The paper considers the direct Fourier methods (DFM's) for reconstructing an image from its given X-ray projections. The main purpose here is to use concepts from numerical analysis to estimate the errors in the methods. We also suggest an alternative to the interpolations involved in the DFM's and estimate the number of terms involved. Realizing that one of the main reasons for the degradation of the quality of the reconstruction is the Gibbs phenomenon, we suggest a way to overcome this difficulty and give numerical evidence of its efficacy.