ROI CT reconstruction combining analytic inversion of the finite Hilbert transform and SVD

In computed tomography, a scan of the whole object may be impossible, leading to truncated projection data. Using differentiated backprojection, the reconstruction problem can be reduced to a set of independent and one-dimensional Hilbert transforms to invert. Depending on the truncation pattern, this inversion problem can either be "zero-", "one-" or "two-endpoint". The zero-endpoint case is known as the interior problem: the field-of-view is completely contained in the object and the reconstruction problem has no unique solution. The two-endpoint case possesses an analytic, numerically stable inverse. The one-endpoint case has a unique and mathematically stable inverse, but no analytic formula for its inverse has been derived so far. A field-of-view (FOV) which is not interior generally contains both one-and two-endpoint sub-regions and we propose here to combine them by using the analytic two-endpoint reconstruction as additional knowledge for the one-endpoint inversion in the rest of the FOV. We hence obtain two reconstructed regions, which we chose to slightly overlap to partially correct for a small residual error appearing in the one-endpoint reconstructions.