Spectral x-ray imaging: Conditions under which propagation-based phase-contrast is beneficial

Energy-resolved attenuation data in spectral x-ray imaging enables material decomposition, in which the different materials inside an object can be identified and separated virtually. Material decomposition has the drawback of increased noise in the resulting material images relative to the measured images. Recently, spectral x-ray imaging was combined with propagation-based x-ray phase-contrast imaging, an x-ray technique that has the potential to greatly reduce image noise by utilizing wave-optical effects. The net combined effects on image noise of performing spectral material decomposition with phase-contrast are not yet well understood, and we provide a detailed theoretical investigation of this topic here. In particular, we investigate how the addition of phase-contrast in spectral imaging affects material decomposition compared to using conventional spectral attenuation data. We show how the underlying equations can be rearranged into parts that resemble low- and high-pass filters on the input images, from which we are able to identify different energy-dependent cases where phase-contrast is or is not advantageous. Our results suggest that the benefits of phase-contrast in the context of material decomposition are primarily restricted to x-ray energies under a certain threshold, where that threshold depends on the given material combination, and sits in a region where photoelectric absorption dominates x-ray attenuation. Additionally, we show that decomposition of the electron density using an image basis spanned by functions of the Alvarez-Macovski model benefits from phase-contrast, regardless of the x-ray energies. All our findings are based purely on theoretical considerations, and can, therefore, be used to determine the feasibility and utility of propagation-based phase-contrast in spectral x-ray imaging ahead of any data collection.