# Improved Color Attenuation Prior for Single-Image Haze Removal
## 4 Evaluations
### Quantitative Evaluation
#### 4.1.2 Synthetic Datasets
FSIMc is calculated by:  
\[FSIM_{c}(\mathbf{X},\mathbf{Y})=\frac{\sum_{i\in\Omega}\mathbf{S}_{L}(i)\mathbf{S}_{\mathbf{C}}(i) ^{\lambda}\mathbf{PC}_{\mathbf{m}}(i)}{\sum_{i\in\Omega}\mathbf{PC}_{\mathbf{m}}(i)}, \tag{29}\]  
where \(\mathbf{S}_{\mathbf{L}}(i)\) denotes the combined similarity measure of the phase congruency and the gradient magnitude similarities between two images, \(\mathbf{S}_{\mathbf{C}}(i)\) the chrominance similarity measure, \(\mathbf{PC}_{\mathbf{m}}(i)\) the coefficient weighting the importance of \(\mathbf{S}_{\mathbf{L}}(i)\) in the overall similarity between \(\mathbf{X}\) and \(\mathbf{Y}\), and \(\lambda\) is a positive constant for adjusting the importance of the chrominance component. Looking at Equation (29), \(\Omega\) means the whole image domain. Since both SSIM and TMQI are designed to evaluate grayscaleimages only, an additional FSIMc is used to conduct a thorough assessment. Similarly, FSIMc lies between 0 and 1, where the higher FSIMc means that the compared image resembles the reference image to a greater degree. FADE is a referenceless evaluation metric based on natural scene statistics (NSS), fog aware statistical features (FASFs), and Mahalanobis distance. The feature domain is defined based on a total of twelve features, including NSS and FASFs [44]. [PERSON] et al. [44] first collected 500 hazy images and another 500 haze-free images. The features extracted from these two image groups were then individually fitted to the multivariate Gaussian (MVG) model to calculate their corresponding mean vector (\(\mu_{f}\), \(\mu_{f\!\!f}\)) and covariance matrix (\(\Sigma_{f}\), \(\Sigma_{f\!\!f}\)), respectively. To access the perceptual haze density of a particular input image, its extracted features also are fit to MVG model. The mean vector (\(\mu_{i}\)) and covariance matrix (\(\Sigma_{i}\)) are then employed to calculate the Mahalanobis distances from input image to hazy image group (\(D_{f}\)) and haze-free image group (\(D_{ff}\)), as shown in Equations (30) and (31). Finally, FADE is calculated as the ratio in Equation (32) and then has positive values, where the smaller is the better. \[D_{f}=\sqrt{\left(\mu_{i}-\mu_{f}\right)^{T}\left(\frac{\Sigma_{i}+\Sigma_{f}} {2}\right)^{-1}\left(\mu_{i}-\mu_{f}\right)}, \tag{30}\]  
\[D_{ff}=\sqrt{\left(\mu_{i}-\mu_{f\!\!f}\right)^{T}\left(\frac{\Sigma_{i}+ \Sigma_{f\!\!f}}{2}\right)^{-1}\left(\mu_{i}-\mu_{f\!\!f}\right)}, \tag{31}\]  
\[FADE=\frac{D_{f}}{D_{ff}+1} \tag{32}\]  
Tables 2 and 3 display the average SSIM, TMQI, FSIMc, and FADE results on the FRIDA2 and D-HAZY datasets, respectively. The best result is marked with bold. Since the luminance information contributes toward the calculation of SSIM, TMQI, and FSIMc, the noticeable background noise of HPM and the halo artifacts of FFD increase these quantitative scores to a certain extent. Thus, they exhibit fairly good performance in terms of SSIM, TMQI, and FSIMc, but not FADE. Alternately, CAP possesses the best FADE score and high TMQI and FSIMc. This is because CAP often excessively removes haze from hazy images, bringing about better colorfulness restoration. Removing too much haze also causes loss of image details, however, resulting in CAP's low SSIM. Regarding DCP, as FRIDA2 dataset contains images of a road scene with sky regions, DCP shows poor performance. The proposed ICAP is the best performing method under the SSIM, TMQI, and FSIMc metrics and the second-best method in terms of FADE. It is attributed to the adaptive constraints on the lower limit of a transmission map, which avoids the excessive haze removal that occurs in CAP. According to Table 3, however, ICAP is only the third best under the SSIM and FSIMc metrics, and the best under the FADE metric. This is because D-HAZY is a kind of biased dataset that solely consists of indoor images, while our proposed ICAP has been tuned to gain best performance in both outdoor and indoor scenes.