## 3 Experiments and Results
### Results  
The proposed scheme assumes that the filtering process is a linear combination of the product of three SMs with membership functions. The average filters with two different windows (3 \(\times\) 3, 5 \(\times\) 5) and a Gaussian filter (\(\sigma\) = 0.5) were a priori applied to this noisy image to make three SMs. In this work, SM\({}_{5}\) was used for heavy noise, SM\({}_{3}\) for medium, and SM\({}_{0.5}\) for less noisy or edge areas. Six frequently used images were selected in this work, as shown in Figure 5 (images were download from [[https://ccia.ugr.es/cvg/dbimages](https://ccia.ugr.es/cvg/dbimages)]([https://ccia.ugr.es/cvg/dbimages](https://ccia.ugr.es/cvg/dbimages)), accessed on 1 September 2022, Computer Vision Group, University of Granada). These test images were all sized 512 \(\times\) 512 and 9 bits deep. Various Gaussian noise was a priori added to these images and they were then divided into three groups: (1) padded randomly on 20%, 50%, 75% of the entire image with \(\mu\) = 0 and \(\sigma\) = 20, 30, 50, 60; (2) padded randomly on 50% of the entire image with \(\mu\) = 30 and \(\sigma\) = 20, 30, 50, 60; (3) padded randomly on 20% of the entire image with \(\mu\) = 100 and \(\sigma\) = 10, 15, 20, 30, respectively. Figure 4: A flow chart of the proposed image de-noise scheme. The average filter with window (\(3\times 3\), \(5\times 5\)), a Gaussian filter (\(\sigma=0.5\)), the bilateral filter (36), the non-local means scheme (37), and the proposed algorithm were applied to these noisy images. The PSNR and SSIM (38) were calculated and are shown in Figures 6-11. Figures 6-8 show the PSNR results. Figures 9-11 show the SSIM results on the de-noised images. In these figures, (a, f): added 20%, \(\mu=0\) with various \(\sigma\), (b, g): added 50%, \(\mu=0\) with various \(\sigma\), (c, h): added 75%, \(\mu=0\) with various \(\sigma\), (d, i): added 50%, \(\mu=30\) with various \(\sigma\), (e, j): added 20%, \(\mu=100\) with various \(\sigma\). The image qualities depend on the PSNR and SSIM values. For both indices, the higher values correspond to better quality. The higher \(\sigma\) indicates noise variations in the image. This variation can reduce the image quality. The abscissas in Figures 6-11 show variations in \(\sigma\). Both the PSNR and SSIM indices show declines in response to the increasing variation in \(\sigma\). In some of the proposed scheme's results, a linear combination of SM\({}_{5}\), SM\({}_{3}\), and SM\({}_{0.5}\) is superior to the single de-noise filter. We can see this in Figures 8-11. Most of the proposed scheme's results are better than the bilateral filter and some are even better than NLM. The quality indices resulting from this novel scheme are not always superior to the single filter but most are satisfactory. Figure 5: Six images used in this work: (top left to right) Barbara, boat, hill, (bottom left to right) Lena, man, and Pentagon. Figure 6: PSNR after de-noising Barbara (left) and boat (right), (**a,f**) added 20%, \(\mu=0\) with various \(\sigma\), (**b,g**) added 50%, \(\mu=0\) with various \(\sigma\), (**c,h**) added 75%, \(\mu=0\) with various \(\sigma\), (**d,i**) added 50%, \(\mu=30\) with various \(\sigma\), (**e,j**) added 20%, \(\mu=100\) with various \(\sigma\). Figure 7: PSNR after de-noising hill (left) and Lena (right): (**a,f**) added 20%, \(\mu=0\) with various \(\sigma\), (**b,g**) added 50%, \(\mu=0\) with various \(\sigma\), (**c,h**) added 75%, \(\mu=0\) with various \(\sigma\), (**d,i**) added 50%, \(\mu=30\) with various \(\sigma\), (**e,j**) added 20%, \(\mu=100\) with various \(\sigma\). Figure 8: PSNR after de-noising man (left) and Pentagon (right): (**a,f**) added 20%, \(\mu=0\) with various \(\sigma\), (**b,g**) added 50%, \(\mu=0\) with various \(\sigma\), (**c,h**) added 75%, \(\mu=0\) with various \(\sigma\), (**d,i**) added 50%, \(\mu=30\) with various \(\sigma\), (**e,j**) added 20%, \(\mu=100\) with various \(\sigma\).