Alternative Fan-beam Backprojection and Adjoint Operators

We present in this work alternative analytic formulations for the fan-beam tomographic backprojection operation and its associated adjoint transform in standard (equiangular) and linear (equidistant) detector geometries. The proposed formulations are obtained from a recent backprojection theorem in parallel tomography. Such formulations are written as a Bessel-Neumann series in the frequency domain that can be implemented as an O(N^2.3729) matrix multiplication. Proofs are provided together with numerical simulations compared with conventional fan-beam O(N^3) backprojection representations showing more robustness when dealing with highly noisy data.