## 2 Materials and Methods
### Loss Function  
The loss function of the proposed MTUW-GAN in this paper comprises three primary components: the underwater image enhancement (UIE) loss, the knowledge distillation (KD) loss, and the channel distillation (CD) loss. When image translation losses [37] are applied to underwater image enhancement, they often encounter challenges, resulting in deficiencies such as blurred details, a low contrast, and overall darkness in the generated images. To address these issues and generate high-quality enhanced underwater images, we propose the underwater image enhancement  
Figure 3: The Channel Distillation module. (UIE) loss. This loss formulation effectively eliminates noise and color distortion caused by underwater light scattering by restoring the visual perception quality of underwater images. The restoration process is achieved by utilizing the global content, color, and local texture information of the images. Among them, for the teacher model, the GAN objective is formalized as:  
\[\min_{G_{T}}\max_{D}V(D,G_{T})=\mathbb{E}_{y\sim p_{\text{train}}(y)}[\log D(y) ]+\mathbb{E}_{x\sim p_{\text{gen}(x)}}[\log(1-D(G_{T}(x)))] \tag{3}\]  
In the context of the image dehazing task, the \(\mathcal{L}_{1}\) loss demonstrates superior capability in aligning the feature distribution between pixels in hazy and clear images compared to the \(\mathcal{L}_{2}\) loss [38]. Moreover, the \(\mathcal{L}_{1}\) loss mitigates the risk of introducing blurred artifacts. Therefore, we employ the \(\mathcal{L}_{1}\) loss to enhance the dehazing performance of the network. The calculation formula for the \(\mathcal{L}_{1}\) loss is expressed as follows:  
\[\mathcal{L}_{1}(G_{T})=\mathbb{E}_{x,y}[\|y-G_{T}(x)\|_{1}] \tag{4}\]  
To enhance the generation of images with advanced features and details, we calculate the content loss [19] by utilizing the output of the middle layer of VGG-19 [39] as a feature representation. This approach assists the generator in producing high-quality images. The content loss is defined as the Euclidean distance between the generated underwater image from the teacher model and the corresponding ground truth. It can be expressed as follows:  
\[\mathcal{L}_{content}(G_{T})=\mathbb{E}_{(x,y)}[\|\varphi_{j}(y)-\varphi_{j}( G_{T}(x))\|_{2}] \tag{5}\]  
The UIE loss is represented as follows:  
\[\mathcal{L}_{\text{UIE}}(G_{T}^{*})=\min_{G_{T}}\max_{D}V(D,G_{T})+\lambda_{1 }\mathcal{L}_{1}(G_{T})+\lambda_{content}\mathcal{L}_{content}(G_{T}) \tag{6}\]  
To optimize the \(G_{S}\), it is necessary to simulate the outputs of multiple \(G_{T}\) models. To accomplish this, we introduce the knowledge distillation (KD) loss. For the sake of clarity, we denote the outputs of the \(G_{T}\) and \(G_{S}\) models as \(t\) and \(s\), respectively. The optimization process of \(G_{S}\) involves simulating the outputs of multiple \(G_{T}\) models. The SSIM index [40] quantifies the similarity between two images by considering their brightness, contrast, and structure. It serves as a valuable metric for measuring image similarity. In our study, we utilize the SSIM index to enhance the local structure and detail information of the generated image. The SSIM loss for \(t\) and \(s\) is defined as follows:  
\[\mathcal{L}_{\text{SSIM}}(t,s)=1-SSIM(t,s) \tag{7}\]  
Relying solely on the SSIM loss during training may lead to significant geometric distortions in the generated images. Furthermore, since the SSIM loss is primarily designed for grayscale images, it can fall short in accurately assessing the quality of color images, resulting in color bias within the generated outputs. To mitigate these issues, we integrate perceptual loss [41] into the training process. By leveraging a pre-trained VGG network, we extract high-level feature representations from \(t\) and \(s\) to approximate the perceptual information between them.