# AdaptiveConv2d: A Novel Convolutional Module for Medical Image Segmentation
## 2 Materials and Methods
### Channel Weight Generation and Optimization  
As shown in Figure 4, after introducing the energy function, the channel weight generation function is formulated as  
\[\alpha_{c}=\sigma(W_{1}\times\text{ReLU}(W_{2}\times w_{c}+\lambda\times \mathcal{L}_{\text{total}}(X_{c}))-\gamma\times|w_{c}-\sigma_{c}|) \tag{5}\]  
where  
* \(W_{1}\) and \(W_{2}\) are the weight matrices of the fully connected layers, which map the global context information, \(w_{c}\), to the channel weight space. These matrices are learned during training to capture the essential relationships between feature maps and global context, enabling the model to adaptively adjust the channel weights based on the overall context. * \(\lambda\) is the regularization coefficient for the energy function, controlling the contribution of the energy function, \(\mathcal{L}_{\text{energy}}(X_{c})\), to the overall channel weight generation. This term ensures that the energy function's influence is appropriately balanced with the main loss function, thereby preventing excessive regularization that could hinder the model's ability to learn from the data. * \(\gamma\) is a hyperparameter that adjusts the balance between global context information, \(w_{c}\), and channel feature deviation, \(|w_{c}-\sigma_{c}|\). This term plays a critical role in regulating how much emphasis is placed on preserving the global contextual features versus minimizing the deviation between the actual channel weight and the expected variance, \(\sigma_{c}\). * \(\sigma\) represents the Sigmoid function, which maps the channel weights to values between 0 and 1. The Sigmoid function introduces non-linearity into the model, allowing the channel weights to be dynamically scaled according to the context and feature deviation, which ensures that the model can adapt to different feature distributions. This mechanism allows the channel weights to be dynamically adjusted according to the trade-off between global and local information. By combining the energy function with the primary loss function, the model can simultaneously optimize feature extraction and context preservation. As a result, the channel weights are not fixed but are dynamically updated during training, leading to improved performance and robustness. Thus, the channel weight generation process ultimately combines the primary loss function with the energy function, enabling the model to adjust the channel weights dynamically during training. This approach improves the accuracy and effectiveness of feature extraction, as it helps the model focus on the most relevant features while maintaining the global coherence of the input data [37]. The total optimized loss function is  
\[\mathcal{L}_{\text{total}}=\mathcal{L}_{\text{primary}}+\lambda_{\text{reg}} \times\mathcal{L}_{\text{energy}}(X) \tag{6}\]  
where \(\mathcal{L}_{\text{primary}}\) is the main loss function used to optimize the model's primary task, such as the cross-entropy loss for classification tasks or the mean squared error for regression tasks. This loss function is designed to guide the model in achieving the specific goals of the task, such as distinguishing between classes or minimizing the prediction error. \(\lambda_{\text{reg}}\) is the regularization coefficient associated with the energy function, \(\mathcal{L}_{\text{energy}}(X)\). The purpose of this regularization term is to encourage the model to maintain coherent and consistent feature maps across channels. The coefficient \(\lambda_{\text{reg}}\) controls the strength of this regularization, essentially determining how much influence the energy function has on the overall optimization process. By adjusting \(\lambda_{\text{reg}}\), we can balance the trade-off between the model's primary objective, which focuses on task-specific performance (e.g., classification accuracy), and the energy regularization, which enforces smoothness and coherence in the feature maps.