## 3 Proposed Dehazing Method
### Transmission Medium Estimation Based on Cascaded Multi-scale CNN
#### 3.2.1 Rough Transmission Map Subnetwork CMCNN\({}_{tr}\)  
CMCNN\({}_{tr}\) is composed of four sequential cascaded MCNN D-units. Each D-unit is defined as follow:  
1. _D-Unit Structure:_ The overall design of a D-unit is shown in Figure 8 (Left) and comprises three multi-scale convolutional layers pursued by a Bilateral Rectified Linear Unit (BReLU) [28], except for the last convolutional layer. Each convolutional layer represents a fusion of convolutions with a different kernel size (\(3\times 3\), \(5\times 5\), and \(7\times 7\)), thus it is called multi-scale convolution. Regarding the number of filters \(Fn_{i_{t}}\), for the first and second layers \(C_{1}\) and \(C_{2}\), we use 16 filters, and for the last layer we use eight filters. The idea of these D-units is inspired by the DenseNet [39] architecture, where layers of each D-unit are densely connected. The output feature maps of each layer \(C_{n}\) are concatenated with those of all succeeding layers; feature maps of the first layer \(C_{1}\) are concatenated with that of the second and third layers \(C_{2}\) and \(C_{3}\). Moreover, the same for feature maps of \(C_{2}\) are concatenated with \(C_{3}\) feature maps. The proposed D-units-based architecture has significant benefits, e.g. it can avoid the gradient vanishing problem that deep CNNs have with fewer required parameters. In addition, it can maximize the information flow with non-redundancy of feature maps. 2. _Multi-scale CNN:_ It is a general truth that human visual perception is a multi-scale neural system. In contrast, using CNN models to solve low-level computer vision tasks always requires the best choice of kernel size. The small size of the filter window can highlight only low-frequency features and neglect the high-frequency content or other important  
Figure 7: General proposed architecture CMT\({}_{nt}\). information. Likewise, big kernels can extract only high-frequency details and disregard low-frequency image information. Besides, employing CNNs with successive single filter sizes produces a deeper CNN architecture, which makes the computations complicated and thus hinders the speed of the training process. Therefore, the success of multi-scale CNN representation [26; 28] motivates us to apply multi-scale CNN layers in our work to obtain both low- and high-frequency details. In this work, each convolutional layer can be defined as a concatenation of three convolutions with different filter sizes (\(3\times 3\), \(5\times 5\), and \(7\times 7\)), as shown in Figure 8 (right), and can be expressed as: \[F_{n}^{[i]}=max\left(0,concat(f_{1}^{[i]},f_{2}^{[i]},f_{3}^{[i]})\right)\] (7) where \(F_{n}^{[i]}\) represents the i-th layer generated feature maps, \(n\) is the layer depth (\(n\) = 3), and \(f_{1}^{[i]},f_{2}^{[i]},f_{3}^{[i]}\) are the output feature maps attained by the three convolutions (\(3\times 3\), \(5\times 5\), and \(7\times 7\)) in each multi-scale CNN layer. * _BReLU:_ Recently, most of the deep learning methods [26; 40] have employed the ReLU as a nonlinear transfer function to solve the problem of vanishing gradient instead of the previous weak sigmoid function [41] that makes the learning convergence very slow. However, the ReLU function [42] is especially designed for classification problems. In this work, we adopt the BReLU function: [PERSON] [28] proposed a sparse representation, benefiting from local linearity of ReLU and preserving the bilateral control of Sigmoid function and considering the restoration problem. Figure 9 shows the difference between the conventional ReLU and Bilateral ReLU; note that \(t_{min}\)=0 and \(t_{max}=1\). * _Concatenation:_ After generating the rough transmission map through CMCNN\({}_{tr}\) subnetwork, it is transferred to CMCNN\({}_{t}\) subnetwork as support information, where it is merged with the feature maps extracted by the first unit to explore new features. Exploring new features by using concatenation augments the performance of the CMCNN\({}_{t}\) to predict efficient refined transmission maps.