# Improved Color Attenuation Prior for Single-Image Haze Removal
## 4 Evaluations
### Quantitative Evaluation
#### 4.1.2 Synthetic Datasets  
Foggy Road Image DAtabase 2 (FRIDA2) [36] and D-HAZY [38] are two synthetic datasets that contain pairs of ground truth reference images and hazy images of the same scene. There are 264 synthetic hazy images of 66 road scenes in the FRIDA2 dataset and they are generated completely by computer graphics. Each road scene comprises a ground truth reference and four associated hazy images corresponding to four different kinds of haze: homogeneous, heterogeneous, cloudy homogeneous, and cloudy heterogeneous. This dataset is designed for Advanced Driver Assistance Systems (ADAS) [38]. Since FRIDA2 solely contains computer graphic generated scenes, it may not be valid for real scenarios. Accordingly, an additional D-HAZY dataset consisting of real ground truth reference images and synthesized hazy ones has come into use. It contains over 1400 images of real scenes and their corresponding depth maps captured by a Microsoft Kinect camera. Then, the atmospheric scattering model is used with the assumptions that the atmospheric light and haze density are both uniform to synthesize hazy images. As these datasets have ground truth reference images, Structural SIMilarity (SSIM) [41], Tone-Mapped image Quality Index (TMQI) [42], and Feature SIMilarity extended to color image (FSIMc) [43] are employed in an evaluation. Additionally, Fog Aware Density Evaluator (FADE) [44], a learned metric predicting the haze density in a scene, also is used to assess the visibility of the scene after removing haze. The smaller FADE is, in this context, the larger the amount of haze that has been removed. It is supposed that \(\mathbf{X}\) and \(\mathbf{Y}\) are two nonnegative image luminance signals. The SSIM measure of two images is calculated as follows:  
\[SSIM(\mathbf{X},\mathbf{Y})=\frac{\Big{(}2\mu_{x}\mu_{y}+C_{1}\Big{)}\Big{(}2\sigma_{ xy}+C_{2}\Big{)}}{\Big{(}\mu_{x}^{2}+\mu_{y}^{2}+C_{1}\Big{)}\Big{(}\sigma_{x}^{2}+ \sigma_{y}^{2}+C_{2}\Big{)}}, \tag{27}\]  
where (\(\mu_{x}\), \(\mu_{y}\)) and (\(\sigma_{x}\), \(\sigma_{y}\)) are the local average and standard deviation of \(\mathbf{X}\), \(\mathbf{Y}\), respectively; \(C_{1}\) and \(C_{2}\) are constants for avoiding the instability when (\(\mu_{x}^{2}+\mu_{y}^{2}\)) and (\(\sigma_{x}^{2}+\sigma_{y}^{2}\)) are very close to zero; and \(\sigma_{xy}\) is the correlation coefficient between (\(\mathbf{X}-\mu_{x}\)) and (\(\mathbf{Y}-\mu_{y}\)). The value of SSIM ranges from 0 to 1, in which higher SSIM means two images are more structurally similar. Moreover, Equation (28) gives the formula of TMQI, where \(\mathbf{X}\) and \(\mathbf{Y}\) now denote two color images. \(S(\mathbf{X},\mathbf{Y})\) is the statistical naturalness measure, \(N(\mathbf{X},\mathbf{Y})\) is the overall structural fidelity score, \(0\leq a\leq 1\) is used to adjust the relative importance of these two components, and \(\alpha\) and \(\beta\) determine their sensitivities. Even though \(\mathbf{X}\) and \(\mathbf{Y}\) are two color images, TMQI only evaluates their grayscale channel to assess the structural information preserving characteristics. TMQI takes on values within the range [0, 1], in which the higher, the better:  
\[TMQI(\mathbf{X},\mathbf{Y})=a\times S(\mathbf{X},\mathbf{Y})^{\alpha}+(1-a)\times N(\mathbf{X},\bm {Y})^{\beta} \tag{28}\]  
FSIMc is developed based on the observation that a human visual system perceives an image mainly according to its low-level features, such as the phase congruency, the image gradient magnitude, and the chrominance similarity.