Inversion of the 3D exponential x-ray transform for a half equatorial band and other semi-circular geometries

This work presents new mathematical results on the inversion of the exponential x-ray transform. It is shown that a reconstruction formula can be obtained for any dataset whose projection directions consist of a union of half great circles on the unit sphere. A basic example of such a dataset is the semi-equatorial band. The discussion in the paper is mostly focused on this example. The reconstruction formula takes the form of a Neumann (geometric) series and is both exact and stable. The exponential x-ray transform has been mainly studied in SPECT imaging. In this context, our results demonstrate mathematically that fully 3D image reconstruction in SPECT with non-zero attenuation does not always require symmetric datasets (opposing views).