# FF-YOLO: An Improved YOLO11-Based Fatigue Detection Algorithm for Air Traffic Controllers
## 3 Materials and Methods
### CA-C3K2 Module
This enables dynamic channel-wise weight adjustment of fatigue features, thereby enhancing cross-channel feature extraction and representation capabilities under complex lighting interference. The structure of CA-C3K2 is illustrated in Figure 7c. The shallow branch captures low-level detail features such as edges and textures, while the deep branch extracts high-level semantic information. Equation (6) provides the mathematical expression of the C3K module, where \(X\in\mathbb{R}^{H\times W\times C}\) denotes the input feature map, \(X_{1},X_{2}\in\mathbb{R}^{H\times W\times\frac{C}{2}}\) represent the two parts split along the channel dimension, and \(Seq_{3\times 3}\) signifies a bottleneck sequence using 3 \(\times\) 3 convolution kernels. \[C3K(X)=F_{conv}(Concat[X_{1},Seq_{3\times 3}(X_{2})]) \tag{6}\]  
CA-C3K2 incorporates the channel attention mechanism via the squeeze-and-excitation (SE) module [33]. This module adaptively adjusts the response intensity of feature channels, thereby enhancing the model's sensitivity to critical fatigue features. Compared to traditional convolution operations that treat channel information equally, the SE module achieves channel-wise focusing capability for fatigue features in the detection task without significantly increasing computational complexity. The structure of the SE module is shown in Figure 8. The workflow of the module can be divided into three stages: squeeze, excitation, and scale. In the squeeze stage, the input feature map \(U\in\mathbb{R}^{H\times W\times C}\) aggregates spatial dimension information via global average pooling (GAP) to generate the channel-wise descriptor \(z\in\mathbb{R}^{C}\). Equation (7) provides the mathematical expression of the squeeze operation, where \(u_{c}(i,j)\) represents the feature value at position (i,j) for channel c, and \(z_{c}\) denotes the global statistical descriptor for channel c.  
\[z_{c}=\frac{1}{H\times W}\sum_{i=1}^{H}\sum_{j=1}^{W}u_{c}(i,j) \tag{7}\]  
Figure 8: SE module working mechanism. The excitation stage captures nonlinear interdependencies among channels through dimensionality reduction and restoration via a bottleneck structure. First, a fully connected layer reduces the dimensionality of the compressed features. After introducing nonlinear transformations via the ReLU activation function, a second fully connected layer restores the original channel dimensionality. Finally, the Sigmoid function generates the channel attention weight vector \(s\in\mathbb{R}^{\mathbb{C}}\). The mathematical formula of the excitation operation, which is a Sigmoid function, is expressed as Formula 8, where \(W_{1}\in\mathbb{R}^{\overleftarrow{\gamma}\times\mathbb{C}}\) and \(W_{2}\in\mathbb{R}^{\mathbb{C}\times\overleftarrow{\gamma}}\) are the weight matrices of the two fully connected layers, r is the dimensionality reduction ratio, and \(\delta\) is the ReLU activation function. \[s=\sigma(W_{2}\cdot\delta(W_{1}\cdot z)) \tag{8}\]  
In the feature recalibration stage, the learned weight vector s is multiplied channel by channel with the original input feature map U to obtain the final output. Formula 9 is the mathematical expression of the SE module, where represents channel-by-channel multiplication and GAP is global average pooling. \[\tilde{U}=U\odot\sigma(W_{2}\cdot ReLu(W_{1}\cdot GAP(U))) \tag{9}\]  
The proposed CA-C3K2 module adopts a dual-branch channel attention mechanism. The SE module in the shallow branch enhances channel responses to low-level texture features, improving the detection capability for subtle edges and local contrast, while the SE module in the deep branch focuses on suppressing redundant channels weakly correlated with semantic categories. CA-C3K2 modifies the original C3K2 to employ two fixed C3K modules to enhance the model's feature extraction capability while ensuring computational efficiency, and connects an SE module after each bottleneck layer to incorporate channel attention. Equation (10) presents the mathematical computation of the entire CA-C3K2 module.