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In this paper we consider a novel technique for the restoration of noise corrupted images, using order-constrained least squares methods. The restoration process uses a moving cross-shaped filter window, within which two operations are combined. The first operation consists of simple hypothesis tests for the presence of an edge of some minimal height δ, crossing the center of the window. The second operation computes the window output as the order-constrained least squares fit of the windowed values (if an edge is deemed to be present), or simply the average (if no edge is present). The new technique is applied to some actual noise corrupted images, and the results are compared to the results of applying similarly configured median and averaging filters. Some computational considerations and comparisons are discussed at the end of the paper.