Inversion of the 3D exponential parallel-beam transform and the Radon transform with angle-dependent attenuation

Abstract

The inversion problem for the 3D parallel-beam exponential ray transform is solved through inversion of a set of the 2D exponential Radon transforms with complex-valued angle-dependent attenuation. An inversion formula for the latter 2D transform is derived; it generalizes the known Kuchment–Shneiberg formula valid for real angle-dependent attenuation. We derive an explicit theoretically exact solution of the 3D problem which is valid for arbitrary closed trajectory that does not intersect itself. A simple reconstruction algorithm is described, applicable for certain sets of trajectories satisfying Orlov's condition. In the latter case, our inversion technique is as stable as the Tretiak–Metz inversion formula. Possibilities of further reduction of noise sensitivity are briefly discussed in the paper. The work of our algorithm is illustrated by an example of image reconstruction from two circular orbits.