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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.
X-ray scattering is a weak linear probe of matter. It is primarily sensitive to the position of electrons and their momentum distribution. Elastic X-ray scattering forms the basis of atomic structural determination while inelastic Compton scattering
The recent development of energy-resolving cameras opens the way to new types of applications and imaging systems. In this work, we consider computerized tomography (CT) with fan beam geometry and equipped with such cameras. The measured radiation is
We propose a new acquisition geometry for electron density reconstruction in three dimensional X-ray Compton imaging using a monochromatic source. This leads us to a new three dimensional inverse problem where we aim to reconstruct a real valued func
Here we present new $L^2$ injectivity results for 2-D and 3-D Compton scattering tomography (CST) problems in translational geometries. The results are proven through the explicit inversion of a new toric section and apple Radon transform, which desc
The article presents an efficient image reconstruction algorithm for single scattering optical tomography (SSOT) in circular geometry of data acquisition. This novel medical imaging modality uses photons of light that scatter once in the body to reco