## 2 Principle
### Phase Recovery Theory
The mean squared error (MSE) serves as the loss function, defined as the average squared discrepancy between the real and imaginary components of the recovered signal and the original signal:  
\[L=\frac{1}{2M}\text{$\sum_{m=1}^{M}$}\Big{(}\Big{|}\Big{|}\mathbf{s}^{m}- \mathbf{s}^{m}\Big{|}\Big{|}^{2}_{2}\Big{)}+\lambda\text{$\sum_{\theta}$}|| \theta||_{1} \tag{6}\]  
Let M denote the batch size, \(\mathbf{s}^{m}=\Big{[}\hat{a}^{m},\hat{b}^{m}\Big{]}\) represent the reconstructed signal, \(\mathbf{s}^{m}=[a^{m},b^{m}]\) signify the original signal, and \(\lambda\) indicate the \(L_{1}\) regularization coefficient. Figure 2 shows the flowchart of 2D-CNN. The network has convolutional, batch normalization, activation, pooling, and fully connected layers. The initial and subsequent convolutional layers are succeeded by a batch normalization layer (BN) to standardize feature maps, mitigate internal covariate shift, and accelerate training convergence. RELU introduces non-linearity, enabling the network to develop intricate patterns. POOL layers downsample features by max-pooling, decreasing dimensionality while preserving essential information. Subsequent to the convolutional and pooling stages, the feature maps are transformed into a one-dimensional vector and input into a fully connected layer (FCL), which integrates the accumulated spatial-temporal characteristics to reconstruct the signal. The FCL transforms high-dimensional feature space into output space, enabling the network to generate final predictions or indicate adjustments. This architecture enables the 2D-CNN to integrate spatiotemporal data, therefore diminishing phase noise and nonlinear distortions. In backpropagation, the gradient of the loss function concerning the convolution kernel \(\frac{\partial\mathcal{L}}{\partial\mathbf{K}^{(l)}}\) is computed using the following steps:  
The gradient of the loss concerning the output feature map is:  
\[\frac{\partial\mathcal{L}}{\partial\mathbf{Y}^{(l)}}=\delta^{(l+1)}*\mathbf{K}^ {(l+1)} \tag{7}\]  
Here, \(\delta^{(l)}\) denotes the error value of the \(l\)-th layer. \(\delta^{(l+1)}=\frac{\partial\mathcal{L}}{\partial\mathbf{Y}^{(l+1)}}=\hat{ \mathbf{s}}-\mathbf{s}\), representing the error value of the subsequent layer. The derivative of the activation function concerning the linear transformation is:  
\[\frac{\partial\mathbf{Y}^{(l)}}{\partial(\mathbf{X}^{(l)}*\mathbf{K}^{(l)})}= \text{ReLU}^{\prime}(\mathbf{Y}^{(l)})=\mathbb{I}(\mathbf{Y}^{(l)}>0) \tag{8}\]  
Here, \(\mathbb{I}(\cdot)\) denotes an indicator function that equals 1 when the input exceeds 0 and 0 otherwise. The gradient of the feature map concerning the convolution kernel is:  
\[\frac{\partial(\mathbf{X}^{(l)}*\mathbf{K}^{(l)})}{\partial\mathbf{K}^{(l)}}= \mathbf{X}^{(l)} \tag{9}\]  
Consolidated using the chain rule:  
\[\frac{\partial\mathcal{L}}{\partial\mathbf{K}^{(l)}}=\frac{\partial\mathcal{ L}}{\partial\mathbf{Y}^{(l)}}\odot\frac{\partial\mathbf{Y}^{(l)}}{\partial( \mathbf{X}^{(l)}*\mathbf{K}^{(l)})}*\frac{\partial(\mathbf{X}^{(l)}*\mathbf{ K}^{(l)})}{\partial\mathbf{K}^{(l)}} \tag{10}\]  
Here, \(\odot\) denotes the element-wise product (Hadamard product), where corresponding elements of two matrices are multiplied together. Deriving the formula for error term propagation:  
\[\delta^{(l)}=\Big{(}\delta^{(l+1)}*\mathbf{K}^{(l+1)}\Big{)}\odot\sigma^{ \prime}\Big{(}\mathbf{Y}^{(l)}\Big{)} \tag{11}\]  
\[(a+jb)(\cos\theta+j\sin\theta)=a\cos\theta-b\sin\theta+j(a\sin\theta+b\cos\theta) \tag{3}\]0  
Here, \(sign(\cdot)\) indicates the sign function. The forward propagation of the fully linked layer is as follows:  
\[(a+jb)(\cos\theta+j\sin\theta)=a\cos\theta-b\sin\theta+j(a\sin\theta+b\cos\theta) \tag{3}\]1  
\[(a+jb)(\cos\theta+j\sin\theta)=a\cos\theta-b\sin\theta+j(a\sin\theta+b\cos\theta) \tag{3}\]2 Schematic flowchart of 2D-CNN. where \(\hat{\mathbf{s}}= \begin{bmatrix}\hat{a}(n)\\ \hat{b}(n)\end{bmatrix} \).