Cone-Beam Pair-Wise Data Consistency Conditions in Helical CT

Data consistency conditions (DCC) are mathematical equations characterizing the redundancy in X-ray projections. They have been used to correct inconsistent projections before computed tomography (CT) reconstruction. This article investigates DCC for a helical acquisition with a cylindrical detector, the geometry of most diagnostic CT scanners. The acquired projections are analyzed pair-by-pair. The intersection of each plane containing the two source positions with the corresponding cone-beams defines two fan-beams for which a DCC can be computed. Instead of rebinning the two fan-beam projections to a conventional detector, we directly derive the DCC in detector coordinates. If the line defined by two source positions intersects the field-of-view (FOV), the DCC presents a singularity which is accounted for in our numerical implementation to increase the number of DCC compared to previous approaches which excluded these pairs of source positions. Axial truncation of the projections is addressed by identifying for which set of planes containing the two source positions the fan-beams are not truncated. The ability of these DCC to detect breathing motion has been evaluated on simulated and real projections. Our results indicate that the DCC can detect motion if the baseline and the FOV do not intersect. If they do, the inconsistency due to motion is dominated by discretization errors and noise. We therefore propose to normalize the inconsistency by the noise to obtain a noise-aware metric which is mostly sensitive to inconsistencies due to motion. Combined with a moving average to reduce noise, the derived DCC can detect breathing motion.