Cramér–Rao lower bound of basis image noise in multiple-energy x-ray imaging

We present an analytical method to compute the basis image noise in the context of multi-energy x-ray imaging based on the Cramér-Rao lower bound (CRLB). The proposed formalism extends the original idea of Alvarez and Macovski (1976 Phys. Med. Biol. 21 733) to estimate the noise in the photo-effect and Compton-effect basis images in the case of dual-energy imaging. It includes an arbitrary number of independent, spectrally distinct attenuation measurements and also goes beyond the two-dimensional decomposition of the attenuation, including, e.g., a contrast agent as a third basis material. To illustrate our method, we consider three simple applications. The first application is to study the influence of the exact values for the energy thresholds on the basis image noise for a binned photon-counting system. The second application relates to the same detector system as the first and is an investigation of the dependence of the basis image noise on the energy resolution of the detector system. Finally, the third application provides an example for the case of an energy-integrating detector: the aim is to optimize the front-scintillator layer thickness of a dual-crystal detector for dual-energy imaging. The CRLB is used to minimize the noise of a photo-effect/Compton-effect basis material decomposition.