3D image reconstruction from exponential X-ray projections: a completeness condition and an inversion formula

This work concerns the problem of reconstructing a 3D image from exponential X-ray (parallel-beam) projections. It is shown that exact reconstruction can be achieved when the projections are known on a set of directions Ω which satisfies the Orlov condition for non-attenuated projections. More specifically, it is shown that exact reconstruction can be achieved when the set Ω is intersected by every great circle on the unit sphere, provided the product μR is sufficiently small, where R is the radius of the region where the image is non-zero and μ is the attenuation coefficient. A reconstruction method is suggested and simulation results are provided to demonstrate the exactness and usefulness of the method.