# Multiple Optimizations-Based ESRFBN Super-Resolution Network Algorithm for MR Images
## 2 Algorithm Architecture
### Cascade Optimization of Loss Function Based on Joint PSNR-SSIM  
In order to improve the PSRN and SSIM indexes of MR super-resolution, a cascade optimization method based on the joint loss function of PSNR-SSIM is proposed in this section. SSIM as a loss function is not particularly sensitive to uniform deviation. This leads to changes in brightness or color, which usually become duller. However, SSIM can preserve contrast and is better than other loss functions in the high-frequency region. On the other hand, the L1 loss function can keep the brightness and color unchanged. Therefore, in order to obtain the best characteristics of the two loss functions, this section combines them to form a new loss function, and the name is Mix loss function. Its formula is as follows  
\[L^{\text{Mix}}=\alpha\cdot L^{\text{MS$-$SSIM}}+(1-\alpha)\cdot G_{\sigma_{C}^ {M}}\cdot L^{\ell_{1}} \tag{1}\]  
The cascade optimization calculation steps of the loss function based on joint PSNR-SSIM are as follows. Step 1. Calculate \(L^{\text{MS$-$SSIM}}\)  
SSIM/MS-SSIM is an index that measures the similarity of two images, mainly from three aspects: brightness, contrast and structure. Therefore, the SSIM index can better characterize the structural information of the image. As a loss function, it can better retain the high-frequency information of the image, so that the super-resolution image has more detailed information, which is useful for MR images in clinical diagnosis. Its expression is as follows. Where \(y\) is the original high-resolution image; \(y^{\prime}\) is the high-resolution image output by the deep network; \(\mu_{y}\) is the mean of the image \(y\); \(\mu_{y^{\prime}}\) is the mean of the image \(y^{\prime}\); \(\sigma_{y}^{2}\) is the variance of the image \(y\); \(\sigma_{y^{\prime}}^{2}\) is the variance of the image \(y^{\prime}\); \(\sigma_{y^{\prime}y}\) is the covariance of the sum of the image \(y\) and \(y^{\prime}\); \(c_{1}\) and \(c_{2}\) are two constants, avoiding the denominator to be 0. \[SSIM(y,y^{\prime})=\frac{\Big{(}2\mu_{y^{\prime}}h_{y}+c_{1})(\sigma_{y^{ \prime}y}+c_{2})}{(\mu_{y^{\prime}}^{2}+\mu_{y}^{2}+c_{1})(\sigma_{y^{\prime} }^{2}+\sigma_{y}^{2}+c_{2})} \tag{2}\]  
Figure 1: The architecture of deep super-resolution algorithm based on multiple optimizations. The index range of SSIM is 0-1. The larger the SSIM is, the higher the similarity of the two images. Therefore, SSIM as the loss function can be expressed as the following formula [20]. \[L^{SSIM}=1-SSIM(y,y^{\prime}) \tag{3}\]  
In order to better transmit the error, the SSIM formula is deformed, and the dependence of the mean and standard deviation on the pixel \(p\) is ignored here. The mean and standard deviation are calculated using a Gaussian filter with a standard deviation \(G_{\sigma G}\). \[SSIM(y,y^{\prime})=\frac{(2\mu_{y^{\prime}}\mu_{y}+c_{1})(\sigma_{y^{\prime}} +c_{2})}{(\mu_{y^{\prime}}^{2}+\mu_{y}^{2}+c_{1})(\sigma_{y^{\prime}}^{2}+ \sigma_{y}^{2}+c_{2})}=l(p)\cdot cs(p) \tag{4}\]  
At the same time, the SSIM loss function can be re-characterized as  
\[L^{SSIM}(P)=1-SSIM(\widetilde{p}) \tag{5}\]  
\(\widetilde{p}\) represents the central pixel. Even if the network learns the weight to maximize the SSIM value of the central pixel, the learned kernel will be applied to all pixels in the image.