# Improved Color Attenuation Prior for Single-Image Haze Removal
## 2 Related Works
### Color Attenuation Prior  
The transmission map in the atmospheric scattering model is exponentially proportional to the scene depth, and this relationship can be expressed as:  
\[\mathbf{f}(x,y)=e^{-\beta\mathbf{d}(x,y)} \tag{13}\]  
where \(\beta\) is the scattering coefficient of the atmosphere, and \(\mathbf{d}\) is the scene depth. The large \(\mathbf{d}\) results in a small \(\mathbf{t}\) in the distant regions. Accordingly, a decrease in the image's saturation caused by the multiplicative direct attenuation term \(\mathbf{J}(x,y)\mathbf{t}(x,y)\) and an increase in the image's brightness caused by the additive airlight term \(A[1-\mathbf{t}(x,y)]\) lead to a big difference between the image's saturation and brightness. To contrast, the small \(\mathbf{d}\) in the close regions results in a large \(\mathbf{t}\), therein lies the cause for the small difference between the image's saturation and brightness. Hence, [PERSON] et al. (2018) proposed a linear model describing the correlation between the scene depth and the difference between the image's saturation and brightness:  
\[\mathbf{d}(x,y)=\theta_{0}+\theta_{1}\mathbf{v}(x,y)+\theta_{2}\mathbf{s}(x,y)+\epsilon(x,y), \tag{14}\]  
where \(\mathbf{v}\) and \(\mathbf{t}\) are the image's brightness value and saturation, respectively; \(\theta_{0}\), \(\theta_{1}\), \(\theta_{2}\) are the model's constant parameters; and \(\epsilon\) is a random variable that denotes the model's error. The authors employed a simple and efficient supervised learning method for the synthetic training data to learn the coefficients \(\theta_{0}\), \(\theta_{1}\), and \(\theta_{2}\).  
The method for estimating the atmospheric light is similar to that of DCP. The depth calculated using Equation (14) is filtered by a minimum filter to reduce the influence of white objects on the accuracy of the estimation process. Then, the top 0.1 percent brightest pixels in the depth map are chosen. Among these pixels, the one with the largest L-2 norm in the input image is selected as the atmospheric light. Since the scene depth is blurred as the result of passing through the minimum filter, it is refined by the guided image filter ([PERSON] et al., 2016) before substituting into Equation (2) to get the haze-free image \(\mathbf{J}\). The use of a minimum filter is beneficial to the estimation of atmospheric light, but it creates the unfortunate byproduct of blurring the depth map; therefore, a very large guided image filter must be used, and therein causes CAP's high computational cost. The block diagram of CAP is depicted in Figure 1, in which the red circle shows the minimum filter effectively reduces the influence of white objects (e.g., a train's headlight), whilst the pale orange circle indicates that it also blurs the depth map, leading to the use of a very large guided image filter.  
According to our experiments, CAP works well in most cases but it still suffers some drawbacks, such as background noise, color distortion, and intensive computation. We will delve deeply into these drawbacks and then propose solutions for them in Section 3.