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Two methods are introduced for reducing additive noise in approximately piecewise constant images. In the first method, the correct gray level of a point P is estimated by a weighted average of the gray levels of P's immediate neighbors. To assign these weights for a given P so that the local averaging operation does not blur the image, the presence of any edges or lines passing through P's neighborhood must be determined. For example, when an edge is present, only those points on P's side of the edge should be used. In general, the distribution of gray levels in P's neighborhood will suggest many possible line and edge configurations which can be used to compute the weight distribution for the averaging operation. The averaging is performed in parallel over the entire image and is iterated for additional smoothing. The second method is a relaxation labeling formulation which can be psed in cases of more severe noise. Each point is now assigned a vector of probabilities over its allowable gray level set. These probabilities are updated as weighted averages of the gray level probabilities at neighboring points. Once again the presence of lines and edges determines how the contribution from each neighboring point is weighted. Examples demonstrating both methods are given.