Sufficient statistics as a generalization of binning in spectral X-ray imaging

It is well known that the energy dependence of X-ray attenuation can be used to characterize materials. Yet, even with energy discriminating photon counting X-ray detectors, it is still unclear how to best form energy dependent measurements for spectral imaging. Common ideas include binning photon counts based on their energies and detectors with both photon counting and energy integrating electronics. These approaches can be generalized to energy weighted measurements, which we prove can form a sufficient statistic for spectral X-ray imaging if the weights used, which we term μ-weights, are basis attenuation functions that can also be used for material decomposition. To study the performance of these different methods, we evaluate the Cramér-Rao lower bound (CRLB) of material estimates in the presence of quantum noise. We found that the choice of binning and weighting schemes can greatly affect the performance of material decomposition. Even with optimized thresholds, binning condenses information but incurs penalties to decomposition precision and is not robust to changes in the source spectrum or object size, although this can be mitigated by adding more bins or removing photons of certain energies from the spectrum. On the other hand, because μ-weighted measurements form a sufficient statistic for spectral imaging, the CRLB of the material decomposition estimates is identical to the quantum noise limited performance of a system with complete energy information of all photons. Finally, we show that μ-weights lead to increased conspicuity over other methods in a simulated calcium contrast experiment.