# Image Dehazing and Enhancement Using Principal Component Analysis and Modified Haze Features
## 3 Haze Removal Using PCA and Haze-Relevant Features
### Estimation of the Atmospheric Light Using PCA  
Since existing dehazing methods estimate the atmospheric light for every candidate pixel in the input haze image, they require a very large amount of computation. To solve this problem, the proposed method estimates the atmospheric light vector using PCA on the the reduced number of the candidate haze patches. Figure 1: The block diagram of the proposed method. [PERSON] et al. estimated the atmospheric light by estimating the intersection of the principal vectors of the haze patches, but they performed PCA twice on all haze patches, which requires more computation [11]. In the proposed method, we employ the additional constraints to efficiently estimate the atmospheric light using the reduced number of the candidate pixels regarded as haze particles, as shown in Figure 2.  
More specifically, the proposed method reduces the number of candidate haze patches by dividing the input image into four sub-images and then selects the sub-image with the maximum mean brightness value. Since the haze region smoothly varies with the bright pixels, we assume that the selected sub-image with the maximum average includes more haze components. Figure 2 shows the comparison of the region of interests containing the candidate atmospheric light to be estimated. In addition, the proposed method can reduce the probability of inaccurate estimation which regards a bright object as the atmospheric light, since the region of maximum average in the image is usually the sky area. For that reason, the proposed method selects the non-edge candidate pixels that have higher intensity values than the average of the selected sub-image, as shown in Figure 2c. As a result, the proposed method can estimate the candidate atmospheric light at the reduced processing time, because the candidate region is selected, providing the highest mean brightness value. Since the haze patch containing the haze pixel smoothly varies with similar intensity values and principal vector, [PERSON] et al. proposed constraints to select a haze patch having the positive principal component, a single large eigenvalue, and a PCA matrix of rank-one [11]. However, since [PERSON] et al. performed PCA twice to remove outliers by discarding the farthest principal component vector, this method requires more computation. On the other hand, the proposed method can estimate the atmospheric light vector using the constraints proposed in reference [11] in a computationally efficient manner, because the constraints of the proposed method reduce the number of haze patches to be computed. The optimum haze patches containing the atmospheric light are selected at the intersection of the principal component vectors. The candidate vector of atmospheric light is determined to give the smallest error among the intersection points and principal vectors of the optimum haze patches. For that reason, the magnitude of atmospheric light of a hazy image can be estimated using the candidate vectors as  
\[m_{i}=\left\|\widetilde{A}_{i}\right\|_{2}^{2}, \tag{5}\]  
where \(\widetilde{A}_{i}\) represents the \(i\)-th candidate vector of the atmospheric light. Since the magnitude of the atmospheric light is estimated by averaging the magnitude of \(\widetilde{A}_{i}\), the proposed method can provide more accurate atmospheric light in a haze image. Figure 3 shows the dehazing results using the atmospheric light values estimated by three different haze removal methods, including the proposed method. The dehazing results were obtained by solving  
Figure 2: Region of interests for the candidate atmospheric light pixels: (**a**) input hazy image [11], reproduced with permission from [11], IEEE, 2014, (**b**) [PERSON]’s method [11], and (**c**) the proposed method. the degradation model of a hazy image in (3).