# Self-Supervised Remote Sensing Image Dehazing Network Based on Zero-Shot Learning
## 3 Methodology
### Loss Function Design  
In this section, we provide a detailed description of the loss function. The elaborated loss function, \(L\), comprises four terms:  
\[L=L_{rec}+L_{TV}+\alpha L_{DCP}+\beta L_{\min} \tag{11}\]  
where \(L_{rec}\) represents the reconstruction loss, \(L_{TV}\) denotes the total variance (TV) loss, \(L_{DCP}\) is the \(DCP\) loss, and \(L_{\min}\) represents the minimum pixel intensity loss. \(\alpha\) and \(\beta\) are balancing factors that weigh the contributions of each term in the loss function.  
**Reconstruction loss.** As shown in Figure 1, if the transmission map, atmospheric light, and scene radiance are accurately estimated, the input hazy image, \(I\), and the regenerated hazy image, \(I^{\prime}\), should be identical. Therefore, we propose the reconstruction loss \(L_{rec}\) to minimize the difference between \(I\) and \(I^{\prime}\), as follows:  
\[L_{rec}=\left\|I-I^{\prime}\right\|_{2} \tag{12}\]  
where \(\left\|\cdot\right\|_{p}\) indicate \(L_{p}\) loss. The \(L_{rec}\) ensures that \(I\) and \(I^{\prime}\) are cycle consistent, thereby allowing the entire network to be optimized in a zero-shot learning strategy.  
**TV loss.** The TV loss is incorporated to prevent the model from generating images with high-frequency artifacts or noise and to produce visually pleasing results. It is computed as the sum of the absolute differences between neighboring pixel values in the image. Mathematically, the TV loss can be expressed as:  
\[L_{TV}=\left\|\
abla_{h}I^{\prime}\right\|_{1}+\left\|\
abla_{v}I^{\prime} \right\|_{1} \tag{13}\]  
where \(\
abla_{h}\) and \(\
abla_{v}\) represent the differential operations in the horizontal and vertical dimensions, respectively. Accordingly, the TV loss encourages smoothness in the output images.  
_DCP_ **loss.** As discussed in Section 3.2, the dark channel of a clean image is typically close to zero. Therefore, we use the \(L_{1}\) loss to minimize the dark channel of the recovered \(J^{\prime}\), as follows:  
\[L_{DCP}=\left\|I^{\prime dark}\right\|_{1} \tag{14}\]  
where \(J^{\prime dark}\) can be computed using Equation (7). By leveraging the \(DCP\) loss \(L_{DCP}\), the model is optimized to remove haze more effectively.  
**Minimum pixel intensity loss.** Although the \(DCP\) loss can effectively remove haze, the dehazed result may have a dark color. Furthermore, if the model is overfitted, the intensity of some image pixels may be clipped to zero, causing the loss of dark details.  
Therefore, we propose the minimum pixel intensity loss \(L_{\text{min}}\) to prevent the image pixels from being over-saturated due to the minimization of \(L_{DCP}\). The \(L_{\text{min}}\) is defined as follows:  
\[L_{\text{min}}=-\frac{1}{N}\sum_{x=1}^{N}\left(\min(J^{\prime}(x),0.1)+\min(t^{ \prime}(x),0.1)\right) \tag{15}\]  
where \(x\) represents the image pixel, and \(N\) denotes the total number of image pixels. The \(L_{\text{min}}\) ensures that the pixel intensity of \(J^{\prime}\) and \(t^{\prime}\) is not lower than 0.1, compensating for the over-clipping of pixel intensity caused by \(L_{DCP}\).