## 3 Rotational Feature Enhancement FCOS
### Rotation-Invariant Learning Module  
In the process of multi-scale feature extraction, a common strategy is to disregard shallow features in order to minimize computational complexity. However, shallow features are particularly effective at capturing edge and texture information, which is critical to learning rotation invariance. Motivated by this observation, our approach incorporates a rotation operation applied to the feature map C2, which facilitates the extraction of rotation-invariant features, thereby enhancing the robustness of the network to geometric transformations. Given the shallow feature C2 \(\in\mathbb{R}^{\mathbf{H}\times\mathbf{W}\times\mathbf{256}}\), we first generate three adaptive rotation angles through a lightweight angle prediction module. Specifically, the global average pooling (GAP) condenses spatial information into a channel descriptor \(z\in\mathbb{R}^{\mathbf{256}}\), which is then passed through a compact multilayer perceptron (MLP) composed of two fully connected layers. The resulting output is further constrained by a scaled \(tanh\) activation (bounded by \(\pm 45^{\circ}\)) to ensure that rotation angles remain within a reasonable range. Equation (1) shows the process of generating the adaptive rotation angles. \[[\theta_{1},\theta_{2},\theta_{3}]=45^{\circ}\cdot tanh(MLP(GAP(C2)))\quad \theta\in[-45^{\circ},45^{\circ}], \tag{1}\]  
where \(\theta_{i}\) (\(i=1,2,3\)) denote the learned rotation angles, adaptively determined from the input feature statistics. With the learned angles \(\theta_{i}\), we perform explicit rotation transformations on C2 through a grid sampling procedure, producing three rotated feature maps \(R_{i}\):  
\[R_{i}=GS(\beta(\theta_{i}),C2),\quad i=1,2,3, \tag{2}\]where GS denotes the grid sampling operation, and the affine rotation matrix \(\beta(\theta)\) is defined as:  
\[\beta(\theta)= \begin{bmatrix}cos\theta&-sin\theta&0\\ sin\theta&cos\theta&0\end{bmatrix} . \tag{3}\]  
Each rotated feature \(R_{i}\) is further individually processed by our proposed rotated feature refinement (RFR) module (Section 3.1.1), which aims to mitigate redundant channels while accentuating rotation-sensitive details, thereby delivering more discriminative representations for subsequent processing. The resulting refined characteristics \(R_{i}^{{}^{\prime}}\in\mathbb{R}^{\mathbf{H}\times\mathbf{W}\times\mathbf{128}}\) are concatenated along the channel dimension, forming an intermediate aggregated characteristic \(F_{ext}\in\mathbb{R}^{\mathbf{H}\times\mathbf{W}\times\mathbf{384}}\). Subsequently, a \(1\times 1\) convolution is employed to revert the original channel dimensions, yielding the fused feature map \(F_{mix}\), which not only preserves the global semantic information but also effectively captures the local rotational variations at multiple directions and scales. To selectively highlight informative features crucial for rotation invariance, \(F_{mix}\) is conveyed to our Cross-Gated Channel-Spatial Attention (CG-CSA) module (Section 3.1.2),  
Figure 2: Overall architecture of our proposed RFE-FCOS, which integrates the backbone network, rotational feature enhancement (RIL and RFF) modules, FPN, and the prediction head. Starting from an input image, the backbone generates multi-scale feature maps. Next, the shallow feature map C2 is directed to the RIL module for rotation-invariant learning, while the remaining deep feature are blended with the extracted rotation features in RFF module before being passed to the FPN. In both the classification and regression branch, centerness and classifier predictions follow FCOS-O defaults, while the localization branch utilizes the proposed DARIoU loss for improved OBB regression. The RFR and CG-CSA components are utilized as sub-modules within the RIL module. where cross-gating reinforces complementary interactions between channel and spatial attention maps.