# Pgds-YOLOv8s: An Improved YOLOv8s Model for Object Detection in Fisheye Images
## 3 Materials and Methods
### The Proposed Approach
#### 3.2.1 Novel Downsampling Modules
First, on one branch, the Max Pooling operation is used to extract the key feature information of the objects in the input feature map. Max Pooling does not care about the specific location of the features but only abstracts the critical features of the region, so it effectively preserves the key features of distorted and edge-discontinuous objects. At the same time, it sparsifies the error, suppresses noise, and reduces information redundancy. Subsequently, the Ghost module expands the receptive field to extract the essential features further and generate redundant features. The Ghost module generates intrinsic features using a (\(1\times 1\) convolutional kernel with a step size of 1) CBS module. Then, a (\(5\times 5\) convolutional kernel with a step size of 1) CBS module is used to expand the receptive field. Second, to further preserve the original feature information of the fisheye image, another branch uses a (\(1\times 1\) convolution kernel with a step size of 1) CBS module to extract comprehensive feature information, and then, a (\(3\times 3\) convolution kernel with a step size of 2) CBS is used for the downsampling operation. Finally, the branch information with local key features and global features is concatenated. The MPGD module acquires rich essential feature information during the downsampling process, and it maximizes the retention of critical features in the object in the fisheye image. Figure 6: Illustration of the convolutional layer and Ghost module generating the same number of feature maps separately. (**a**) Convolutional layer; (**b**) Ghost module. In addition, the APGD module is proposed based on the MPGD module to obtain more comprehensive fisheye image feature information. As shown in Figure 8, the APGD module is to replace the Max Pooling of the MPGD module with Average Pooling, and the other structures are the same. The Average Pooling can equalize the error in the downsampling process so that the APGD module can obtain rich global features. The MPGD module can effectively extract the key features in the fisheye image, and the APGD module can obtain rich global features in the fisheye image. The MPGD and APGD modules are used alternately in Backbone, which can effectively improve the overall feature extraction ability of the improved model for fisheye images. As shown in Figures 7 and 8, given the input data \(X\in\mathbb{R}^{C\times H\times W}\), \(C\) denotes the number of input channels, and \(H\) and \(W\) denote the height and width of the input data, respectively. First, MaxPool and AvgPool are uniformly denoted as Pool. Figure 7 adopts (pooling kernel size of 2 x 2) MaxPool, and Figure 8 adopts (pooling kernel size of 2 x 2) AvgPool. \(Y\in\mathbb{R}^{C\times(H/2)\times(W/2)}\) denotes the feature map of the output of the pooling operation:  
\[Y=Pool(X) \tag{1}\]  
Subsequently, the pooled output feature map \(Y\) is taken as input; \(m\) intrinsic feature maps \(Y^{\prime}\) are generated by primary convolution [41]; and \(Y^{\prime}\in\mathbb{R}^{H^{\prime}\times W^{\prime}\times m}\), where \(*\) is the  
Figure 8: The structure of the APGD module. Figure 7: The structure of the MPGD module. convolution operation, \(f\in\mathbb{R}^{C\times k\times k\times m}\) is the filters in the convolution layer, \(k\times k\) indicates the size of the filter \(f\), and the bias terms are omitted. The equation is expressed as follows:  
\[Y^{\prime}=Y*f \tag{2}\]  
The \(s\) ghost feature maps \(Y^{\prime\prime}\) are obtained for each intrinsic feature in \(Y^{\prime}\) using cheap linear operations, where \(y^{\prime}_{i}\) is the \(i\)-th intrinsic feature map in \(Y^{\prime}\), \(\Phi_{i,j}\) is the cheap linear operation, \(y_{ij}\) is the \(j\)-th ghost feature map generated by the \(i\)-th intrinsic feature map, and \(\Phi_{i,s}\) is the constant mapping used to preserve the intrinsic feature maps. The linear operation \(\Phi\) is performed for each channel, and its computational cost is much lower than that of ordinary convolution.