## 2 Methods of Research
### Improvements of the Approach
Atrous convolution increases the receptive field of view of convolution without increasing the convolution kernel and the image output feature size, and provides a better method for identifying image details. This method is used to make up for the detailed feature information of foreign objects lost by the pooling layer. Figure 5 is a one-dimensional comparison diagram of atrous convolution. \(a\) is the original convolution, and \(b\) and \(c\) are atrous convolutions. Suppose the size of the convolution kernel is k, the expansion rate is d, and the size of the convolution kernel after the dilation of the dilated convolution is O, then the size of the dilated convolution can be obtained by the following calculation formula: O = \(k+(k-1)(d-1)\)[25]. Atrous convolution [26] with dilation rate 2 is introduced, considering the size of the foreign object. 4. Design of Loss Function  
The loss function of the model is calculated by detecting the target recognition frame and the prediction result of the corresponding picture. According to the core steps of the Center-Net algorithm, the loss functions are as follows: 1 loss of center point; 2 loss of heat map (classification confidence loss); 3 loss of recognition frame width and height; 4 total loss function. When the image is input to the feature extraction network, the downsampling will lead to the offset of the pixel points and then the accuracy error. Center point loss \(L_{C}\):  
\[L_{c}=\frac{1}{N}{\sum_{P}}\hat{O}_{P}-\left(\frac{P}{R}-\hat{P}\right) \tag{1}\]  
where \(N\) is the total number of target center points predicted by the heat map, \(P\) is the real coordinates of the original image, \(\hat{P}\) is the image center coordinate predicted by the heat map, \(R\) is the multiple set for downsampling in the feature extraction network, and \(\hat{O}_{P}\) is the predicted offset. For the loss of classification confidence \(L_{H}\), due to the working principle of the model, a detection target is predicted by only one feature point, which leads to an extreme imbalance of samples. This phenomenon is avoided by the loss function of the target classification confidence. Figure 5: One-dimensional contrast. Classification confidence \(L_{H}\):  
\[L_{H}=\alpha_{1}L_{n}+\alpha_{2}L_{p} \tag{2}\]  
\[L_{n}=-\left(1-\overset{\wedge}{Y}_{xyc}\right)^{\beta_{1}}\log\left(\overset{ \wedge}{Y}_{xyc}+\delta\right) \tag{3}\]  
\[L_{p}=-\left(1-\overset{\wedge}{Y}_{xyc}\right)^{\beta_{2}}\log\left(\overset{ \wedge}{Y}_{xyc}\right) \tag{4}\]  
where \(\hat{Y}_{xyc}\) is the predicted value of the detected image; in order to increase the loss of positive samples and reduce the loss of negative samples, there are three parameters set in the loss function, \(\alpha\), \(\beta\), and \(\delta\). The specific values of the three parameters were obtained through multiple experimental searches. Negative sample \(L_{n}\) and positive sample \(L_{p}\) are set, respectively, as \(\beta_{1}\); \(\delta\) and \(\beta_{2}\) adjust. The loss of the final positive and negative samples will be reflected on the parameter \(\alpha\) to obtain the classification confidence function \(L_{H}\), where \(\hat{Y}_{xyc}\) is the predicted value of the detected image. In the Center-Net target detection algorithm, the target size frame is generated based on the predicted center point. There will be a certain error in the width and height of the target size box. The loss of the width and height of the recognition box \(L_{S}\):  
Figure 2:0  
where \(\hat{S}_{PK}\) is the size of the predicted image and \(S_{K}\) is the size of the real image. Total loss function \(L_{E}\):  
Figure 2:1  
In the formula, \(\phi_{C}\), \(\phi_{S}\) are the ratio of center point loss and recognition frame width, and height loss, respectively.