A complete scheme of empirical beam hardening correction using Grangeat consistency condition

The polychromatic X-ray spectrum and the energy-dependent attenuation coefficients of the imaged object introduce beam hardening artifacts in CT reconstructed volumes. Cupping, streak, and spill-over are the different manifestations of beam hardening artifacts. Cupping or first order artifacts can be corrected by the transformation of polychromatic projections with a univariate polynomial model. The residual spectral artifacts due to bone beam hardening can be reduced by iterative beam hardening correction methods. Here, the bone corrected projections are modeled with water corrected projections, forward projections of the segmented bone volume and a multivariate polynomial. In this paper, we present a calibration-free beam hardening correction using consistency conditions derived from Grangeat's fundamental relation. Our multi-pass algorithm corrects first and higher order artifacts sequentially. During the first pass of the algorithm, the polynomial for water correction is estimated by enforcing consistency condition on the uncorrected projection data. Subsequently, a prior volume is reconstructed from water corrected projections, and the forward projections of the segmented bone volume are computed. During the second pass of the algorithm, the polynomial coefficients for bone beam hardening correction are estimated using the same consistency condition. Our results from the simulation and clinical datasets show that the first and higher order artifacts are significantly reduced after the proposed correction. The algorithm does not require calibration or prior knowledge about the X-ray spectrum, material attenuation properties, or the detector response function.