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.

Figure 1. A Shepp-Logan phantom, its Radon transform (the sinogram) filled by a rotating gantry, and the image recovered by filtered back-projection with the Ram-Lak ramp filter or by the iterative MLEM algorithm. With few angles the back-projection is streaky; the error falls as the number of projections grows and as MLEM iterates. Method: Radon transform, discrete Ram-Lak FBP and Shepp-Vardi MLEM; Canvas2D, deterministic.

projection angles90

FBP filter

method

WHAT TO TRY

Raise the number of projection angles: the filtered back-projection reconstruction sharpens and the RMSE drops along the diagnostic curve. Too few angles and streak artifacts smear the Shepp-Logan phantom.
Switch the FBP filter (Ram-Lak, Shepp-Logan, none): the unfiltered back-projection is a blurry mess, since the ramp filter is what undoes the 1/r smearing of the Radon transform.
Switch the method to MLEM: the iterative statistical reconstruction handles sparse angles differently from analytic FBP, trading streaks for a slower converging but noise-aware estimate.