Spectral computed tomography (CT) is a technology that utilizes the differences in attenuation coefficients of substances across different channels, which can demonstrate significant capabilities in material identification and analysis. Particularly, photon-counting spectral CT, which significantly curtails electronic noise and enhances resolution, signifies the latest technological advancements in CT imaging. However, effects such as photon starvation, charge sharing, and pulse pile-up engender severe noise in photon-counting spectral CT, directly undermining the image reconstruction quality and hampering the applications of photon-counting spectral CT technology. Our paramount research focus lies in accurately characterizing the statistical properties of projection data noise in photon-counting detectors, designing precise spectral CT reconstruction algorithms, and suppressing noise.

Initially, a theoretical analysis is conducted on the statistical noise characteristics in the projection data of photon-counting detectors. Specifically, by comprehensively considering the statistical distribution of photon flux and electronic noise in the projection data, where photon flux can be characterized by a compound Poisson distribution and approximated by a Gamma distribution, and electronic noise follows a Gaussian distribution. A theoretical noise distribution model of projection data is derived by the Bayesian formula. Subsequently, a statistical inference is carried out on the proposed theoretical noise distribution model of projection data. On the one hand, the probability distribution of the noise is fitted via actual data experimentation. On the other hand, a goodness-of-fit test is conducted on the theoretical noise distribution model. Ultimately, by adopting time series analysis for prediction, the predicted values are employed to restore outliers in the projection data.

We derive a rigorous theoretical noise distribution model in photon-counting spectral CT projection data (Eq. 9), bearing a similar expression to the univariate p-norm distribution. The rationality of characterizing the noise distribution of projection data using univariate p-norm distribution is then analyzed from three perspectives. By fitting the probability distribution of the actual data, the proposed univariate p-norm noise distribution model aligns more closely with the actual data than Gaussian, Poisson, and Gamma distributions, especially under extremely low photon flux, and the fitting degree of the proposed noise distribution model is optimal (Fig. 2). A goodness-of-fit test is conducted on the proposed noise distribution. The results are shown in Table 1. The proposed noise distribution is consistent with various collected datasets and consistency is the best in datasets with low photon flux. Lastly, the restoration of outliers using predicted values shows clear improvement from both visual images (Fig. 4) and quantitative assessments (Table 2). The proposed univariate p-norm distribution aptly characterizes the statistical properties of photon-counting spectral CT. However, the probability density function of the univariate p-norm distribution is challenging to calculate, and it should be transformed into a linear combination of Gaussian distribution and Laplace distribution for approximation, according to the p-value selection.

We investigate the statistical noise characteristics in the projection data of photon-counting spectral CT, and propose to employ univariate p-norm distribution to model the projection data noise. The distribution is verified by fitting actual data probability density functions and statistical inference tests. The univariate p-norm distribution can fully characterize the statistical law of observational errors. Especially under the insufficient number of photons, the univariate p-norm distribution can reach optimal when fitting the actual data distribution. The statistical probability model of projection data from the devised photon-counting detection system allows for an in-depth analysis of the system performance and accurate noise simulation during simulation experiments, and provides an accurate objective function for optimizing the likelihood functions in statistical iteration reconstruction. We explore the statistical noise characteristics of projection data in photon-counting detectors, enrich the theoretical results of X-ray spectral CT imaging systems, and provide theoretical support for the design and optimization of multi-spectral image reconstruction.