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Purpose: In-Depth photon counting detectors (PCXD) use an edge-on configuration and have multi-layer segmentations. The benefit of this configuration for additional spectral information depends on the energy response. Also, inter-layer cross-talk introduces correlation to the signal collected from each layer, which makes the independent Poisson model no longer valid for estimating the Cramér-Rao lower bound (CRLB) of the material decomposition variance. We proposed to use a multivariate Gaussian model as the substitute address the data correlation. Methods: A 120 kVp incident spectrum was simulated and transmitted through 25cm of water and 1cm of calcium. 5- layer In-Depth and 1-layer Edge-On PCXDs with full energy resolution were simulated using Monte Carlo methods. We selected Si, GaAs and CdTe as detector materials. The detectors were defined to have 1mm wide pixels and thickness of 70mm (Si), 10.5mm (GaAs) and 3mm (CdTe). Geant4 was used and energy response functions (ERF) capturing secondary events were obtained, together with the Gaussian parameter estimates. We evaluated the CRLBs of the In-Depth and Edge-On detectors for each material and the systematic variance bounds were compared. Results: For uncorrelated data, the CRLB can assume Poisson statistics. As the data becomes more correlated, the Poisson CRLB fails to capture the cross-talk effect, but a Gaussian model can, and is accurate if the number of photons is not small. The CRLB analysis shows that the effects of the ERF and the noise correlation are significant. If cross-talk can be corrected, the depth information proves to be beneficial and can reduce the variance lower bound by 3% to 10% depending on the detector material. Conclusions: The multivariate Gaussian model was validated to be a good substitute to the Poisson model for PCXD CRLB estimation. It can avoid the errors that would otherwise be caused by correlated measurements.
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