  Material decomposition in X-ray imaging uses the energy-dependence of
attenuation to virtually decompose an object into specific constituent
materials. X-ray phase contrast imaging is a developing technique that can
enhance image contrast seen from weakly attenuating objects. In this paper, we
combine spectral phase contrast imaging with material decomposition to both
better visualise weakly attenuating features and separate them from overlying
objects in radiography. We derive an algorithm that performs both tasks
simultaneously and verify it against numerical simulations and experimental
measurements of ideal two-component samples composed of pure aluminium and
poly(methyl methacrylate). Additionally, we showcase first imaging results of a
rabbit kitten's lung. The attenuation signal of a thorax, in particular, is
dominated by the strongly attenuating bones of the ribcage, which combined with
the weak soft tissue signal make it difficult to visualise the fine anatomical
structures across the whole lung. In all cases, clean material decomposition
was achieved, without residual phase contrast effects, from which we generate
an un-obstructed image of the lung, free of bones. Spectral propagation-based
phase contrast imaging has the potential to be a valuable tool, not only in
future lung research, but also in other systems for which phase contrast
imaging in combination with material decomposition proves to be advantageous.
