## 4 Design of Decoupled Detection Head for Elliptical Center Sampling
### Ellipse Center Sampling and Anchor-Free Implementation Approach  
In the oriented object detection head, the positional offset of the object detection box is defined by Equation (10), the dimensional offset is defined by Equation (11), and the tilt-angle offset is defined by Equation (12).  
\[offset_{xy}=reg_{xy}\times k\times s \tag{10}\]  
\[wh=(relu(reg_{wh}\times k)+1)\times s \tag{11}\]  
\[\theta=Mod(reg_{\theta},90) \tag{12}\]  
where \(reg_{xy}\), \(reg_{wh}\), and \(reg_{\theta}\) represent the direct outputs of the final layer of the regression branch. \(k\) is a learnable adjustment parameter, and \(s\) is the downsampling factor of multi-level feature extraction. To align with the positive and negative sample assignment rules of YOLO, the anchor-free detection head selects only one positive sample for  
Figure 10: Oriented object detection box representation.  
each object, namely, the object's center point coordinates, while ignoring other predicted coordinates with higher confidence scores. To mitigate the positive-negative sample imbalance issue introduced by the anchor-free decoupled head, the central \(3\times 3\) region is typically designated as the positive sample region, also referred to as horizontal center point sampling in FCOS [29]. However, the rotated detection boxes used by FCOS, being non-horizontal rectangles, affect the sampling range. The short edges further reduce the number of sampling points for targets with large aspect ratios. The most intuitive center sampling should involve a rectangular region within a certain range around the target center, as shown by the dashed rectangle in Figure 11, but the short edges limit the range of rectangular center sampling. To mitigate these effects, this paper proposes an ellipse center sampling method based on a two-dimensional Gaussian distribution, as shown by the solid ellipse in Figure 11. Compared to the horizontal center sampling method, the ellipse center sampling method is better suited for rotated target detection, with the sampling region for targets with large aspect ratios being more concentrated by contracting the major axis. The two-dimensional Gaussian distribution [30] is defined using the parameters \((w,h,\theta)\), as specified by Equation (13).  
\[R_{\theta}=\begin{pmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{pmatrix},\quad e_{\theta}=\frac{\min(w,h)}{12} \begin{pmatrix}w^{2}&0\\ 0&h^{2}\end{pmatrix} \tag{13}\]  
where \(R_{\theta}\) is the rotation transformation matrix. Equation (14) defines the probability density of a two-dimensional Gaussian distribution [31] under general conditions. \(x\) represents the coordinates of the target detection box, with \(e\) as the natural constant, set to 2.7 in the experiments.  
\[f(x)=\frac{1}{2.5\sqrt{e}}e^{-\frac{1}{2}[(x-\mu)^{T}e^{-1}(x-\mu)]} \tag{14}\]  
When \(f(x)\in(0,1)\), the elliptical contour of the two-dimensional Gaussian distribution can be represented as \(f(x)=c\). The contour lines of the ellipse increase only as \(c\) decreases, with the effective range of \(c\) being \([c_{0},1]\). Considering the prevalence of small targets in dense rotated images, to prevent insufficient sampling due to a small sampling area, \(c_{0}\) is set to 0.3 in this paper. The central sampling area of a target can be determined by \(f(x)\geq c\)[32]. If \(f(x)\) is greater than \(c\), the point \(x\) lies within the sampling region.