## 2 Materials and Methods
### Adaptive Nonlocal Total Variation (NL-TV) Filter Modeling  
Figure 1 shows a simplified framework of the adaptive search window NL-TV filter for low-dose CT images. Briefly, an NL-TV filter is used to remove noise after acquiring low-dose CT images. The NL-TV filter was designed by [PERSON] et al. [43], based on Equations (1)-(3):  
\[\mathcal{O}(x)=\underset{x}{\text{argmin}}\int_{\Omega}|D_{NL}x|+\frac{\beta }{2}\|x\|_{2}^{2}+\frac{1}{2}\|x-f\|_{2}^{2}, \tag{1}\]  
\[D_{NL}x(s,t)=(x(s)-x(t))\sqrt{w(s,t)}, \tag{2}\]  
\[w(s,t)=\text{exp}\Bigg{(}-\frac{\int_{\Omega}G_{h}(d)|f(s+d)-f(t+d)|_{2}^{2}}{ 2p^{2}}\Bigg{)}. \tag{3}\]In the equations, \(x\) represents the noise-free image and \(f\) is the observed image with a noise component. The domain \(\Omega\in\mathbb{R}^{2}\) is a bounded open region with a Lipschitz continuous boundary. \(\beta\) is a parameter that balances different terms in the model and \(\beta\) = 0.01 is used in this study, empirically. \(D_{NL}x\) is the nonlocal gradient of \(x\). The function \(w(s,t)\) acts as a weighting function to estimate the similarity between two pixels, \(s\) and \(t\). \(G_{h}\) is a Gaussian kernel with a standard deviation, \(h\), of the number of background pixels, \(d\). Here, \(d\) was set to 3 and Gaussian kernel size was set to \(11\times 11\) pixels. \(|f(s+d)-f(t+d)|_{2}^{2}\) is the difference between adjacent pixels based on the Euclidean distance. \(p\) corresponds to the smoothing scale in relation to the noise level, and we used the \(p\) = 3 empirically. To solve the optimization problem in Equation (1), the split Bregman method [44] was used. A smoothed image was obtained by providing more smoothing parameters to the NL-TV filtered image, although the sharpness was reduced. A Gaussian distribution was used as the filter for smoothing, and the kernel size was set to 11 \(\times\) 11 pixels.  
The absolute error for each pixel was computed to determine the gray-level difference (GLD, \(d\hat{X}\)) [40] in the Gaussian filtering image. The first threshold was set as the average value (\(\mu\)) of the GLD, whereas the second threshold was determined by adding the \(\sigma\) of the GLD, multiplied by a weight factor (\(\alpha\)), which can be adjusted to determine the optimal window, to the average GLD value. In this study, the value of \(\alpha\) was set to 0.5, based on common practice. The optimal search window, (\(p_{i}^{opt}\)), was designed as follows [45]:  
\[p_{i}^{opt}=\left\{\begin{array}{c}c_{1},\ d\hat{X}_{i}<T_{1}\\ c_{2},\ T_{1}\leq d\hat{X}_{i}<T_{2}\\ c_{3},\ d\hat{X}_{i}>T_{2}\end{array}\right\}. \tag{4}\]  
The value \(c_{1}\) was assigned if \(d\hat{X}_{i}\) was less than \(T_{1}\), \(c_{2}\) was assigned if \(d\hat{X}_{i}\) was between \(T_{1}\) and \(T_{2}\), and \(c_{3}\) was assigned if \(d\hat{X}_{i}\) was greater than \(T_{2}\) at pixel \(i\). The values of \(c_{1}\), \(c_{2}\), and \(c_{3}\) were multiplied by 0.7, 0.9, and 1.1 times the \(p\) values used in the NL-TV method. The multiplication constant was set as required. The value of each constant can be chosen by the user, adjusted by experience. Finally, the optimal search window derived from the NL-TV algorithm for each pixel was applied to the \(p\) value to achieve optimal noise reduction in the low-dose CT image.