Exact cone beam reconstruction formulae for functions and their gradients for spherical and flat detectors

We derive unified inversion formulae for the cone beam transform similar to the Radon transform. Reinterpreting Grangeat's formula we find a relation between the Radon transform of the gradient of the searched-for function and a quantity computable from cone beam data. This gives a uniqueness result for the cone beam transform of compactly supported functions under much weaker assumptions than the Tuy–Kirillov condition. Furthermore this relation leads to an exact formula for the direct calculation of derivatives of the density distribution; but here, similar to the classical Radon transform, complete Radon data are needed, hence the Tuy–Kirillov condition has to be imposed. Numerical experiments reported in Hahn B N et al (2013 Meas. Sci. Technol. 24 125601) indicate that these calculations are less corrupted by beam-hardening noise. Finally, we present flat detector versions for these results, which are mathematically less attractive but important for applications.