well suited to a decision approach to smoothing, but it enables computer simulations of images thereby permitting a statistical analysis of restoration techniques. Smoothing procedures that are near optimal in the sense of approaching a nonrealizable bound are demonstrated and evaluated. The achievable reduction in mean-square error is considerable for coarsely quantized pictures. This reduction, for the four-level pictures considered, is somewhat greater than that achievable by linear techniques. The approach actually minimizes the probability of error which may be important for preserving picture features.