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A new method for the exact reconstruction of any grayscale image from a limited number of projection samples is proposed. The original image is projected into certain projection angles using a proper projection angle step and a proper number of ray samples on the projection axis. If these conditions are fulfilled the proposed method ensures that for each image pixel a characteristic sample exists which allows the determination of the pixel's grayscale value during the reconstruction phase. Thus, instead of calculating all the projection samples of the image on several projection angles, in the proposed method only a specific number of samples is required that equals the number of pixels in the original image. During the reconstruction procedure the characteristic samples are examined consequently starting from those corresponding to pixels further from the origin and continuing with the closer image pixels. This approach leads to significant reduction of memory and processing time requirements since a simple sequence of samples is used instead of all the data from the image projections.