  We lay the foundations for a new fast method to reconstruct the electron
density in x-ray scanning applications using measurements in the dark field.
This approach is applied to a type of machine configuration with fixed energy
sensitive (or resolving) detectors, and where the X-ray source is
polychromatic. We consider the case where the measurements in the dark field
are dominated by the Compton scattering process. This leads us to a 2D inverse
problem where we aim to reconstruct an electron density slice from its
integrals over discs whose boundaries intersect the given source point. We show
that a unique solution exists for smooth densities compactly supported on an
annulus centred at the source point.
  Using Sobolev space estimates we determine a measure for the ill posedness of
our problem based on the criterion given by Natterer ("The mathematics of
computerized tomography" SIAM 2001). In addition, with a combination of our
method and the more common attenuation coefficient reconstruction, we show
under certain assumptions that the atomic number of the target is uniquely
determined.
  We test our method on simulated data sets with varying levels of added pseudo
random noise.
