## 2 Materials and Methods
### Analysis of Optimization Principles  
In the process of license plate recognition, the input image first undergoes convolutional layer processing [25], where convolutional kernels extract feature information from the image. The feature information processed by the ReLU activation function is then fed into the next convolutional layer for similar operations. The feature maps generated by the convolutional operations are input into the pooling layer for further feature extraction. This study uses Max Pooling to enhance the ability of convolutional kernels to extract image information [26]. Finally, the features are integrated through a fully connected layer [27], and the final classification result is obtained through the classifier. By observing the heat maps generated during the convolutional operations, as shown in Figure 2b, in complex environments, the convolutional layers have difficulty effectively extracting the key features of license plates. To address the nonlinearity issue in convolutional neural networks, this study employs the ReLU activation function [28], which is defined by the following formula:  
\[\text{ReLU}(\mathbf{w})= \begin{cases}w&w\geq 0\\ 0&w\leq 0\end{cases}  \tag{1}\]  
\(w\) represents the input value of the activation function, which specifically refers to the output of the previous convolutional layer in this study. However, relying solely on the ReLU activation function is insufficient to effectively address the challenges faced by the convolutional layers when processing license plates in complex environments. To overcome these issues, BN is introduced into the traditional AlexNet model. The BN layer normalizes each feature channel and introduces two learnable parameters to enhance the model's expressive capability. Figure 1: AlexNetBN model structure diagram. Figure 2: Image processing effect diagram. (**a**) Input image; (**b**) heat map after convolutional layer processing; (**c**) heat map after convolutional and BN layer processing. During normalization, the BN layer computes the mean and variance of the input features within each mini-batch [21]. Let \(\mu_{batch}\) represent the mean, and the formula for calculating the mean is as follows:  
\[\mu_{batch}=\frac{1}{m}\underset{i=1}{\overset{m}{\sum}}X_{i} \tag{2}\]  
where \(m\) is the number of samples in the mini-batch, and \(X_{i}\) represents the input feature of the \(i\) sample. Let \(\sigma^{2}_{batch}\) represent the variance, and the formula for calculating the variance is as follows:  
\[\sigma^{2}_{batch}=\frac{1}{m}\underset{i=1}{\overset{m}{\sum}}(X_{i}-\mu_{ batch})^{2} \tag{3}\]  
Each input feature is normalized according to the following formula [21]:  
\[\hat{X}_{i}=\frac{X_{i}-\mu_{batch}}{\sqrt{\sigma^{2}_{batch}+\varepsilon}} \tag{4}\]  
where \(\hat{X}_{i}\) is the normalized input, and \(\varepsilon\) is a small constant added to prevent division by zero. The BN layer normalizes the input of each layer to reduce the impact of internal covariate shift. This mechanism keeps the mean and variance relatively stable during training, effectively reducing disturbances in the input features [29]. This stability allows the network to better handle conditions like sandstorm blurring and strong light reflections, resulting in more uniform illumination of license plate images, effectively resisting noise interference, and facilitating subsequent feature extraction. In addition, the BN layer introduces two learnable parameters, whose formulas are as follows:  
\[y_{i}=\gamma\hat{X}_{i}+\beta \tag{5}\]  
where the scaling parameter \(\gamma\) adjusts the scale of the features, allowing the normalized features to be adapted to the range suitable for the current task. The shift parameter \(\beta\) translates the normalized features, further adjusting their distribution. The introduction of these parameters enables the network to flexibly recover the original feature distribution during training, reducing sensitivity to the initial values of the network weights [21].