Alternative fan-beam backprojection and adjoint operators

Abstract

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) {O(N^{2.3729})} matrix multiplication. Proofs are provided together with numerical simulations compared with conventional fan-beam O ⁢ (N 3) {O(N^{3})} backprojection representations showing more robustness when dealing with highly noisy data.