## 2 Materials and Methods  
### Application of Noise Reduction Algorithms  
#### 2.1.1 Optimization of \(\sigma\)-Value for the Block Matching and 3D Filtering Algorithm  
The BM3D algorithm is one of the fusion-type noise reduction methods based on non-local means (NLM) [19]. The process of the BM3D algorithm can be largely divided into two steps, and in each step, noise reduction is performed through the wavelet transform-based hard-thresholding technique and Wiener filtering, respectively. The wavelet transform-based hard-thresholding technique treats pixel values below a set threshold value as 0 to increase sparsity, enabling simple and rapid calculation during algorithm processing. In addition, Wiener filtering is an adaptive filter commonly used to remove noise in medical images and has the advantage of less boundary information loss than other linear filters because it is adjusted appropriately for regional variance. Below is the equation for hard-thresholding [20]. \[\mathrm{ht}(x)= \begin{cases}x&if\ |x|\geq\lambda\\ 0&otherwise\end{cases} , \tag{1}\]  
Here, \(x\) and \(\lambda\) denote the input pixel value and the threshold. NLM is a noise reduction algorithm that is the basis of the grouping principle in the BM3D algorithm. The processing of entire image with the same weights or parameters is inefficient and inaccurate, since the noise distribution in a general image is not uniform. To compensate for this problem, NLM was devised [21]. NLM works on the principle of evaluating the similarity between kernels within the search window, and noise is removed by averaging similar image pixels according to the distance to the center pixel of each kernel. The entire image is divided into search windows since the pixel which is the most similar to the given pixel is not absolutely close. The weight for the center pixel p of the kernel set within the search window is equally applied to all pixels in the same kernel. The weights are determined by the square of the Euclidean distance and introduced to average similar kernels. \[\text{NLM}=\frac{1}{C(p)}\int f(d(B(p),(B(q)))u(q)dq, \tag{2}\]  
where \(d(B(p),~{}B(q))\) denotes the Euclidean distance of the two kernels centered at \(p\) and \(q\); \(f\) and \(C(p)\) mean the decreasing function and the normalization factor, respectively. The BM3D algorithm works through two key principles: grouping and collaborative filtering [22]. Grouping means collecting n-dimensional patches similar to the given signal (reference) in the original image in n+1 dimensions. In this study, it was used in the process of forming a 3D block by stacking patches extracted from 2D images. The similarity between patches is judged through the reciprocal of the distance measurement value using the L2 norm. If the distance is less than the predefined threshold, the patches are considered to be mutually similar and grouped together as a single 3D block. The patches in the 3D block are listed in descending or ascending order in a similar order. \[P_{2D}^{ht}=x\in X:\frac{\left\|Y^{\prime}\Big{(}\tau_{2D}^{ht}(P_{x_{R}}) \Big{)}-Y^{\prime}\Big{(}\tau_{2D}^{ht}(P_{x})\Big{)}\right\|_{2}^{2}}{\left( N_{1}^{ht}\right)^{2}}\leq\tau_{match}^{ht}, \tag{3}\]  
where \(P_{x_{R}}\) and \(P_{x}\) denote the reference block and similar block in the noisy image \(X\), \(Y^{\prime}\) is the hard-thresholding factor, \(\tau_{2D}^{ht}\) means the 2D linear transform, \(N_{1}^{ht}\) represents the block size of the first step, and \(P_{2D}^{ht}\) is the set of coordinates of all blocks similar to \(P_{x_{R}}\). \[E_{P_{2D}^{ht}}^{ht}=\tau_{3D}^{ht^{-1}}\Big{(}Y\Big{(}\tau_{3D}^{ht}\Big{(}A _{P_{2D}^{ht}}\Big{)}\Big{)}\Big{)}, \tag{4}\]  
where \(E_{P_{2D}^{ht}}^{ht}\) denotes the estimate acquired through inverse 3D transform, \(\tau_{3D}^{ht}\) and \(\tau_{3D}^{ht^{-1}}\) mean the 3D transform and inverse 3D transform, respectively. \(Y\) is the 3D hard-threholding factor, and \(A_{P_{2D}^{ht}}\) represents the formed 3D block.