Inversion of the Attenuated Geodesic X-Ray Transform over Functions and Vector Fields on Simple Surfaces

We derive explicit reconstruction formulas for the attenuated geodesic X-ray transform over functions and, in the case of nonvanishing attenuation, vector fields, on a class of simple Riemannian surfaces with boundary. These formulas partly rely on new explicit approaches to construct continuous right-inverses for backprojection operators (and, in turn, holomorphic integrating factors), which were previously unavailable in a systematic form. The reconstruction of functions is presented in two ways, the latter being motivated by numerical considerations and successfully implemented in the last section of the paper (before the appendix). Constructing the aforementioned right-inverses requires that certain Fredholm equations, first appearing in [L. Pestov and G. Uhlmann, Int. Math. Res. Not., 2004 (2004), pp. 4331--4347], be invertible. Whether this last condition reduces the applicability of the overall approach to a strict subset of simple surfaces remains open at present.