# Adaptive Beamforming Damage Imaging of Lamb Wave Based on CNN
## 3 Adaptive Beamforming Model Based on CNN  
### Network Model Architecture  
Figure 1 gives a schematic diagram from damage scattering data to final imaging. First, the scattering data are delayed according to the positions of different imaging points, and a 3D matrix is generated by time-space mapping, where the third dimension is the number of channels. Then, the adaptive beamforming method is utilized to calculate the channel weights corresponding to each pixel point. Inspired by the adaptive MVDR method, the designed neural network adaptively computes the weight factors by inputting delayed channel data. Thus, the costly MVDR method for weight calculation is replaced by a more computationally efficient neural network, and this design aims to achieve the adaptive adjustment of weights for different imaging points in the imaging region through network prediction. For different imaging points, the pixel value is obtained by computing the inner product of the delay-corrected array channel data \(\mathbf{z}\) and the weights \(f(\mathbf{z})\) predicted by the neural network:  
\[I_{\mathrm{CNN}}=f\big{(}\mathbf{z}\big{)}^{\mathrm{T}}\cdot\mathbf{z} \tag{8}\]  
The CNN network architecture is shown in Figure 2. The architecture of the CNN beamformer consists of an input layer, four hidden convolutional layers, and an output layer. The dimension of the input layer is (_Nx_, _Ny_, _Nc_), where _Nx_, _Ny_, and _Nc_ are the x, y coordinates and the number of transducers (channels) in the imaging region, respectively. In order to ensure the stability of the training, the input channel data are subsequently subjected to L2 regularization, which penalizes excessively large weight values to prevent overfitting [26]. Specifically, a regularization term \(\lambda\sum w_{i}^{2}\) is incorporated into the loss  
Figure 1: Workflow from delay compensated data to damage imaging in conventional and CNN based beamformers. function, where \(\lambda\) controls the strength of the penalty and \(w_{i}\) denotes the network weights. This approach enhances model robustness by discouraging over-reliance on individual channel features while maintaining the integrity of the training process. In this case, 8-array received signals are used in both simulations and experiments, and the real and imaginary parts of the received resolved signals are considered. The number of points in the imaging region is 201 by 101. Therefore, the value of (_Nx_, _Ny_, _Nc_) in the text is (201, 101, 16). The input layer is followed by four hidden layers, each containing a 2D convolutional layer, an activation layer, and a batch regularization layer. Each convolutional layer has a 7 \(\times\) 7 filter kernel size and a convolutional output, and the output feature map has the same height and width as the input. The activation function used in the hidden layers is an anti-rectifier unit, which prevents negative input values from being mapped to zero. This design mitigates the risk of 'dead neurons'--a scenario where neurons cease updating their weights during training due to prolonged zero outputs--by ensuring continuous gradient flow and maintaining neuronal activity even for negative inputs, thereby enhancing the network's learning stability and generalization capabilities. Batch regularization is used in each convolutional layer to further improve stability during training. In the hidden layer, the number of filters is increased from 32 to 64 in two incremental steps at a rate of 2, and finally back to 16 channel counts. The output layer of the network provides individual channel adaptive apodization weights corresponding to each pixel in the input layer, so the output dimension matches the input dimension, which is (_Nx_, _Ny_, _Nc_). To ensure that the sum of the apodization weights for all channels at each pixel in the output layer equals unity, a Softmax activation function is employed [27]. This choice aligns with the unity constraint (\(\underline{\Gamma}\,w_{i}=1\)) imposed in the MVDR algorithm--a critical requirement for beamforming.