Optimal energy thresholds and weights for separating materials using photon counting x-ray detectors with energy discriminating capabilities

It is well known that decomposing an object into attenuation or material basis functions provides additional imaging benefits such as contrast enhancement or material subtraction. This can be accomplished with photon counting x-ray detectors (PCXDs) with energy discriminating capabilities, which enable us to count x-ray photons and classify them based on their energies. The richness of the information contained in these measurements can depend heavily on how these photons are binned together. In this paper, our goal is to identify a method that yields the optimal energy thresholds and/or weights for binning data from energy discriminating PCXDs. Additional energy information from these PCXDs allows us to use maximum-likelihood to estimate the amount of the basis materials penetrated by the beam. However, due to inherent quantum noise, these estimates are themselves noisy. We show that for PCXDs that discriminate between low and high energy photons, it is beneficial to have a gap between the thresholds. Photons with energies that fall into this gap should either be discarded or counted separately to improve material separability. Furthermore, if the PCXD can discern the energy of each photon, we show that when estimating the amount of each of two material basis functions, two appropriately weighted sums of the photon counts provide as much information as knowing the number of counts at each energy.