CT fan-beam parametrizations leading to shift-invariant filtering

The problem of two-dimensional tomographic image reconstruction from fan-beam projections via shift-invariant filtering (convolution) followed by backprojection has a solution for two well known fan-beam parametrization classes. These parametrizations are associated either with equidistant collinear detector cells or with equi-angular fan rays. In this paper, the problem of finding all such fan-beam parametrizations is solved. Two new parametrization classes are found, which define new CT reconstruction algorithms. Two interpretations are given for the new parametrizations. First, the associated loci of equidistant detector cells are found by solving numerically the differential equations that characterize them. Application of the numerical approach also unveils a new detector geometry associated with one of the two previously known parametrization classes. Secondly, by mapping the parametrizations onto the familiar third-generation CT geometry, variable resolution samplings of the scan field of view are found. Also, the problem of CT image reconstruction on a fourth-generation scanner is addressed. Although it is found that the corresponding parametrization does not belong to the convolution solution set, a `natural' shift-invariant filter decomposition is described that provides a close approximation to the reconstruction filter.