Our method involves a post-acquisition software correction that applies a beam-hardening correction curve to remap the linearised projection intensities. The curve is modelled by an eighth-order polynomial and assumes an average material for the object. The process to determine the best correction curve requires precisely 8 reconstructions and re-projections of the experiment data. The best correction curve is defined as that which generates a projection set p that minimises the reprojection distance. Reprojection distance is defined as the L2 norm of the difference between p, a set of projections, and RR†p, the result after p is reconstructed and then reprojected, i.e., ║RR†p − p║2. Here R denotes the projection operator and R† is its Moore-Penrose pseudoinverse, i.e., the reconstruction operator.
This technique was designed for single-material objects and in this case the calculated curve matches that determined experimentally. However, this technique works very well for multiple-material objects where the resulting curve is a kind of average of all materials present. We show that this technique corrects for both cupping and streaking in tomographic images by including several experimental examples. Note that this correction method requires no knowledge of the X-ray spectrum or materials present and can therefore be applied to old data sets.