## 2 Materials and Methods  
### Proposed Restoration Scheme for Microscopic Images  
[PERSON] and [PERSON] [35] introduced the contourlet transform (CT), which is divided into flexible multiscale and multidirectional expansions. However, it has a limitation--the shift-invariance property is unfulfilled while performing up- and downsampling. NSCT overcomes this problem. Figure 1 shows a simplified NSCT flowchart for a microscopic image. The NSCT decomposition process is classified into two phases: (1) non-subsampled pyramid (NSP) and (2) non-subsampled directional fan filter bank (NSDFB). The NSP separates the low- and high-frequency subband images while maintaining the shift-invariant relationship, and then decomposes the high-frequency subband images at each level [36]. Subsequently, the NSCT can be reconstructed without data loss to ensure the shift-invariant property and detect detailed features according to direction. More information is available in the literature [37]. We implemented noise reduction in a suitable frequency region via NSCT. Figure 2 shows some of examples; noisy and denoising image among the high-frequency subbands images of R-channel degraded image using NSCT. Wiener filtering [38] was used to reduce the noise component. By using the NSCT, we can obtain a successful noise reduction by assigning appropriate parameters of the denoising algorithm in each image, which are divided according to frequency and filter direction. Here, we used the 2D modified Wiener filtering (MMWF) method to reduce the noise component. MMWF has advantage of mean  
Figure 1: Simplified flowchart of the proposed non-subsampled contourlet transform (NSCT). and Wiener filtering in terms of maintaining resolution, even though noise reduction has been implemented [39] and it can be written in Equation (2)  
\[MMWF(x,y)=\widetilde{g_{\Omega}}+\frac{\sigma_{\Omega}^{2}-n_{\Omega}^{2}}{\sigma _{\Omega}^{2}}\Big{(}g_{\Omega}(x,y)-\widetilde{g_{\Omega}}\Big{)}, \tag{2}\]  
where \(\widetilde{g_{\Omega}}\) and \(\sigma_{\Omega}\) are the mean value and standard deviation in mask \(\Omega\), respectively, and \(n_{\Omega}\) is the standard variation of noise. We investigated the size of \(\Omega\) according to the [PERSON] et al. [40] and empirically set to \(11\times 11\) pixels in simulation and experiment. In addition, \(n_{\Omega}\) was calculated by finding the accurate noise level using the homogeneous blocks. The previous studies [41; 42; 43] introduce structure-oriented noise estimation the method that eight directional high-pass filters were applied to the microscopy image and then the resulting image was added. The 10% of pixels with the smallest sum were selected. This is assumed to be a homogeneous block containing only noise. This method has the advantage of effectively extracting homogeneous block, but the disadvantage is that the process of selecting homogeneous block reflects human error, making it difficult to produce consistent results. [PERSON] et al. [44], has introduced an effective method to estimate the noise parameters. The input image was divided into small blocks and the homogenous block selected (e.g., 10 \(\times\) 10 pixels, empirically). Here, the homogeneous blocks are an assumption that noise variation is the dominant factor. [PERSON]'s paper indicates that the larger the block size, the better the accuracy in predicting the noise level. However, within the proposed image size, a larger block size means fewer blocks extracted. [PERSON]'s study used a block of 16 \(\times\) 16 pixels in 512 \(\times\) 512 pixels in optical image. Applying these conditions to our experiments, we extracted fewer than 30 homogeneous blocks on average, despite the much larger image size (3264 \(\times\) 2448 pixels). Through trial and error, we selected a block size of 10 \(\times\) 10 pixels that reliably predicted the noise level in each NSCT domain. Note that the block size chosen is not absolute, as it can change depending on the conditions of the light microscopy imaging system.