Sparse X-ray CT image reconstruction and blind beam hardening correction via mass attenuation discretization

We develop a nonlinear sparse X-ray computed tomography (CT) image reconstruction method that accounts for beam hardening effects due to polychromatic X-ray sources. We adopt the blind scenario where the material of the inspected object and the incident polychromatic source spectrum are unknown and apply mass attenuation discretization of the underlying integral expressions that model the noiseless measurements. Our reconstruction algorithm employs constrained minimization of a penalized least-squares cost function, where nonnegativity and maximum-energy constraints are imposed on incident spectrum parameters and negative-energy and smooth l₁-norm penalty terms are introduced to ensure the nonnegativity and sparsity of the density map image. This minimization scheme alternates between a nonlinear conjugate-gradient step for estimating the density map image and an active set step for estimating incident spectrum parameters. We compare the proposed method with the existing approaches, which ignore the polychromatic nature of the measurements or sparsity of the density map image.