# Establishing Two-Dimensional Dependencies for Multi-Label Image Classification
## 3 Method
### SFDM
#### 3.2.1 Feature Fusion Module  
The Feature Fusion Module (FFM) performs cross-layer and cross-scale fusion, as shown in Figure 2. To enhance the model's representation power, feature tensors \(X_{3}\in\mathbb{R}^{1024\times 28\times 28}\) and \(X_{4}\in\mathbb{R}^{2048\times 14\times 14}\) from 'Layer3' and 'Layer4' of ResNet are extracted and processed. Tensor \(X_{3}\) passes through a convolutional layer to produce \(u_{1}\in\mathbb{R}^{512\times 28\times 28}\), and \(X_{4}\) undergoes two convolutional layers to generate \(u_{2}\in\mathbb{R}^{512\times 14\times 14}\) and \(u_{3}\in\mathbb{R}^{512\times 7\times 7}\). These tensors are unified using bilinear operations, resulting in \(u_{1}^{\prime}\), \(u_{2}^{\prime}\), and \(u_{3}^{\prime}\). The feature fusion factor \(B_{f}\) is then computed as the sum of these unified tensors:  
\[B_{f}=u_{1}^{\prime}+u_{2}^{\prime}+u_{3}^{\prime}. \tag{5}\]  
Subsequently, the fusion factor \(B_{f}\) is added to \(u_{1}^{\prime}\), \(u_{2}^{\prime}\), and \(u_{3}^{\prime}\), respectively, to achieve information fusion of the information among them:  
\[u_{1}^{\prime\prime}=u_{1}^{\prime}+B_{f},\quad u_{2}^{\prime\prime}=u_{2}^{ \prime}+B_{f},\quad u_{3}^{\prime\prime}=u_{3}^{\prime}+B_{f}. \tag{6}\]  
For the abstract tensor \(u_{3}^{\prime}\), an information compensation step is applied to improve feature fusion. This involves element-wise multiplication of \(u_{1}^{\prime}\), \(u_{2}^{\prime}\), and \(u_{3}^{\prime}\), followed by multiplication with a compensation coefficient \(C_{c}\), resulting in the compensation factor \(C_{f}\):  
\[C_{f}=C_{c}\cdot(u_{1}^{\prime}\odot u_{2}^{\prime}\odot u_{3}^{\prime}), \tag{7}\]where \(\odot\) denotes element-wise multiplication. The tensor \(C_{f}\) is then multiplied with \(u^{\prime\prime}_{3}\) to accomplish information compensation:  
\[u^{\prime\prime\prime}_{3}=C_{f}\odot u^{\prime\prime}_{3}. \tag{8}\]  
At this stage, cross-layer information fusion was completed. By interpolation, the tensors that had been fused with each other's information were restored to their original dimensions, and they are denoted as \(U_{1}\), \(U_{2}\), and \(U_{3}\), respectively:  
\[U_{1}=\text{Interp}(u^{\prime\prime}_{1}),\quad U_{2}=\text{Interp}(u^{\prime \prime}_{2}),\quad U_{3}=\text{Interp}(u^{\prime\prime}_{3}), \tag{9}\]  
where \(\text{Interp}(\cdot)\) represents the interpolation operation. Subsequently, a global maximum pooling operation is performed on each tensor to transform their dimensions to \([B,512]\), where \(B\) represents the batch size:  
\[U^{pool}_{1}=\text{MaxPool}(U_{1}),\quad U^{pool}_{2}=\text{ MaxPool}(U_{2}),\quad U^{pool}_{3}=\text{MaxPool}(U_{3}). \tag{10}\]  
The three feature tensors are then concatenated, resulting in a feature tensor of dimension \([B,3\times 512]\):  
\[U_{f}=\text{Concat}(U^{pool}_{1},U^{pool}_{2},U^{pool}_{3}). \tag{11}\]  
Finally, the feature tensor undergoes a fully connected layer to obtain a feature tensor of dimension \([B,C]\), where \(C\) represents the desired number of output classes:  
\[f_{FFM}=f_{c}(U_{f}), \tag{12}\]  
where \(f_{c}(\cdot)\) represents the fully connected layer. The computation of the FFM module can be represented by the following equation:  
\[f_{FFM}=\xi^{GCL}(\Psi(X_{3},X_{4})), \tag{13}\]  
where \(\Psi(\cdot,\cdot)\) represents the utilization of the \(X_{3}\) and \(X_{4}\) convolutions, interpolation, and feature fusion; and the process \(\xi^{GCL}(\cdot)\) involves applying global max pooling, concatenating the fused tensors, and adjusting the length through a final fully connected layer.  
Figure 2: Schematic diagram of FFM. The tensor \(\mathbf{u}\) represents the output of the \(X_{3}\) and \(X_{4}\) convolutions; \(\mathbf{u}^{\prime}\) denotes the tensor obtained by interpolating and unifying the dimensions of \(\mathbf{u}\); \(B_{f}\) represents the feature fusion factor; \(C_{f}\) represents the compensation factor; and \(\mathbf{U}\) represents the tensor that undergoes feature fusion and is restored to its original dimensions.