# Image Noise Reduction by Means of Bootstrapping-Based Fuzzy Numbers
## 6 Fuzzy Amount of Pixel's Color  
One of the important points in using the algorithm presented in the previous section is to assign or calculate a fuzzy number for the color of each pixel. Since the color value of each pixel is expressed as a crisp number, different approaches to their fuzzy expression can be suggested, which can provide a field of study for future research. For a better explanation of the subject, it should be noted that just as the colors of a rainbow cannot be easily separated and the colors are slowly converted, the difference between the color of a pixel and the color of its neighborysis difficult. Fuzzy sets can easily do this, while in existing algorithms, a non-negative integer is considered for the color of each pixel. Our approach in this paper is to use the neighborhoods of a pixel to express the fuzzy color of the desired pixel. In other words, we assumed that the color of each pixel is an asymmetric triangular fuzzy number, the color of each pixel being determined by the degree of membership 1 and its left or right width by the mean of \(k\) means generated by the bootstrap resampling method where \(k\) is the number of sampling repetitions of the desired pixel neighborhoods. We used the bootstrap resampling method to reduce the effect of the outliers in the selected window. One of the main reasons for using the bootstrap is that this sampling is based on the sample we have. Often, this sample is the only source we have for research, and this adds to the importance of the bootstrap method. Since the bootstrap reliability improves with the number of bootstrap repetitions, we can obtain a more reliable estimate of the mean value of the central pixel neighborhoods by increasing the number of samples. The bootstrap method provides an estimate of a quantity of a population. This is achieved by collecting small samples repeatedly, calculating the statistics, and averaging the calculated statistics. This procedure can be summarized in the following terms:  
* Select a number of bootstrap samples to perform;
* Select a sample size;
* For each bootstrap sample: a Draw a sample with replacement with the chosen size; b Calculate the statistic on the sample;
* Calculate the mean of the calculated sample statistics. Figure 4: _BCFM_ matrix. Figure 3: Matrix neighborhood with noise. For example, if the image is grayscale, we have only one color number for each pixel. Consider a \(3\times 3\) window of its neighborhoods of each pixel (except in the pixels of the image margin), and calculate the mean of the bootstrap means. The obtained number is a triangular fuzzy number with a right or left width. For example, suppose the pixel color is 50 and the mean of the bootstrap means is 60. Consider the color of a pixel as a fuzzy number \((50,0,10)\), with the left width of zero (because the mean of the bootstrap means is larger than the desired pixel) and the right width of 10 (which is equal to  
_Now, we eliminated the noise by performing the algorithm described in this section. We know that_  
\[\|\hat{u}_{i}-\hat{u}_{j}\|=d_{2}(\hat{u},\hat{v})=(\int_{0}^{1}f(\alpha)d^{2}(L_ {\alpha}\tilde{u},L_{\alpha}\tilde{v})d\alpha)^{\frac{1}{2}},\]  
_and \(\tilde{u},\tilde{v}\) are two asymmetric triangular fuzzy numbers as follows:_  
\[\tilde{u}=(a,sl_{a},sr_{a})_{T},\] \[\tilde{v}=(b,sl_{b},sr_{b})_{T},\]  
_where \(sl\) is the left width and \(sr\) is the right width.