In computed tomography, a whole scan of the object may be impossible, generally because the object is larger than the scanner field-of-view. Such a set up leads to truncated projections. Using differentiated backprojection, the reconstruction problem can be reduced to a set of one-dimensional problems consisting of the inversion of the Hilbert transform. When the object partly overlaps the scanner field-of-view, this problem is commonly referred to as the "one-sided truncated Hilbert transform". Our work investigates this situation and proposes a novel approach to address it. Using differentiated backprojection, and the object extent supposedly known a priori, a pseudo-inverse of the truncated Hilbert transform is computed by truncated singular value decomposition, and its truncated singular values are replaced by a simple estimation. The estimation is calculated using the singular value decomposition of the known convex hull filled with a constant value per line computed from the corresponding projection in the direction of the Hilbert transform. Experiments illustrate the image quality improvements resulting from this approach compared to a simple truncation of the singular values and the reconstruction speed improvement compared to two-dimensional iterative reconstruction solving penalized least squares with the conjugate gradient algorithm.

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Objective: The aim of this work is to investigate the feasibility of the Jagiellonian Positron Emission Tomography (J-PET) scanner for intra-treatment proton beam range monitoring.Approach: The Monte Carlo simulation studies with GATE and PET image reconstruction with CASToR were performed in order to compare six J-PET scanner geometries (three dual-heads and three cylindrical). We simulated proton irradiation of a PMMA phantom with a Single Pencil Beam (SPB) and Spread-Out Bragg Peak (SOBP) of various ranges. The sensitivity and precision of each scanner were calculated, and considering the setup's cost-effectiveness, we indicated potentially optimal geometries for the J-PET scanner prototype dedicated to the proton beam range assessment.Main results: The investigations indicate that the double-layer cylindrical and triple-layer double-head configurations are the most promising for clinical application. We found that the scanner sensitivity is of the order of 10 −5 coincidences per primary proton, while the precision of the range assessment for both SPB and SOBP irradiation plans was found below 1 mm. Among the scanners with the same number of detector modules, the best results are found for the triple-layer dual-head geometry.Significance: We performed simulation studies demonstrating that the feasibility of the J-PET detector for PET-based proton beam therapy range monitoring is possible with reasonable sensitivity and precision enabling its pre-clinical tests in the clinical proton therapy environment. Considering the sensitivity, precision and costeffectiveness, the double-layer cylindrical and triple-layer dual-head J-PET geometry configurations seem promising for the future clinical application. Experimental tests are needed to confirm these findings.

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