## 4 Design of Decoupled Detection Head for Elliptical Center Sampling
### Implementation Method of the Oriented Object Detection Head  
This study improves upon the one-stage horizontal detection head of the YOLO series [24] by incorporating a rotation dimension. The structure of the horizontal object detection head is shown in Figure 5. The FAS feature, which is the fused feature map extracted in Section 3, serves as the input, with \((x,y,w,h,\theta)\) representing the detection box position vector and \(c\) denoting the classification category vector. Figure 4: The four-dimensional horizontal detection head’s performance degrades when the tilt angle surpasses a specific threshold, rendering it unable to detect closely spaced targets. Figure 5: Structure diagram of the horizontal object detection with added rotational dimension. Figure 3: Difference between categorical and locational feature maps: (**a**) Texture information feature map. (**b**) Edge information feature map. First, a \(3\times 3\) convolutional kernel is applied to the input feature map to enhance its robustness. Then, a \(1\times 1\) convolutional kernel is used to convolve the feature map, generating a tensor with 18 dimensions [25]. Subsequently, the tensor is reshaped to a size of \((15,15,18)\) through a Reshape operation. Finally, it is fed into a Softmax network for binary classification, separating the background from the foreground. Since the generated anchors have relatively blurry boundaries, further bounding box regression is required. During bounding box regression, only the anchors classified as foreground by the Softmax function are selected for regression. A four-dimensional vector, typically represented as \((x,y,w,h)\), is used to define a box, where \(x\) and \(y\) denote the coordinates of the center point of the bounding box, and \(w\) and \(h\) represent the length and width of the bounding box, respectively. The regression process is illustrated in Figure 6.  
The red box \((GT)\) represents the target box, and the numerous green boxes \((A)\) are the source boxes. The goal is to find a mapping from anchor box \(A\) to \(GT\) to obtain a detection box \(G^{\prime}\) that is very close to \(GT\). Assuming anchor A = \((P_{x},P_{y},P_{w},P_{h},)\), GT = \((G_{x},G_{y},G_{w},G_{h},)\),  
find a mapping \(Func\) such that \(Func(A)=G^{\prime}\). 1. **Panning the box:** \[\left\{ \begin{aligned} \Delta x,\Delta y&=P_{w}d_{x}(P),P_{h}d_{ y}(P)\\ G^{\prime}_{x}&=P_{w}d_{x}(P)+P_{x}\\ G^{\prime}_{y}&=P_{h}d_{y}(P)+P_{y}\end{aligned} \right.\] (2)
2. **Scaling the box:** \[\left\{ \begin{aligned} (S_{w},S_{h})&=(\exp(d_{w}(P)), \exp(d_{h}(P)))\\ G^{\prime}_{w}&=P_{w}\exp(d_{x}(P))\\ G^{\prime}_{h}&=P_{h}\exp(d_{h}(P))\end{aligned} \right.\] (3)  
Based on Equations (2) and (3), it is necessary to regress the four values \(d_{x}(P),d_{y}(P),d_{w}(P),d_{h}(P)\). A linear regression model [26] is used for this process, where \(Func(A)=G^{\prime}\) represents the parameters to be learned. For the regression of these four  
Figure 6: Bounding box regression process. variables, the offset \((t_{x},t_{y})\) and scaling factors \((t_{w},t_{h})\) from \(A\) to \(GT\) must first be calculated. Their calculation formulas are as follows:  
\[ \begin{cases}t_{x}=(x-x_{a})/w_{a},t_{y}=(y-y_{a})/h_{a}\\ t_{w}=\log(w/w_{a}),t_{h}=\log(h/h_{a})\end{cases}  \tag{4}\]  
The next step is to calculate the positional offsets \((t_{x}^{*},t_{y}^{*})\) and scaling factors \((t_{w}^{*},t_{h}^{*})\) between the predicted box and the label. Their calculation formulas are as follows:  
\[ \begin{cases}t_{x}^{*}=(x^{*}-x_{a})/w_{a},t_{y}^{*}=(x^{*}-y_{a})/h_{a}\\ t_{w}^{*}=\log(w^{*}/w_{a}),t_{h}^{*}=(h^{*}/h_{a})\end{cases}  \tag{5}\]  
The regression process involves finding the optimal parameters \(K\) for the function \(Func\). For the choice of loss function, this paper uses the smooth L1 loss function [27]. It is calculated using Equation (6). \[Loss=\sum_{i}^{N}\left|t_{*}^{i}-W_{*}^{T}\phi(A^{i})\right| \tag{6}\]  
This completes the design of the horizontal object detection head.