# Fast Fourier Asymmetric Context Aggregation Network:
## 3 Proposed Method
### The Content Inference Stage
To address this, the first FFC in the FF-CDRB applies adaptive channel weighting to the outputs of both branches using the Squeeze-and-Excitation (SE) attention [65], thereby effectively adjusting the response levels of feature branches. The feature input to the \(n\)-th FF-CDRB block is denoted as \(f^{\prime}_{n}\in\mathbb{R}^{H\times W\times C}\), \(n\in\{1,2,\ldots,R\}\). The output of \(f^{\prime}_{n}\) after processing through the first FFC is represented as \(f^{1}_{n}\). The specific operation for adaptive channel weighting on feature \(f^{1}_{n}\) is  
1. Apply global average pooling to compress \(f^{1}_{n}\) into a \(1\times 1\) vector \(V_{f}\) to capture global information: \[Squeeze\Big{(}f^{1}_{n}\in\mathbb{R}^{H\times W\times C}\Big{)}\to V_{f}\in \mathbb{R}^{1\times 1\times C},\] (3) where _Squeeze_ represents the result of compressing the features using global average pooling. 2. Map the features of \(V_{f}\) through a learnable fully connected layer, and finally apply the Sigmoid function to obtain the adaptive weight vector \(w\): \[Sigmoid\Big{(}FullyConnectedLayer\Big{(}V_{f}\in\mathbb{R}^{1\times 1\times C} \Big{)}\Big{)}\to w\in\mathbb{R}^{1\times 1\times C},\] (4) where \(FullyConnectedLayer\) represents a learnable fully connected layer used to capture feature information in the compressed vector, while _Sigmoid_ represents the sigmoid function, which normalizes the final result into the weight vector \(w\). 3. Use the channel-adaptive weights \(w\) to adjust the response levels of the local and global branch features in \(f^{1}_{n}\), ultimately obtaining the feature representation with adaptive weight adjustments: \[F_{\text{scale}}\left(f^{1}_{n},w\right)\to f^{1}_{n}\in\mathbb{R}^{H\times W \times C},\] (5) where \(F_{\text{scale}}\left(f^{1}_{n},w\right)\) represents the result of element-wise multiplication, where the weight \(w\) is applied to \(f^{1}_{n}\). The adaptively weighted feature \(f^{1}_{n}\) is fed into the second FFC in the FF-CDRB block for further content inference. The output after processing by the second FFC is denoted as \(f^{2}_{n}\). Inspired by the residual design in ResNet, a dual-path residual mechanism is incorporated in the FF-CDRB to simplify module learning and mitigate potential feature degradation caused by the serial FFC structure, thereby enhancing the module's expressive capability. Specifically, in the FF-CDRB, the initial input \(f^{\prime}_{n}\) and the feature \(f^{1}_{n}\) processed by the first FFC are merged through skip connections into the final output to optimize feature representation. The final output of the \(n\)-th FF-CDRB block can be expressed as  
\[f^{F}_{n}=f^{\prime}_{n}\oplus f^{1}_{n}\oplus f^{2}_{n}, \tag{6}\]  
Figure 5: Structure of FFC. where \(\oplus\) represents element-wise addition. In summary, the content inference process in FCAN can be described as follows:  
\[ \begin{split} f_{1}^{\mathrm{F}}&=\mathbf{F}-\mathbf{CDRB_{1} }(f_{N})\\ f_{2}^{\mathrm{F}}&=\mathbf{F}-\mathbf{CDRR_{2}}\left(f_{1}^ {\mathrm{F}}\right)\\ &\ldots\\ f_{R}^{\mathrm{F}}&=\mathbf{F}-\mathbf{CDRB_{R}}\left(f_{R-1}^ {\mathrm{F}}\right),\end{split}  \tag{7}\]  
where \(\mathbf{F}\mathbf{F}-\mathbf{CDRB_{n}},n\in\{1,2,\cdots,R\}\) denotes the \(n\)-th FF-CDRB block in the FCAN.