## 3 Research Method
### Cbam_Ista-Net\({}^{+}\) Compression and Reconstruction
#### 3.3.1 Iterative Shrinkage-Thresholding Algorithm Network (ISTA-Net\({}^{+}\))  
The optimization problem of compressive sensing usually involves finding a sparse signal close to the original signal under a certain transform. It is shown as follows:  
\[\underset{x}{\text{min}}\frac{1}{2}\|\Phi x-y\|\binom{2}{2}+\lambda\|\Psi x\| _{1} \tag{5}\]where \(y\) is the measured value, \(\Phi\) is the measurement matrix, \(x\) is the original signal, \(\Psi\) is the sparse transform, and \(\lambda\) is the regularization parameter. \(\|.\|_{2}\) and \(\|.\|_{1}\) represent the Euclidean norm (also known as the L\({}_{2}\) norm) and the L\({}_{1}\) norm, respectively.  
The traditional ISTA solves the aforementioned problem through iterative updates, with each step consisting of two main operations of gradient descent and proximal mapping. The formula is as follows:  
\[r^{(k)}=x^{(k-1)}-\rho\Phi^{T}\Big{(}\Phi x^{(k-1)}-y\Big{)} \tag{6}\]  
\[x^{(k)}=\underset{x}{\text{argmin}}\frac{1}{2}\Big{\|}x-r^{(k)}\Big{\|}_{2}^{2 }+\lambda\|\Psi x\|_{1} \tag{7}\]  
where, \(r^{(k)}\) is the residual of the \(k\)th iteration, \(x^{(k-1)}\) is the solution from the previous iteration, \(\rho\) is the step size, and \(x^{(k)}\) in Equation (7) is the solution after the \(k\)th iteration.  
ISTA-Net is a deep neural network based on Iterative Shrinkage-Thresholding Algorithm (ISTA), which maps each iteration of ISTA to a layer of the network and learns the network parameters through end-to-end training. A non-linear transform \(\mathcal{F}(\cdot)\) is introduced to replace the traditional linear sparse transform \(\Psi\), which is learned by the convolutional neural network. Therefore, the above proximal mapping formula can be further written as follows:  
\[x^{(k)}=\underset{x}{\text{argmin}}\frac{1}{2}\ ||\ \mathcal{F}(x)-\mathcal{F} \Big{(}r^{(k)}\Big{)}\ ||_{2}^{2}+\theta\ ||\ \mathcal{F}(x)||_{1} \tag{8}\]  
where \(\lambda\) and \(\alpha\) are merged into one parameter, i.e., \(\theta=\lambda\alpha\), and \(\alpha\) is a scalar.  
Finally, the soft thresholding function is used to solve \(\mathcal{F}\Big{(}r^{(k)}\Big{)}\), that is  
\[\mathcal{F}\Big{(}x^{(k)}\Big{)}=\text{soft}\Big{(}\mathcal{F}\Big{(}r^{(k)} \Big{)},\theta\Big{)} \tag{9}\]  
where soft (\(\cdot\), \(\theta\)) is the soft thresholding function used to achieve sparsity.  
ISTA-Net\({}^{+}\) is an enhanced version of ISTA-Net, and its structure is shown in Figure 4. In the figure, \(D^{(k)}\), \(G^{(k)}\), \(H^{(k)}\), and \(\overset{\sim}{H}^{(k)}\) are learnable linear convolutional operators. Replacing Equation (8) with \(H\circ D\) results in the following:  
\[\underset{x}{\text{min}}\frac{1}{2}\ ||\ H(D(x))-H\Big{(}D\Big{(}r^{k}\Big{)} \Big{)}\ ||_{2}^{2}+\theta\ ||\ H(D(x))\ ||_{1} \tag{10}\]  
Following the same strategy as in ISTA-Net, the left inverse of \(H\) is defined as \(\overset{\sim}{H}\), which has the same structure as the symmetry, and satisfies the symmetry constraint \(H\circ\overset{\sim}{H}=I\). For ISTA-Net\({}^{+}\), \(x^{(k)}\) is updated as follows:  
\[x^{(k)}=r^{(k)}+G(\overset{\sim}{H}(soft(H(D(r^{k})),\theta))) \tag{11}\]  
It is sparse in the residual domain rather than the original image domain. Residual images are easier to compress and improve the quality of CS reconstruction. Furthermore,  
Figure 4: ISTA-Net\({}^{+}\) framework.  
ISTA-Net\({}^{+}\) introduces a skip connection, which adds the input and output of each stage together, thus speeding up the training and convergence of the network.