CT fan-beam parameterizations leading to shift-invariant filtering

The problem of reconstructing a tomographic image from fan-beam projections via shift-invariant filtering followed by back-projection (without rebinning the projection rays) has a solution for two well-known parameterization classes. These parameterizations are associated either with equidistant collinear detector cells or with equiangular fan rays. Here, the problem of finding all the tomographic fan-beam parameterizations that lead to image reconstruction via shift-invariant filtering (convolution) followed by backprojection is solved in generality, under a simple symmetry condition. Two new parameterization classes are found, that define new reconstruction algorithms and provide the mathematical framework describing new CT detector geometries. Numerical solutions are found for the differential equations characterizing the detector geometries. Application of the numerical approach also uncovers a new detector geometry associated with one of the two previously known parameterization classes. Also, by mapping the fan-beam parameterizations onto the familiar third-generation geometry, new variable resolution samplings of the field of view are obtained. Finally, the question of finding a shift-invariant filter decomposition for the fourth generation geometry is addressed.