## 2 Materials and Methods
### Quantitative Evaluation for the Results of Applying Noise Reduction Algorithms  
We measured the contrast-to-noise ratio (CNR) and coefficient of variation (CV) to quantitatively evaluate the results of optimization and application of the BM3D algorithm in this simulation study [30; 31]. ROIs for the quantitative evaluation were set in cerebrospinal fluid (CSF) and white matter (WM).  
The CNR is a factor representing the contrast ratio between adjacent tissues and is one of the methods for evaluating contrast resolution in medical images. It represents the similar results to visual evaluation and is often used as an index for quantitative evaluation such as image quality control and equipment performance comparison. The CNR can be obtained by dividing the signal intensity by the noise standard deviation, and the value decreases as the image contains more noise. The set ROIs in CSF and WM were used to calculate the CNR. The CV is a metric that quantitatively assesses the amount of noise present in an image by dividing the standard deviation within the defined ROI by the signal value. Therefore, CV was employed to measure the noise reduction rate after restoring the noisy image in this study. A smaller value means that the image contains less noise. The final value used for quantitative evaluation was acquired by averaging the CV of the two ROIs.  
\[\text{CNR}=\frac{|S_{CSF}-S_{WM}|}{\sqrt{\sigma_{CSF}^{2}+\sigma_{WM}^{2}}}, \tag{9}\]  
\[\text{CV}=\frac{\sigma_{ROI}}{S_{ROI}}, \tag{10}\]  
where \(S_{CSF}\) and \(S_{WM}\) denote the signal intensity in the ROIs of the CSF and WM, \(\sigma_{CSF}\) and \(\sigma_{WM}\) represent the standard deviations.  
In addition, the structural similarity index measurement (SSIM) and root mean square error (RMSE), which are similarity evaluation factors, were measured to evaluate how closely the degraded image was reconstructed from the noise-free reference [32; 33]. The characteristic of simulation studies lies in the ability to obtain an ideal reference image with zero noise. The objective of noise reduction research is to make the degraded image most similar to the reference through algorithms. For this purpose, the structural similarity index measure (SSIM) was utilized to compare the reference and resulting images. SSIM enables a quantitative evaluation similar to human vision in terms of brightness, contrast, and structure. In addition, the RMSE is also for evaluating the similarity between the reference and noisy image. The difference is measured by calculating the pixels in the corresponding coordinates in both images. The RMSE is used as an index to determine the degree of distortion in the entire image. As the SSIM and RMSE assess the similarity between the restored noisy images and the reference images, the entire region of the images was utilized to evaluate the overall structure and pixel values.  
\[\text{SSIM}(x,\;y)=\frac{\big{(}2\mu_{x}\mu_{y}+c_{1}\big{)}\big{(}2\sigma_{xy }+c_{2}\big{)}}{\Big{(}\mu_{x}^{2}+\mu_{y}^{2}+c_{1}\big{)}\Big{(}\sigma_{x}^{ 2}+\sigma_{y}^{2}+c_{2}\Big{)}}, \tag{11}\]  
where given the images \(x\) and \(y\), \(\mu_{x}\) and \(\mu_{y}\) denote the mean value of \(x\) and \(y\), \(\sigma_{x}\) and \(\sigma_{y}\) represent the standard deviation, \(c_{1}\) and \(c_{2}\) represent the stabilization variables.  
\[\text{RMSE}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}[I_{1}(m,\;n)-I_{2}(m,\;n)]^{2}}, \tag{12}\]  
where \(N\) denotes the number of pixels; \(I_{1}\) and \(I_{2}\) denote the reference and noisy images, respectively; and \((m,\;n)\) are the coordinates of the pixel.