Characterizing a discrete-to-discrete X-ray transform for iterative image reconstruction with limited angular-range scanning in CT

Iterative image reconstruction in computed tomography often employs a discrete-to-discrete (DD) linear data model, and many of the aspects of the image recovery relate directly to the properties of this linear model. While much is known about the properties of the continuous X-ray, the corresponding DD version can be more difficult to characterize due to non-standardization and wide variation in model parameters in the image expansion set and the integration model. For this work, we analyze in detail the DD model for fan-beam CT with a limited scanning range, namely less than 180 degrees plus the fan-angle. The analysis is performed by specifying the class of system matrices considered and computing their condition number. A scaling is observed that aids in relating the condition number for large system matrices to that of more easily analyzed small matrices.