Beam hardening correction via mass attenuation discretization

We develop a beam-hardening correction method for polychromatic x-ray computed tomography (CT) reconstruction based on mass attenuation coefficient discretization. We assume that the inspected object consists of an unknown single material and that the incident x-ray spectrum is unknown. In this case, the standard photon-energy discretization of the Beer´s law measurement equation leads to an excessive number of unknown parameters and scaling ambiguity. To obtain a parsimonious measurement model parametrization, we first rewrite the measurement equation in terms of integral expressions of the mass attenuation rather than photon energy. The resulting integrals can be discretized easily thanks to the fact that the range of mass attenuations is bounded and, in practice, fairly narrow. We then develop a constrained least-squares optimization approach for reconstructing the underlying object from log-scale measurements, where we impose the nonnegativity constraint to both the signal and the x-ray spectrum density estimates. We demonstrate the performance of the proposed method via a numerical example where we compare it with the standard filtered backprojection (FBP), which ignores the polychromatic nature of the measurements.