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.
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