# An Advanced AFWMF Model for Identifying High Random-Valued Impulse Noise for Image Processing
## 4 Experiment Results and Discussions  
Initially, the estimate matrix for filters, including the mean square error (MSE) and the PSNR, is defined and determined using Equations (18) and (19), respectively. The MSE can be used to compute the PSNR, and the PSNR is a quantitative indicator for evaluating the performance of different filters. Importantly, the smaller the value of MSE is, the better the performance is. Conversely, the PSNR is bigger for a better performance. Moreover, this study uses another quantitative indicator, structural similarity index measure (SSIM), for further measuring the filtering performance, and its formula is defined as Equation (20) below. The SSIM value is an index that is mainly used to predict the perceived quality of digital images and measure the similarity between two images. The difference between the PSNR value and the SSIM value is that the former method estimates absolute errors,  
Figure 10: Input attribution function of Figure 9.  
and the latter has the structural information that the pixels have strong inter-dependencies when they are spatially close to each other. More importantly, both the PSNR and SSIM values are effective evaluation standards from past studies.  
\[MSE=\frac{1}{R\times C}{\sum_{i=1}^{R}}{\sum_{j=1}^{C}}{\left(O_{i,j}-Y_{i,j} \right)^{2}}, \tag{18}\]  
\[PSNR=10 log_{10}\left(\frac{255^{2}}{MSE}\right), \tag{19}\]  
\[SSIM=\frac{\left(2\mu_{O}\mu_{Y}+e_{1}\right)\left(2\sigma_{CY}+e_{2}\right)} {\left(\mu_{O}{}^{2}+\mu_{Y}{}^{2}+e_{1}\right)\left(\sigma_{O}^{2}\sigma_{Y} ^{2}+e_{2}\right)}, \tag{20}\]  
where \(R\) and \(C\) represent the dimensions of the image and \(R\times C\) is the size of an image. \(O\) represents the original image and \(Y\) is the image that has been restored. The SSIM gives the similarity measurement between the original and restored image as in Equation (20), where \(\mu_{O}\) and \(\mu_{Y}\) are the mean intensities of the original and restored images, \(\sigma_{O}\) and \(\sigma_{Y}\) are the standard deviations of the original and restored images, \(\sigma_{OY}\) is the covariance of the original image and restored image, \(e_{1}=\left(k_{1}L\right)^{2}\), \(e_{2}=\left(k_{2}L\right)^{2}\), \(L\) is the dynamic range of the pixel values, and \(k_{1}\) and \(k_{2}\) are regularization parameters.  
To further verify the performance of the proposed AFWMF model, five famous testing pictures, including Lena, Peppers, Boat, Gold Hill, and Plane, respectively, are used in this study for the image-processing experiments. They are pictured in Figure 11, which was generated by an 8-bit grey level. All their resolutions are 512 by 512. In order to make the experiments more general and precise, this study conducted two experiment evaluations. One compared the PSRNR of the AFWMF model with the PSNRs of different filters; the other compared the psycho-visual result of the AFWMF model with the psycho-visual results of different filters. The different filters are addressed in BDND [9], DWM [16], EAIF [22], FRDFM [18], SBF [3], SDOOD [29], ROR [25], ASMF [12], and EBDND [30], since they were all identified to have good performances in the literature reviews in this area. The experimental results are described with measurement indexes in the next four subsections, and these results and the different filters are discussed in the last subsection.