CT Reconstruction: Radon, Filtered Back-Projection and MLEM

A computed-tomography reconstruction playground. A Shepp-Logan phantom is projected by the parallel-beam Radon transform into a sinogram, filled angle by angle by a rotating gantry. The image is recovered either by filtered back-projection, applying the discrete Ram-Lak ramp filter (or the Shepp-Logan apodisation, or none) and smearing each projection back across the field, or by the Shepp-Vardi MLEM iteration. Panel A shows the phantom and the sinogram; Panel B the reconstruction; Panel C the error against the number of projection angles and, for MLEM, against iteration. The Radon transform is linear with angle-independent total attenuation, filtered back-projection inverts a point source exactly, the reconstruction error falls as more projection angles are added, and the MLEM iteration converges monotonically.