## 2 Materials and Methods
### Proposed Restoration Scheme for Microscopic Images
This block can detection used the rank correlation such as kendall's \(\tau\)-coefficient method [45; 46]. It can select the homogeneous block with high similarity by correlating them in four-directional expanded kendall's \(\tau\)-coefficient. The noise level can be calculated by averaging the standard deviation of the selected blocks and \(n_{\Omega}^{2}\) was used as the predicted noise level each subband images. Thus, there is an advantage in obtaining an optimal noise removal effect while preserving the un-sharpness compared to performing noise reduction on a single image. After reducing the effect on noise, each high-frequency component was performed using the blind deconvolution scheme to restore the blur artifact resulting from the finite focal spot size and denoising process. First, we used the conjugate gradient-based framework to estimate the PSF between the Laplacian image of \(f_{D}{}^{(k)}\) and g, according to [PERSON] et al. [47]:  
\[\
abla^{2}g(x,y)=\
abla^{2}f_{D}{}^{(k)}(x,y)\bigotimes\bigotimes pgf^{(k)}(x, y), \tag{3}\]  
Figure 2: Examples of noisy and denoising image among the high-frequency subbands images of R-channel degraded image using NSCT. where \(\
abla^{2}\) is the Laplacian operator. Here, differential image was acquired to extract only the edge information. Then, \({f_{D}}^{(k+1)}\) is predicted to solve the object function using the \(l_{1}\)-norm-based regularization term as follows:  
\[{f_{D}}^{(k+1)}(x,y)=\underset{{f}^{(k)}\in Q}{\arg\min}\frac{\alpha}{2}\left\|{ f_{D}}^{(k)}(x,y)\bigotimes\bigotimes\psi{psf}^{(k+1)}(x,y)-g(x,y)\right\|_{2}^{2}+ \left\|{\
abla f_{D}}^{(k)}(x,y)\right\|_{1}, \tag{4}\]  
where \(Q\) is the set of feasible values of \({f_{D}}^{(k)}\) and \(\alpha\) is a parameter that balances the two terms. When matches between the edge information of the degraded image and that of blurred image by convolving the appropriate PSF in clean image, the PSF can be considered as the optimal PSF of the optical microscope imaging system, and the process of repeating Equations (3) and (4) are performed to satisfy this requirement. [PERSON] et al., investigated the performance of proposed algorithm in total variation (TV)/\(l_{1}\) problem [48]. When using an objective function to obtain a deblurred image from a noisy image, it was difficult to give much weight to the regularization term due to increase the possibility of obtained the smoothing results. To overcome this limitation, our previous study demonstrated that proposed approach by setting the regularization term based on weight map and \(l_{1}\)-norm improved the sharpness of image without cartoonish artifact [49]. The \(l_{0}\)-norm-based deconvolution method without weight function was applied to the deblurred image for improving the resolution. In addition, we optimized the balance parameter values between the fidelity and regularization terms through trial and error until the discrepancy between the current and updated results converged to \(10^{-6}\). Therefore, \(\alpha\) is chosen such that the signal-to-noise ratio is maximized (In this study, \(\alpha=500\) was empirically used in the experiment). Finally, the optimization problem is calculated through the alternating direction method of multipliers (ADMM) method to find the optimal solution. This method is discussed in detail in a previous work [48]. Figure 3 shows the proposed restoration scheme flowchart incorporating the \(l_{1}\)-norm-based blind deconvolution and NSCT in light microscopy images. Briefly, a degraded image is acquired from the light microscope system (1), low- and high-frequency coefficients are separated via NSCT, and noise reduction is performed with the appropriate parameter considering each high-frequency coefficient (2). Subsequently, the deblurred images (3), which are calculated by entering the blind deconvolution scheme to improve the sharpness, are inversely transformed with the existing low-frequency coefficient (4).