Recently, my first first-author manuscript related to my thesis work on x-ray phase contrast was published. The paper summarizes artifacts experienced while using a tabletop x-ray phase contrast micro-CT system. While the focus of the paper is limited to grating-based x-ray phase contrast systems, the empirical methods used to solve the beam hardening problem are quite generalizable. I first learned about these interpolation-based empirical corrections from researching previous x-ray corrections that are referenced in the paper, but along the way I enjoyed digging more into the math of solving least squares problems and how they are implemented. The following tutorials document some of my explorations and provide background to the paper.