# Impulsive Noise Removal with an Adaptive Weighted Arithmetic Mean Operator for Any Noise Density
## 3 Weighted Arithmetic Mean Based Fuzzy Filter Algorithm
### Noise Detection Phase Based on Fuzzy Open-Close Operators
The algorithm designed by [PERSON] and [PERSON] in order to perform a fast restoration of natural images corrupted by high-density impulsive noise presented in [PERSON] and [PERSON] (2018). * _[PERSON]_. A two-stage adaptive iterative fuzzy filter for denoising images corrupted with high-density salt-and-pepper noise proposed in [PERSON] and [PERSON] (2018). In this filter, the detection of noisy pixels is performed with an adaptive fuzzy detector followed by denoising using a weighted arithmetic mean filter, with weights expressed as a potential function of the Euclidean distance between the pixel to be restored, and a set of selected noncorrupted pixels in the filter window. * _[PERSON]_. The filter proposed in [PERSON] and [PERSON] (2018) classifies pixels into three categories: uncorrupted pixels, lightly corrupted pixels, and heavily corrupted pixels, through the maximum absolute luminance difference. In the filtering stage, a weighted average value of the weighted mean of some neighboring pixels and the original pixel value is used to reconstruct the corrupted pixels. All of them were implemented following the descriptions and parameter values recommended in the original contributions. The next step is to determine the best filter according to the considered performance measures and to analyse if this filter significantly outperforms the others. Recall that the images in the test set were corrupted using a sequence of 5% to 95% noise levels with a 5% step and 98% noise level. Each considered filter has been applied to all the noisy images. For each filter and noise level, we computed the values of the similarity measures for all the images in the test set. Next, for each pair of filters, the p-value of a paired Wilcoxon signed-rank test based on a 95% confidence level was calculated. From these p-values of these tests, we obtained which filter achieves the finest results, i.e., we found the filterthat surpasses the other filters taking into account the statistical test. Table 2 displays those filters that have achieved the best results. If one filter is included in a row, it means that it stands out (as previously explained) for the performance measure indicated in the column and the noise density included in each cell. The color code indicates whether or not the filter significantly surpasses the competitors. For example, according to all the measures, the AWAM filter significantly overcomes the competitors for the 20% noise density, whereas although it always obtains the greatest mean value for the 35% noise, it does not significantly overcome the Hosseini filter (according to M-SSIM and MSE) or the Wang filter (according to PSNR\({}_{\text{HVS}}\) and VIF). As it can be seen, the AWAM filter stands out in almost all the noise densities and in all the performance measures. Furthermore, in a non-negligible quantity of the cases, it is significantly the best among the considered filters. In 28 of the 80 cases, the AWAM filter significantly outperforms the others; in 25 cases, it stands out but not significantly and in only 6 cases, there is another filter that significantly outperforms the others. These 6 cases correspond to low or medium noise levels: 5% (2 cases), 10%, 40%, 50% and 55% (one case). In short, in approximately 66% of the cases, the AWAM filter outperforms the others and this performance is significant in more than half.