## 5 Conclusions  
This work proposed the estimation of the variation in spatial information in RE images using Moran statistics. The Moran's Z measurements were used as feature data in FCM. Three clusters were pre-assumed for FCM in this work: they were heavy, medium, and less noisy or edge areas. The rates for each position partially belonged to clusters determined using an FCM membership function. The membership function used weights to calculate the weighted sum of each position. Each pixel in the noisy image assumed that it was a linear combination of the product of three SMs with membership functions in the same positions. 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. Six frequently used images were chosen in this work. Various Gaussian noise was a priori added to these images for testing and they were then de-noised using the proposed scheme. The performance of this scheme was evaluated in terms of the PSNR, SSIM, and comparisons made to the bilateral filter and NLM. Results from the proposed scheme were superior to those of the single de-noising filter and most results were better than those of the bilateral filter and some better than the NLM. The quality indices resulting from this novel scheme were not always superior to those of the single filter, but most were good. This scheme was a pilot study on image de-noising. In the future, further research on the optimized number of clusters and better smoothing or structured versions used in linear combination is needed to optimize this scheme. This work was supported by Wuyi University with grant number #YJ202215; Natural Science Foundation of Fujian Province, China (#2022J011204); The project of the Fuxiaquan Innovation Platform, Fujian Province, China (#2022-p-029). **Institutional Review Board Statement:** Not applicable. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Scheme** & **Ave3** & **Ave5** & **Proposed** & **G05** & **Bi** & **NLM** \\ \hline PSNR & 31.1 & 31.1 & 31 & 28.1 & 26.9 & 28 \\ \hline SSIM & 0.34 & 0.27 & 0.4 & 0.4 & 0.36 & 0.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average PSNR and SSIM for Lena using various de-noising schemes. We used random padded Gaussian white noise, \(\mu\) = 0, 0, 0, 30, 100; \(\sigma\) = 10, 15, 20, 30, 50, 60; on 20%, 50%, 75% of the entire image. There were a total of 20 images. Ave3 = 3 \(\times\) 3, Ave5 = 5 \(\times\) 5 average filter, G05 = Gaussian filter (\(\sigma\) = 0.5), Bi = bilateral filter, NLM = non-local means. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Scheme** & **Ave3** & **Ave5** & **Proposed** & **G05** & **Bi** & **NLM** \\ \hline PSNR & 37.3 & 35.1 & 38.6 & 34.1 & 32.5 & 32.8 \\ \hline SSIM & 0.48 & 0.42 & 0.53 & 0.54 & 0.45 & 0.46 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: PSNR and SSIM for Lena using various de-noising schemes. We used random padded Gaussian white noise \(\mu\) = 0, \(\sigma\) = 20, on 30% of the entire image. Ave3 = 3 \(\times\) 3, Ave5 = 5 \(\times\) 5 average filter, G05 = Gaussian filter (\(\sigma\) = 0.5), Bi = bilateral filter, NLM = non-local means. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Scheme** & **Ave3** & **Ave5** & **Proposed** & **G05** & **Bi** & **NLM** \\ \hline PSNR & 28.3 & 27.7 & 28.7 & 27.8 & 26.7 & 27.4 \\ \hline SSIM & 0.34 & 0.27 & 0.4 & 0.4 & 0.36 & 0.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average PSNR and SSIM for Barbara using various de-noising schemes. We used random padded Gaussian white noise, \(\mu\) = 0, 0, 0, 30, 100; \(\sigma\) = 10, 15, 20, 30, 50, 60, on 20%, 50%, 75% of the entire image. There were 20 images in total. Ave3 = 3 \(\times\) 3, Ave5 = 5 \(\times\) 5 average filter, G05 = Gaussian filter (\(\sigma\) = 0.5), Bi = bilateral filter, NLM = non-local means. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable.