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The importance of development of new methods for reconstruction of an object image given its sinogram and some additional information about the object stems from the possibility of artifact presence in the reconstructed image or its insufficient sharpness when the used additional information does not hold. A method based on combining an assumptions-free image of the object and a dummy estimate followed by projection onto the set of plausible images is proposed and developed for the case when this set is the non-negative cone. This additional information corresponds to the natural condition of non-negativeness of the estimated brightnesses. Examples of use of the proposed method with variations in the used iterative solving method and its parameters for processing teeth sinograms are given.