# Saturation-Based Airlight Color Restoration of Hazy Images
## 3 Proposed Dehazing Method
### Depth Map Setting Based on Luminance and Saturation
#### 3.3.2 Depth Map with the Saturation Weight  
Scenes in distant areas create dark shading and noise as visibility increases [5; 32]. Hazy images contain scenes that require visibility due to atmospheric light; thus, a method to distinguish between the sky area and objects is needed to address this. This study uses the color-restored saturation of a and b channels as the method for differentiating haze and objects.  
The proposed method is explained in more detail in the saturation map shown in Figure 6. The saturation used the \(|a|+|b|\) value, multiplied by 10, to visualize the noise in the sky area.  
In Figure 6, the color-restored input image shows the acromatic airlight area mostly via Equation (13). As a result, the airlight area shows the darkest value, and as shown in Figure 6b, airlight and objects can be separated by the color of the objects. Based on this assumption, the saturation map applied with the weighting value \(s(x)\) is proposed as follows.  
\[s(x)=(L(x)-\mu_{w})(|a(x)|+|b(x)|) \tag{16}\]  
Here, \(\mu_{w}\) is the average of the \(L\) channel of the white-balanced total image, showing the standard by which to differentiate close and distant objects. \(s(x)\) outputs a positive value against the distant scene, whose luminance is higher than \(\mu_{w}\), and a negative value against the close scene, whose luminance is smaller than \(\mu_{w}\). As a result, Equation (16) performs stronger dehazing as it is closer to airlight, and dehazing with a smaller weighting value is performed on close objects susceptible to shading.  
Since the saturation of the sky area results in a very low value close to 0, based on the assumption of the saturation of the sky area, the multiplication of \(s(x)\) and saturation in a distant scene leaves only the hazy object components except the sky area. The multiplication of \(s(x)\) and saturation is shown in Figure 6c, which was amplified 10 times for visualization.  
As explained, this study sets \(d(x)\) based on luminance and saturation as follows:  
\[d(x)=a_{1}+a_{2}L(x)+a_{3}s(x)+e_{0} \tag{17}\]  
\(s(x)\) in Equation (17) is the saturation applied with the weighting value set in Equation (16). All parameters in Equation (17), \(a_{1}\), \(a_{2}\), and \(a_{3}\), and scattering \(\sigma\) can be calculated by resetting the input value x3 as \(s(x)\)C in Algorithm 1, and the random error \(e_{0}\) can be calculated by applying the calculated scattering \(\sigma\) to Equation (15).  
Because of executing Equation (17) in Algorithm 1, the values \(a_{1}=-0.2138\), \(a_{2}=0.9103\), \(a_{3}=1.3970\), and \(\sigma=0.0224\) were obtained, and \(\sigma\) was used to generate random error \(e_{0}\), for each location \(x\), through the Gaussian distribution model described in Equation (15). This study uses all of these parameters as generalized parameters for all hazy images. Meanwhile, Figure 6b,c show some noise, and to remove it, we performed guide filtering [3].  
The proposed depth map estimation, as explained above, is performed as follows:  
1. Retrieve the L, a, and b channels of the image resulting from the restoration presented in Section 3.2 and define their pixel values as \(L(x)\), \(a(x)\), and \(b(x)\).  
Figure 6: Saturation map for the color-restored hazy image: (**a**) color-restored input image, (**b**) 20-times amplified saturation map, and (**c**) saturation map of the hazy image multiplied by the weighting value \((L(x)-\mu_{w})\).  
2. \(\sigma=0.0224\) is applied to the Gaussian distribution model of Equation (15), \(N(0,\sigma^{2})\), to produce the random error \(\epsilon_{0}\) to each location \(x\).
3. Enter these into Equation (17) using \(a_{1}=-0.2138\), \(a_{2}=0.9103\), \(a_{3}=1.3970\), and \(\epsilon_{0}\) to estimate \(d(x)\), the depth map to which the weighting value of saturation is applied.
4. To solve the noise of the saturation map, filtering is performed using the guide filter on \(d(x)\).  
Figure 7 illustrates the depth map and the corresponding guide filtering results.