2D directional ramp filter

Usual tomographic reconstruction methods start by filtering projections before backprojecting the data. In some cases, inverting the filtering and the backprojection steps can be useful to preserve spatial information. In this paper, an intermediate between a filter-backproject and a backproject-filter approach is proposed, based on the extension of the usual ramp filter to two dimensions. To this end, an expression for a band-limited 2D version of the ramp filter is derived. We have tested this filter on simulated x-ray CT projections of a Shepp-Logan phantom and on proton CT list-mode data. We accurately reconstructed the x-ray CT and the proton CT data, although the reconstruction can be slightly noisier than a standard filtered backprojection in some cases. A slight improvement of the spatial resolution of proton CT images reconstructed with this 2D filter has been observed.