# Remote Sensing Image Enhancement Based on Non-Local Means Filter in NSCT Domain
## 4 Results and Discussions
### Objective Analysis  
The Entropy (H) shows the overall information of the image, and it can be calculated by the following equation [15]:  
\[H=-\sum_{i=1}^{M}\sum_{j=1}^{N}p(i,j)\log p(i,j) \tag{8}\]  
where \(M\times N\) represents the size of the image. A higher H shows that the image contains more effective information. The Mean Square Error (MSE) is computed by:  
\[MSE=\frac{1}{R\times C}{\sum\
olimits_{x=1}^{C}}{\sum\
olimits_{y=1}^{R}}{ \left({f_{x,y}-g_{x,y}}\right)^{2}} \tag{9}\]  
The MSE shows the difference between the estimated image and the reference image [33]. A lower MSE demonstrates that the enhanced image is closer to the reference image. The Peak Signal to Noise Ratio (PSNR) is computed by the following equation:  
\[PSNR=10\times\log\frac{L^{2}}{\frac{1}{R\times C}{\sum\
olimits_{x=1}^{C}}{ \sum\
olimits_{y=1}^{R}}{\left({f_{x,y}-g_{x,y}}\right)^{2}}} \tag{10}\]  
where \(L\) presents the maximum intensity in the image [38]. The higher the PSNR, the better the image quality. The Gradient Magnitude Similarity Deviation (GMSD) reflects the range of distortion degree in the image; a higher GMSD reflects larger distortion and lower image quality, and can be calculated by [35]:  
\[GMSD=\sqrt{\frac{1}{N}{\sum\
olimits_{i=1}^{N}}{\left({GMS(i)-GMSM}\right)^{2}}} \tag{11}\]  
where \(N\) represents the number of pixels in the image, and GMSM is Gradient Magnitude Similarity Mean. A higher GMSM represents higher image quality, and it is defined as:  
\[GMSM=\frac{1}{N}{\sum\
olimits_{i=1}^{N}}{GMS(i)} \tag{12}\]  
Figure 7: Comparisons on Image 3. (**a**) Original image; (**b**) HE; (**c**) ESIHE; (**d**) FLM; (**e**) LSCN; (**f**) Proposed method. where GMS is gradient magnitude similarity, and the corresponding formula is described as follows:  
\[GMS(i)=\frac{2m_{r}(i)m_{d}(i)+c}{m_{r}^{2}(i)+m_{d}^{2}(i)+c} \tag{13}\]  
where \(c\) is a positive constant, \(m_{r}\) and \(m_{d}\) are the gradient magnitude images, and \(m_{r}\) and \(m_{d}\) can be calculated by:  
\[m_{r}(i)=\sqrt{\left(r\otimes h_{x}\right)^{2}(i)+\left(r\otimes h_{y}\right)^ {2}(i)} \tag{14}\]  
\[m_{d}(i)=\sqrt{\left(d\otimes h_{x}\right)^{2}(i)+\left(d\otimes h_{y}\right)^ {2}(i)} \tag{15}\]  
\[h_{x}=\left[ \begin{array}{ccc}\frac{1}{3}&0&-\frac{1}{3}\\ \frac{1}{3}&0&-\frac{1}{3}\\ \frac{1}{3}&0&-\frac{1}{3}\end{array} \right]h_{y}=\left[ \begin{array}{ccc} \frac{1}{3}&\frac{1}{3}&\frac{1}{3}\\ 0&0&0\\ -\frac{1}{3}&-\frac{1}{3}&-\frac{1}{3}\end{array} \right] \tag{16}\]  
where \"\(\otimes\)\" is the convolution operation. The Mean Absolute Error (MAE) can be defined as:  
\[MAE=\frac{1}{M\times N}\sum_{x=1}^{M}\sum_{y=1}^{N}\left|g_{x,y}-f_{x,y}\right| \tag{17}\]  
where \(g_{x,y}\) and \(f_{x,y}\) denote the original image pixel and the enhanced image pixel, respectively [36]. If the MAE is lower, then the image quality is better. The Multi-scale structural similarity (MSSSIM) is an improved evaluation metric based on structural similarity (SSIM). The higher the value of MSSSIM, the better the quality of the image. It is defined as [39]:  
\[MSSSIM=\left[l_{M}(x,y)\right]^{\sigma_{M}}\prod_{j=1}^{M}\left[c_{j}(x,y) \right]^{\beta_{j}}\left[s_{j}(x,y)\right]^{\tau_{j}} \tag{18}\]  
From Table 1, we notice that the H obtained by the proposed method is the best; for the metrics MSE, PSNR, MAE, and MSSSIM, the values achieved by the ESIHE method are the best, but the values obtained by the proposed algorithm in these four metrics are still second-best; for the index GMSD, the ESIHE method is the best, and the LSCN and proposed methods are second and third, respectively. From the data as shown in Table 2, the H, MSE, PSNR, and MAE values achieved by the proposed technique are the best; for the GMSD, the value achieved by the ESIHE method is the best, but the proposed approach is second; for the metric MSSSIM, the FLM method is the best, and the ESIHE and proposed techniques are the second and third, respectively.