We present new software allowing significantly improved quantitative mapping of the three-dimensional density distribution of objects using laboratory source polychromatic X-rays via a beam characterisation approach (c.f. filtering or comparison to phantoms). One key advantage is that a precise representation of the specimen material is not required. The method exploits well-established, widely available, non-destructive and increasingly accessible laboratory-source X-ray tomography. Beam characterisation is performed in two stages: (1) projection data are collected through a range of known materials utilising a novel hardware design integrated into the rotation stage; and (2) a Python code optimises a spectral response model of the system. We provide hardware designs for use with a rotation stage able to be tilted, yet the concept is easily adaptable to virtually any laboratory system and sample, and implicitly corrects the image artefact known as beam hardening.