## 3 Materials and Methods
### Rotationally Invariant Object Detection  
Currently, there are three kinds of object detection bounding boxes: horizontal bounding box, rotating bounding box, and custom bounding box. One of the characteristics of remote sensing image object detection is that the direction of the object to be detected is random and diverse, and the boundary box labeling method adopted by the object detection method should be changed according to the shape characteristics of the detected object itself. Obviously, in view of the fact that the direction of the object is not fixed in remote sensing image object detection, rotating the bounding box is a better choice. In this paper, because the remote sensing image dataset used has undergone data preprocessing, the image is clipped into a rectangle, and the rotation angle of the target is defined as the angle where the x-axis clockwise rotation meets the longest edge of the detection frame.  
Rotationally invariant object detection needs to introduce new angle loss. Common definition methods of arbitrary rotating frame include the x, y, w, h, and angle \(\theta\) Opencv definition method and the long side definition method of five parameters, including four vertex coordinates sorted counterclockwise and an ordered quadrilateral definition method  
Figure 4: DCE_YOLOX network structure.  
with eight parameters. Due to the periodicity of the angle, boundary discontinuity will occur when calculating the angle loss, resulting in the sudden increase in the loss value. The angle loss was previously calculated using mostly One-hot, but we found that the angle loss of \(-90^{\circ}\) and \(89^{\circ}\) differed significantly from each other, even though they were actually only \(1^{\circ}\) apart.  
In addition to the common horizontal boundary box object detection, this paper also carries out rotation boundary box object detection, adding the angle loss. The loss is mainly divided into four parts: regression loss, confidence loss, classification loss, and angle loss. CSL (circular smooth label) [40] is introduced in the calculation of angle loss, which solves the regression problem of angle in the form of classification. By dividing the angle, the prediction result is limited, and the boundary problem is eliminated. The circular smooth label obtains the angle prediction by classification without affecting the boundary conditions. The cyclic label coding method used by the circular smooth label has periodicity, and the assigned smooth label value has a certain tolerance. The expression of CSL is as follows:  
\[\text{CSL}(x)=\left\{\begin{array}{cc}g(x)\;,&\theta-r<x<\theta+r\\ 0\;,&\text{\emph{otherwise}}\end{array}\right. \tag{6}\]  
Among them, \(g(x)\) is a window function, and its four main characteristics are periodicity: \(g(x)=g(x+kT)\), \(k\in N\), \(T=180/\text{w}\); symmetry: \(0\leq g(\theta+\epsilon)=g(\theta-\epsilon)\leq 1\), \(\left|\,\varepsilon\,\right|<r\); maximum value: \(g(\theta)=1\); and monotonicity: \(0\leq g(\theta\pm\epsilon)\leq g(\theta\pm\zeta)\leq 1\), \(\left|\,\zeta\,\right|<\left|\,\varepsilon\,\right|<r\). The \(r\) represents the radius of \(g(x)\) and \(\theta\) represents the angle of the current bounding box. Due to the setting of the window function, \(g(x)\), the network model can calculate the angular distance between the predicted tag and the real tag, so that the loss value of the predicted value decreases with the improvement of the prediction result. The problem of angular periodicity is solved by the periodicity of \(g(x)\). The CSL is shown in Figure 5.