# Image Noise Reduction by Means of Bootstrapping-Based Fuzzy Numbers
## 6 Fuzzy Amount of Pixel's Color
Then,_  
\[d_{2}(\tilde{u},\tilde{v})=(\int_{0}^{1}\alpha d^{2}(L_{\alpha} \tilde{u},L_{\alpha}\tilde{v})d\alpha)^{\frac{1}{2}} = (\frac{1}{4}(sl_{b}-sl_{a})^{2}+\frac{1}{2}[(a-b)+(sl_{b}-sl_{a}) ]^{2}\] \[- \frac{2}{3}(sl_{b}-sl_{a})[(a-b)+(sl_{b}-sl_{a})]\] \[+ \frac{1}{4}(sr_{b}-sr_{a})^{2}+\frac{1}{2}[(a-b)-(sr_{b}-sr_{a}) ]^{2}\] \[+ \frac{2}{3}(sr_{b}-sr_{a})[(a-b)-(sr_{b}-sr_{a})])^{\frac{1}{2}}.\]  
_and BCFM will be equal to:_  
\[BCFM(\tilde{u}_{i},\tilde{u}_{j},t)=\frac{c}{c+d_{2}(\tilde{u},\tilde{v})}. \frac{t}{t+\|i-j\|_{\infty}}. \tag{6}\]  
_Now, the estimate of the noisy pixel for \(10\) repetitions in bootstrap sampling is shown in Figure 3:0_  
Figure 3:1 Left width of the fuzzy matrix. Figure 3:2 Image matrix after noise reduction. Figure 3:3 Right width of the fuzzy matrix. It can be seen that the value of this pixel (162) is the best estimate for the center of the matrix, which was \(165\) as the actual value. That is, while using the median filter, the alternative value will be equal to \(157\). Of course, it should be noted that to achieve a convergence in estimation, it is necessary to increase the number of iterations. In the following, as a real example, we removed the noise from the gray and color images. First, for the gray image, we examined the lung CT scan image related to the COVID-19 virus. Then, we used the images of [PERSON] and [PERSON] for the color images. For each image, we compared the result of the filter presented in this paper with the median filter and _CFM_ using the mean absolute error (_MAE_) and peak signal-to-noise ratio (_PSNR_). **Example 5**. : _A lung CT scan is a common imaging tool for diagnosing pneumonia. CT scans provide high-quality images of lung tissue, and the radiologist can quickly determine the extent of lung involvement with the disease. A CT scan of the lung reveals common radiological features in patients with COVID-19 pneumonia. Noise in medical images can make it difficult to diagnose COVID-19. In this section, we remove the noise from the lung CT scan image in a patient with the COVID-19 virus by using the BCFM filter. To show the power of the BCFM filter for fuzzy sets in noise removal or reduction, we first added noise (salt and pepper of 5 and 10 percent) to the initial image._  
_After using the filter, the results are in Figure 11 and Figure 3:4 As we can see in Table 1, the BCFM filter has the lowest MAE and the highest PSNR among the three filters BCFM, CFM, and Median. Furthermore, as the number of bootstrap samples increases, MAE and PSNR will converge to the minimum and maximum values, respectively._  
To better compare the two methods CFM and BCFM, we can use the probability density graph of images. Figure 12 shows the density of the original image in black, CFM in green, and BCFM in red. Whichever density of green or red is closer to the density of black indicates the better performance of the method. Careful examination of the densities shows that, except in the range of \(0\) to about \(30\), the red density is closer to black, and in general, it can be said that the red density performs better in most places or is similar to the green density. To better illustrate this point, we see a slice of the graph in Figure 3:5 \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Noise Percentage**} & \multirow{2}{*}{**Filter Evaluation**} & \multirow{2}{*}{**Median Filter**} & \multirow{2}{*}{**CFM Filter**} & \multicolumn{4}{c}{_BCFM Filter_} \\ \cline{5-6}  & & & & \(n=10\) & \(n=20\) & \(n=50\) \\ \hline
5 & MAE & 5.0347 & 1.009 & 0.7990 & 0.7948 & 0.7935 \\
5 & PSNR & 22.5748 & 24.2157 & 24.7815 & 24.7672 & 24.7862 \\
10 & MAE & 5.4470 & 1.7251 & 1.5781 & 1.5621 & 1.5610 \\
10 & PSNR & 21.8976 & 22.7346 & 23.0609 & 23.0460 & 23.0580 \\ \hline \hline \end{tabular} Note: \(n\) is the number of bootstrap resampling.  
\end{table} 
Table 1: Comparison of the performance of noise reduction filters for the COVID-19 image. Figure 11: 1. Original COVID-19 image. 2. Noisy image after adding noise. 3.