Quantitative reconstruction from truncated projections

We prove that some kinds of truncation are entirely admissible for region-of-interest (ROI) reconstructions, contrary to the long-standing folklore that 2D tomography is "all or nothing." The proof is based on a link between the Hilbert transforms of parallel beam and fan beam projections which was recently used by Noo et al to achieve ROI image reconstruction on less than a short scan, provided 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 axes 15 cm and 30 cm. The detector rotates 180 degrees about a point on the long axis at 11.25 cm from the (say, right) 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 accurately reconstructed, as is verified by our simulation. This result brings new understanding to the fundamental mechanisms of tomography. Practical applications of this result appear in SPECT when using multi-segment slant-hole collimators where the effective detector length is determined by the segment dimensions. There may be applications for ROI reconstruction in PET since we now know that not all LORs must be used. Iterative reconstruction algorithms might benefit from the reduced data requirement for ROI reconstruction.