Date Published:

2020 Nov

Abstract:

PURPOSE: To provide a unifying statistical model that characterizes the integrated x-ray intensity at the detector after logarithmic transformation and can be extended to the characterization of computed tomography (CT) numbers in the reconstructed image. METHODS: We study the statistical characteristics of polyenergetic x-ray beams in the detector. Firstly, we consider the characterization of the integrated x-ray intensity at the detector through a probabilistic model (compound Poisson) that describes its statistics. We analyze its properties and derive the probabilistic distribution after the logarithmic transformation analytically. Finally, we propose a more tractable probabilistic distribution with the same features observed in the characterization, the noncentral Gamma (nc-Gamma). This distribution exhibits desirable properties for the statistical characterization across the reconstruction process. We assess the assumptions adopted in the derivation of the statistical models throughout Monte Carlo simulations and validate them with a water phantom and a lung phantom acquired in a Siemens clinical CT scan. We evaluate the statistical similarities between the theoretical distribution and the nc-Gamma using a power analysis with a Kolmogorov-Smirnov test for a 95% confidence level. RESULTS: The Kolmogorov-Smirnov goodness-of-fit test obtained for the Monte Carlo simulation shows an extremely high agreement between the empirical distribution of the post-logarithmic-integrated x-ray intensity and the nc-Gamma. The experimental validation performed with both phantoms confirmed the excellent match between the theoretical distribution, the proposed nc-Gamma, and sample distributions in all situations. CONCLUSION: We derive an analytical model describing the post-log distribution of the linear attenuation coefficient in the sensor for polychromatic CT scans. We also demonstrate that the nc-Gamma distribution approximates well the theoretical distribution. This distribution also approximates well the CT numbers after reconstruction since it naturally extends across linear operations involved in filtered back projection reconstructions. This probabilistic model may provide the analytical foundation to define new likelihood-based reconstruction methodologies for polychromatic scans.