Cramér–Rao lower bound for spectral propagation-based x-ray phase-contrast imaging

Abstract Objective: Spectral x-ray phase-contrast imaging (XPCI) is a promising emerging method for quantitative phase retrieval. With an energy-discriminating detector, the independent measurements needed for quantitative phase retrieval or material decomposition can be acquired in a single shot. The existing Cramér–Rao lower bound (CRLB) model for attenuation-based spectral imaging, however, cannot capture the spatially distributed phase information associated with propagation-based (PB) XPCI. In this work, we develop a generalized CRLB framework for spectral PB-XPCI. Approach: The PB-XPCI forward model was formulated for nonideal, polychromatic photon-counting detector measurements. Two-material toy models were constructed with simple edge geometry to parallel the attenuation-based single-ray CRLB, with Poisson Fisher information computed over both energy-resolved and spatially distributed detector measurements. The conventional single-ray and proposed single-edge CRLBs were compared for ideal dual-monoenergy acquisitions. The single-edge model was then used to optimize detector energy thresholds for a nonideal 30-kVp polychromatic system, and CRLB predictions were validated against simulations. Main results: The single-edge model yielded substantially lower CRLBs relative to the fluence-matched single-ray model, while free-space propagation provided additional energy-dependent improvements. For the idealized monoenergetic single-edge model, the CRLB for each basis material was minimized when both measurements used the same energy, indicating that spatial measurements supplied sufficient independent information for material decomposition. For the polychromatic system, the CRLB exhibited material-dependent optimal energy thresholds, which shifted with the number of available energy bins. The single-edge and 2D-geometry CRLB models identified similar optimal energy thresholds within 0.1–1.9 keV, while simulated maximum-likelihood thickness estimates closely approached the CRLB with mean variance ratios of 1.006–1.020. Significance: The proposed CRLB model retains the interpretability of the familiar attenuation-based model while including propagation-based phase information, providing a useful theoretical foundation for evaluation and optimization of emerging spectral PB-XPCI systems as well as proposed phase retrieval and material decomposition algorithms.