# Convolved Feature Vector Based Adaptive Fuzzy Filter for Image De-Noising
## 4 Proposed Methodology: Convolved Feature Vector Based Fuzzy Filter
### Performance of the Proposed De-Noising Filter
\[MSE=\frac{1}{I\times m}\sum_{i=1}^{l}\sum_{j=1}^{m}\left[O_{i,j}-R_{i,j}\right]^ {2} \tag{20}\]  
\[PSNR=10\times\log_{10}\frac{N^{2}}{MSE} \tag{21}\]  
\[EPI=\frac{\sum_{i=1}^{l}\sum_{j=1}^{m-1}\left|R_{i,j+1}-R_{i,j}\right|}{\sum_{ i=1}^{l}\sum_{j=1}^{m-1}\left|O_{i,j+1}-O_{i,j}\right|} \tag{22}\]  
where \(O_{i,j}\) and \(R_{i,j}\) represent noise-free original and de-noised images of size \(l\times m\) at location (_i,j_) respectively and \(N\) represent maximum possible gray-level value of a 8-bits per pixel. EPI values ranges from 0 to 1, higher value indicates better edge preservation. Figure 6: Shows some of the standard images: (**a**) Lena, (**b**) Boat, (**c**) Baboon and (**d**) Barbara. Figure 7: Shows a comparison in terms of PSNR of different methods for Lena, Bridge and Boat images corrupted with noise ratios of 30, 40, 50 and 60%. A subjective comparison is performed on a small sample of standard images as shown in Figure 6 as well as randomly selected medical images of MedPix dataset and the restoration results are shown in Figures 8-10. In Figure 8, the restored images of NWM [29], SRM [40] and AEPWM [41] have some traces of noise clearly visible on the restored images whereas AFIDM [42], QSAF, TSA and the proposed filter de-noised the noisy image well. It is clear from the visual results that the proposed filter has preserved the edges and detailed regions of the Lena image very well as compared TSA, QSAF and other competing filters. The performance of the proposed filter is tested on detailed images, i.e., Baboon and Barbara, of size 512 \(\times\) 512 at high noise densities, 50% and 60% RVIN, and the results are tabulated in Figure 9. It can be seen from the restored results that the proposed filter has effectively preserved the edges and fine details in both the images. Similarly, when the proposed technique is further tested on medical images of MedPix dataset as shown in Figure 10, it is clear from subjective results that the proposed filter outperforms QSAF and AFIDM filters. Figure 11 shows EPI values of different de-noising methods for Lena image at 30% noise ratio. It is evident from quantitative as well as subjective comparisons that the proposed filter gives best results and outperforms many existing filters against different noise densities ranging from low to high. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Methods** & **DWM** & **NWM** & **ACWM** & **SDOOD** & **AFIDM** & **SRM** & **QSAF** & **AEPWM** & **SBF** & **TSA** & **Proposed** \\ \hline
**Lena** & & & & & & & & & & \\
30\% & 0.914 & 0.831 & 0.936 & 0.920 & 0.938 & 0.918 & 0.948 & 0.932 & 0.913 & 0.923 & **0.950** \\
40\% & 0.886 & 0.814 & 0.917 & 0.887 & 0.906 & 0.889 & 0.930 & 0.901 & 0.895 & 0.904 & **0.933** \\
50\% & 0.847 & 0.762 & 0.860 & 0.849 & 0.869 & 0.836 & 0.879 & 0.866 & 0.846 & 0.868 & **0.881** \\
60\% & 0.747 & 0.709 & 0.794 & 0.809 & 0.797 & 0.787 & 0.805 & 0.775 & 0.728 & 0.817 & 0.759 \\
**Bridge** & & & & & & & & & & \\
30\% & 0.718 & 0.709 & 0.795 & 0.776 & 0.709 & 0.699 & 0.782 & 0.786 & 0.749 & 0.791 & **0.806** \\
40\% & 0.617 & 0.655 & 0.734 & 0.725 & 0.641 & 0.628 & 0.728 & 0.735 & 0.655 & 0.740 & **0.759** \\
50\% & 0.556 & 0.600 & 0.658 & 0.688 & 0.618 & 0.598 & 0.630 & 0.629 & 0.583 & 0.663 & **0.678** \\
60\% & 0.492 & 0.524 & 0.557 & 0.502 & 0.498 & 0.514 & **0.591** & 0.581 & 0.501 & 0.580 & 0.586 \\
**Boat** & & & & & & & & & & \\
30\% & 0.814 & 0.756 & 0.858 & 0.849 & 0.793 & 0.826 & 0.849 & 0.855 & 0.854 & 0.862 & **0.879** \\
40\% & 0.731 & 0.714 & 0.827 & 0.819 & 0.756 & 0.764 & 0.818 & 0.816 & 0.822 & 0.805 & **0.843** \\
50\% & 0.672 & 0.669 & 0.765 & 0.715 & 0.728 & 0.719 & 0.759 & 0.737 & 0.778 & 0.761 & **0.806** \\
60\% & 0.639 & 0.609 & 0.696 & 0.579 & 0.649 & 0.656 & 0.703 & 0.665 & 0.707 & 0.693 & **0.755** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of results in terms of **SSIM** for Lena, Bridge and Boat images corrupted with 30% to 60% RVIN.