Quantitative reconstruction from truncated projections in classical tomography

We prove that some kinds of truncation are entirely admissible for region-of-interest (ROI) reconstructions in classical tomography, contrary to the long-standing folklore that two-dimensional tomography is "all or nothing". The proof is based on a link between the Hilbert transforms of parallel beam and fanbeam projections, which was recently used by Noo et al. to achieve ROI image reconstruction from fanbeam data on less than a short scan, assuming no fanbeam truncation occurs. We extend the use of this link to achieve quantitative ROI reconstruction in the presence of truncated projections. Our results are illustrated with a specific example of a parallel-hole detector of length 24 cm and an elliptical object of 15- and 30-cm axes. The detector rotates 180° about a point on the long axis at 11.25 cm from the right-hand-side boundary. The right half of the ellipse, the ROI in this case, is not truncated, although the left side is truncated in many views. The ROI can be quantitatively reconstructed, as is verified by our simulation. We also show that under certain conditions, ROI reconstruction is possible with truncation on both sides of the field of view. These results bring new understanding to the fundamental mechanisms of tomography. Benefits of this new understanding can be anticipated in many classical tomography applications, particularly when the projections of the object are too wide for the available detector.