# Multi-Color Channels Based Group Sparse Model for Image Restoration
## 4 Experimental Results  
This section provides lots of experimental results to verify the effectiveness of the proposed GSR model based on multi-color channels. In the restoration experiments, four types of corrupted images corresponding to different pixel missing rates are used as examples. The selected four pixel missing rates are 80%, 70%, 60%, and 50%. The pixels are lost randomly under different missing rates. The size of image block, \(\sqrt{b}\times\sqrt{b}\), is set to \(5\times 5\), the overlap width between adjacent blocks is set to 1 pixel, the size of search window, \(W\times W\), is set to \(20\times 20\), the number of similar blocks, \(m\), is set to 75, and the constant, \(\varepsilon\), is set to 0.2.  
The quality of the restored images is evaluated according to objective quantitative metrics and subjective visual effects. Peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) [37] are used as objective metrics. The PSNR is calculated as shown in Formulas (21) and (22),  
\[MSE=\frac{1}{H\times W}\sum_{i=0}^{H-1}\sum_{j=0}^{W-1}\|X(i,j)-Y(i,j)\|^{2} \tag{21}\]  
\[PSNR=10\cdot\log_{10}\left(\frac{(2^{n}-1)^{2}}{MSE}\right) \tag{22}\]  
Here, \(X\) and \(Y\) denote the original image and the repaired image, respectively. \(H\) and \(W\) denote the height and the width of the of the image, respectively. Formula (21) is used to calculate the mean square variance (MSE) of the original image and the repaired image. Formula (22) is used for calculating PSNR, where \(n\) is the number of bits per pixel. The larger the value of PSNR, the smaller the distortion. The calculation of SSIM is shown in Formulas (23)-(27),  
\[l(X,Y)=\frac{2u_{X}u_{Y}+C_{1}}{u_{X}^{2}+u_{Y}^{2}+C_{1}},\ c(X,Y)=\frac{2 \sigma_{X}\sigma_{Y}+C_{2}}{\sigma_{X}^{2}+\sigma_{Y}^{2}+C_{2}},\ s(X,Y)= \frac{2\sigma_{XY}+C_{3}}{\sigma_{X}\sigma_{Y}+C_{3}} \tag{23}\]  
\[u_{X}=\frac{1}{H\times W}\sum_{i=1}^{H}\sum_{j=1}^{W}X(i,j) \tag{24}\]  
\[\sigma_{X}^{2}=\frac{1}{H\times W-1}\sum_{i=1}^{H}\sum_{j=1}^{W}(X(i,j)-u_{X}) ^{2} \tag{25}\]  
\[\sigma_{XY}=\frac{1}{H\times W-1}\sum_{i=1}^{H}\sum_{j=1}^{W}\left((X(i,j)-u_{ X})(Y(i,j)-u_{Y})\right) \tag{26}\]  
\[SSIM(X,Y)=l(X,Y)\cdot c(X,Y)\cdot s(X,Y) \tag{27}\]SSIM is measured according to brightness, \(l\), contrast, \(c\), and image structure, \(s\), in Formula (23). \(H\) and \(W\) denote the height and the width of the original image \(X\) and the repaired image \(Y\). \(u_{X}\) and \(u_{Y}\) denote the mean value of \(X\) and \(Y\). \(\sigma_{X}\) and \(\sigma_{Y}\) denote the variance of \(X\) and \(Y\), respectively, and \(\sigma_{XY}\) denotes the covariance of \(X\) and \(Y\). \(C_{1}\), \(C_{2}\) and \(C_{3}\) are constants to be introduced to avoid the denominator being zero. The range of SSIM is \([0,\ 1]\). The SSIM metric is closer to a human's subjective feelings. The larger SSIM value indicates that the two images are more similar and the restoration effect is better.  
In this paper, 10 images as shown in Figure 2 below are selected as test images. The quantitative experimental results and visual restoration effects given in the subsection are obtained on these images. The names of the images are Bahoon, Butterfly, Cowboy, Flower, Girl, Lake, Lena, Mickey, Mural, and Nanna, respectively.