## 3 Proposed Dehazing Method
### Transmission Medium Estimation Based on Cascaded Multi-scale CNN
#### 3.2.3 Training of CMT\({}_{net}\)  
1. _Training Data:_ A well-performed network cannot be attained only by making a network with good structure and perfect implementation, but it is also based on the training process. Generally, deep learning networks are data-hungry models. However, it is not easy to provide massive data to train the network, especially for providing pairs of real-world hazy images and their truth transmission maps. Based on the assumption that indicates that the transmission map of an image is constant in small patches, and employing the hazy image formation model (Equation (1), by regarding \(A=1\)), we synthesized a massive number of hazy/clear pairs of patches (as shown in Figure 10). First, we collected from the Internet more than 1000 natural clear images for different scenes (mountain, herb, water, clouds, and traffic). Then, we randomly picked 10 patches of \(15\times 15\) from each image, thus totally we had 10,000 training patches. To reduce the overfitting and make the training process more robust, We added a GaussianNoise layer as the input layer. The amount of noise must be small. Given that the input values are within the range [0, 1], we added Gaussian noise with a mean of 0.0 and a standard deviation of 0.01, chosen arbitrarily.
2. _Training Strategy and Network Loss:_ The prediction of the transmission map requires the mapping link \(\mathcal{F}_{t}\) between the color image and its corresponding transmission medium through automatic supervised learning. The optimization of the model is realized by minimizing the loss function  
Figure 10: Learning Process Illustration.  
Figure 9: ReLU and BReLU Representations.  
between \(I\)(x) and its truth transmission, with predicting optimal parameters \(\theta\) (weights and biases). First, the network feeds on the training image patches \(I\)(x) and updates the parameters iteratively, until the loss function reaches the minimum value. The loss function in learning models is a crucial part because it measures the performance of the model to predict the desired result. The extensively used loss function in image dehazing is MSE (Mean Squared Error) that can be expressed as follows: \[\mathcal{L}_{MSE}=\frac{1}{M}\sum_{i=1}^{M}||\mathcal{F}_{t}(I_{i},\theta)-t_{ i}||^{2}\] (8) where \(\mathcal{F}_{t}(I_{i},\theta)\) represents the predicted transmission medium, \(t_{i}\) is the truth transmission medium, and \(M\) is the number of hazy image patches in the training data. As noticed, although the \(l_{2}\) loss \(\mathcal{L}_{MSE}\) can maintain both edges and details well during the reconstruction process, it cannot preserve sufficiently the background texture. Therefore, we also consider the SSIM loss [43] in this work to assess the texture and structure similarity. The structural similarity (SSIM) value of each pixel \(x\) can be calculated as follows: \[SSIM(p)=\frac{2\mu_{x}\mu_{y}+C_{1}}{\mu_{x}^{2}+\mu_{y}^{2}+C_{1}}\cdot\frac{ 2\sigma_{xy}+C_{1}}{\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2}}\] (9) Note that \(C_{1}\) and \(C_{2}\) are constants of regularization set as default 0.02 and 0.03. \(\mu_{x}\), \(\mu_{y}\), \(\sigma_{x}\) and \(\sigma_{y}\) represent the average and standard deviation of the predicted image patch x and ground truth image patch y, respectively. Eventually, the loss \(\mathcal{L}_{SSIM}\) is calculated as: \[\mathcal{L}_{SSIM}=1-\frac{1}{M}\sum_{p=1}^{M}SSIM(p)\] (10) For the proposed network CMT\({}_{net}\), the final loss is defined as the aggregation of \(\mathcal{L}_{MSE}\) and \(\mathcal{L}_{SSIM}\) as follows: \[\mathcal{L}_{f}=\mathcal{L}_{MSE}+\mathcal{L}_{SSIM}\] (11)