Approximate Cramér-Rao lower bound analyses for semi-optimal energy thresholds selection in photon counting CT

Selecting energy thresholds in photon counting CT (PCCT) is crucial for reducing noise in material decomposition. Theoretically, it can be determined by analyzing the Cramér-Rao lower bound (CRLB). However, the CRLB analysis is challenging in practice without complete knowledge of system models.

This study aims to develop a practical method for selecting semi-optimal energy thresholds in PCCT without the system models.

We proposed approximate CRLB approaches, linear and nonlinear. It only needs to generate look-up tables of measured counts using a threshold scan for reference slab phantoms, which roughly model the largest basis line integrals in the object. We also introduced total noise, which summarizes the noise in the basis line integrals and can be calculated from the look-up table; one can select energy thresholds that yield the minimum total noise. The proposed methods were validated using the PcTK toolbox in simulation studies for two- and three-material decomposition for various numbers of energy thresholds. The proposed methods were compared to the theoretically optimal CRLB analysis and the condition of equally distributed counts (EDC). It was also validated using a bench-top PCCT system for two-material decomposition with two energy thresholds.

In simulation studies, all methods showed similar total noises for two-material decomposition. For the three-material decomposition, the proposed methods showed lower or similar noise levels compared to the EDC condition. All methods showed higher noise than the optimal one, except the proposed nonlinear approach, which exhibited the optimal noise level at four energy bins. In experiments, the proposed linear method achieved a near-optimal performance in a two-material decomposition, while the EDC condition increased noise levels by $15\%$ and $18\%$ for epoxy resin and CaHA, respectively.

The proposed energy threshold selection methods are practical and effective: they require only a threshold scan of the reference phantoms and are computationally efficient. It outperformed the existing EDC condition in both simulation and experimental studies. It also showed a near-optimal noise level across various conditions, except in the three-material with three-energy-bin case.