# Evidence-Based Investigation of Coronary Calcium Score in Cardiac Computed Tomography
## 2 Materials and Methods
### Image Analysis  
Many mathematical models have been developed to represent degradation in radiologic images [22], of which the standard noise model is the most commonly used. However, it is limited in its ability to distinguish between Gaussian and Poisson noise. The Poisson-Gaussian mixture model addresses these limitations. It involves dividing the image into roughly uniform regions, estimating the mean and standard deviation for each region, and then using these local estimates to determine the global implicit functional relationship between image intensity and noise standard deviation through regression. In this study, we adopted this model because it better reflects the noise degradation characteristics of radiographic images. The Poisson-Gaussian mixture model is described by Equation (1):  
\[y(x)=p(x)+\eta(p(x))\delta,\ \eta^{2}(p(x))=\alpha p(x)+\beta^{2}, \tag{1}\]  
where \(x\) is the pixel position, and \(y\) and \(p\) are the degraded and ideal images without the noise component, respectively; \(\eta(\cdot)\) denotes the standard deviation of the noise distribution, and \(\delta\) is the zero-mean independent random noise and standard deviation equal to one; \(\alpha\) is the Poisson signal-dependent component, and \(\beta\) is the standard deviation of the Gaussian signal-dependent component. Among the noise level estimation methods used to predict \(\alpha\) and \(\beta\), the patch-based noise parameter prediction method has proven to be effective and accurate [23; 24; 25]. A homogeneous patch is crucial for predicting noise levels because it is assumed that any variation within the patch is solely due to noise. First, homogeneous patches were extracted from the image using the non-parametric test Kendall's tau rank correlation [26; 27]. [PERSON]'s tau was calculated as shown in Equation (2):  
\[\tau=\frac{C-D}{C+D^{\prime}} \tag{2}\]  
where \(C\) is the number of concordant pairs and \(D\) is the number of discordant pairs. \(\tau\) is in the range of \(-1\) to \(1\). When each variable has the same relative order in a comparison, it forms a concordant pair, and [PERSON]'s tau is particularly useful when the sample size is \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Patient Study** & **Phantom Study** \\ \hline kV\({}_{\text{p}}\) & 100 & 120, 100 \\ mAs & 80 & 80 \\ Scan time (s) & 0.14 & 0.14 \\ Rotation time (s) & 0.25 & 0.25 \\ Slice thickness/increment (mm) & 3.0/1.5 & 3.0, 1.0/1.5 \\ Field of view (mm) & 300 & 300 \\ Reconstruction kernel & Sa 36 & Sa 36 \\ Scan mode & Sequence & Sequence \\ Direction & Craniocaudal & Craniocaudal \\ Window & Mediastinum & Mediastinum \\ \hline \hline \end{tabular}
\end{table} 
Table 1: CT scanning parameters for the patient and phantom studies. small or the data show high levels of similarity, extracting only the most similar patches to calculate the noise level. In this study, the homogeneous patches were 16 \(\times\) 16 pixels. The smaller the value of \(\tau\), the more homogeneous the patch. The homogeneous patches can be detected by averaging the four-directional \(\tau\)-values. The obtained homogeneous patches can be expressed in terms of the mean (\(\mu\)) and standard deviation (\(\sigma\)). Second, it can be modeled as shown in Equation (3) to determine the optimal noise levels of \(\alpha\) and \(\beta^{2}\):  
\[f^{*}=\underset{f\in R^{+}}{\operatorname{argmin}}\|\text{U}f-V\|_{1},\ f= \begin{pmatrix}\alpha\\ \beta^{2}\end{pmatrix} ,\ U= \begin{pmatrix}n_{1}&1\\ \mu_{2}&1\\ \vdots&\vdots\\ \mu_{n}&1\end{pmatrix} ,\ V= \begin{pmatrix}\sigma_{1}^{2}\\ \sigma_{2}^{2}\\ \vdots\\ \sigma_{n}^{2}\end{pmatrix} , \tag{3}\]  
where \(f\) denotes the matrix of noise level parameters that satisfies the feasible sets \(f\in R^{+}\), and \(U\) and \(V\) are the mean matrix of \(n\times 2\) and variation matrix of \(n\times 1\), respectively. The minimization problem can be solved using a preconditioned primal-dual algorithm [28].