## 3 Research Method
### Cbam_Ista-Net\({}^{+}\) Compression and Reconstruction
#### 3.3.2 Improved CBAM_ISTA-Net\({}^{+}\) Algorithm  
The proposed method combines spatial and channel block attention module CBAM (Convolutional Block Attention Module) based on ISTA-Net\({}^{+}\), and its formula is as follows:  
\[x^{(k)}=r^{(k)}+G(\widetilde{H}(M_{s}(soft(M_{c}(H(D(r^{k}))),\theta)))) \tag{12}\]  
A channel attention mechanism (Channel Attention, CA) is added behind the second convolutional layer, and a spatial attention mechanism (Spatial Attention, SA) is incorporated after the soft thresholding operation, as shown in Figure 5. Together, they enhance the sensitivity and discriminative ability of the network to important features.  
* Channel Attention  
Channel attention focuses on the importance of features, and the structure is shown in Figure 6.  
After the forward transform module of CBAM_ISTA-Net\({}^{+}\), the maximum pooling and the average pooling are first used for the features \(F_{H}\), and then the multilayer perceptron (MLP) layer is used to obtain the transform results. Finally, they are applied to two channels separately, and the attention results of the channels are obtained by using sigmoid function. The importance of different channels is used to weight the feature mapping of each channel, emphasizing the important channel features and suppressing the unimportant channel features. The formula is shown below:  
\[M_{c}(F_{H})=\sigma(MLP(AvgPool(F_{H}))+MLP(MaxPool(F_{H}))) \tag{13}\]  
Figure 5: Schematic diagram of the \(k\)th stage of CBAM_ISTA-Net\({}^{+}\).  
Figure 6: Channel attention model.  
where \(M_{\text{c}}(F_{H})\) is the weighted result after forward propagation.
* Spatial Attention  
The spatial attention focuses on the importance of spatial positions, and its structure is shown in Figure 7.  
With the soft thresholding operation, a channel-based global max pooling and global average pooling are first performed to obtain two feature maps, Then, they are concatenated into a feature map. It focuses on different spatial locations of the feature map and weights the feature map by learning the importance of each location, making the network pay more attention to the important spatial regions of the image. The formula is shown below:  
\[M_{\text{s}}\Big{(}F_{Soft}\Big{)}=\sigma\Big{(}f^{7*7}\Big{(}\Big{[}AvgPool \Big{(}F_{Soft}\Big{)}\Big{)};AvgPool\Big{(}F_{Soft}\Big{)}\Big{]}\Big{)}\Big{)} \tag{14}\]  
where \(M_{\text{s}}\Big{(}F_{Soft}\Big{)}\) is the weighted result after soft thresholding iterations. \(F_{Soft}\) is the feature after the soft thresholding operation.  
The major steps of the proposed CBAM_ISTA\({}^{+}\) are summarized in Algorithm 1.  
```
Input: Training dataset \(X\), sampling matrix \(\Phi\), number of network layers \(L\), initialization matrix \(Q_{init}\) Output: Reconstruction result \(\hat{Y}\)
1. Compute the initial matrix \(\Phi^{T}\Phi\) and vector \(\Phi\)X;
2. Initialize reconstruction result \(x_{0}=X\Phi_{init}^{T}\);
3. for k = 1; k \(\leq\) L do
4. Updating the estimates \(x^{k}\) by Formulas (6) and (7);
5. Reshape \(x^{k}\) to the image format \(x_{input}\) and apply \(Conv_{D}\) to obtain the feature map \(x_{D}\);
6. Forward propagation convolution to extract deep features \(x=ReLU(Conv_{1}(x_{D}))\), \(x_{forward}=Conv_{2}(x)\);
7. Compute the channel attention weight \(M_{c}\Big{(}x_{forward}\Big{)}\) weighted the feature map \(x_{forward}\);
8. \(x=Soft\theta\Big{(}x_{forward},\theta_{k}\Big{)}\);
9. Compute the spatial attention weight \(M_{\text{s}}(x)\) weighted feature map \(x\);
10. Backward propagation convolution \(x=ReLU(Conv_{3}(x))\), \(x_{backward}=Conv_{4}(x)\);
11. Reconstruction updates : \(x_{pred}=x_{input}+x_{G}\), \(x_{G}=Conv_{G}(x_{backward})\), and flattened to vector form: \(x^{k}=Flatten\Big{(}x_{pred}\Big{)}\), and calculate the symmetric loss \(SymLoss_{k}\);
12. end;
13. Obtaining reconstruction results \(\hat{Y}=x^{k}\).
```  
**Algorithm 1**The procedure of CBAM_ISTA\({}^{+}\)  
Figure 7: Spatial attention model.