*3.3. Robust Atmospheric Light Estimation*

He et al. [5] estimated atmospheric light **A** using the original patch-wise DCP, which is a robust method because it ignores small white objects by using a large patch size (e.g., 15 × 15). In addition, Liu et al.'s method [19] segments the sky region and uses the average value of that region

as atmospheric light **A**. In our proposed method, because the dark-channel image is calculated for each pixel, white regions (represented by the blue '+' mark in Figure 5) are misinterpreted as atmospheric light **A**. On the basis of Figure 5, we found that our proposed method cannot use the He et al. [5] approach directly. In addition, their method requires extra computation time for sorting the top 0.1% brightest pixels in the dark channel of haze image **I**. To solve this problem, we propose a method that robustly estimates atmospheric light **A** by using a coarse-to-fine search strategy. Figure 6 shows the flow of the coarse-to-fine search strategy. In this strategy, initially the resolution of the dark-channel image is reduced step by step and the position of the largest dark-channel value is obtained at the lowest resolution; next it recalculates the position of the largest dark-channel value in the second-lowest resolution and continues to recalculate the position of the largest dark-channel value until the original image size is attained. In Figure 5, the red '×' mark (coarse-to-fine search strategy) is the correctly estimated atmospheric light **A**.

**Figure 5.** Effectiveness of coarse-to-fine search strategy. Blue '+' mark is result of atmospheric-light estimation by pixel-wise dark-channel image without using coarse-to-fine strategy. Red '×' is result of pixel-wise dark-channel image using coarse-to-fine strategy.

**Figure 6.** Flow of coarse-to-fine search strategy for estimting atmospheric light **A**.

### **4. Results and Discussion**

In this section, we compare our method with Tarel et al.'s method [6], He et al.'s method [5] and Cai et al.'s method [10] for qualitative visual evaluation and quantitative evaluation. Ref. [10] is used trained network provided by [20]. We used haze and haze-free images downloaded from the Flickr website [21] (all collected images are public domain or creative commons zero license) and MATLAB source codes [20,22,23].

Initially, we generated five uniform and nonuniform haze images (Figures 7b and 8c) from the haze-free image (Figure 7a) by applying Equation (1). In order to do these simulations, we had to set the transmission map, for which we experimented with setting to uniform and nonuniform medium transmissions. In the uniform medium transmission *t*, it is set to 0.5 directly. On the other hand, in the nonuniform medium transmission *t*(**x**), we set depth by manually segmenting four to five classes for the each image (Figure 8b) and then fixing the depth for each class; we determined the medium transmission *t*(**x**) by applying Equation (2).

**Figure 7.** Comparison of proposed method with conventional method using simulated haze images generated with uniform transmission map (*t* = 0.5). (**a**) Original haze-free image, (**b**) Simulated haze image, (**c**) Tarel et al. [6], (**d**) He et al. [5], (**e**) Cai et al. [10], (**f**) Pixel-wise DCP without normalisation, (**g**) Proposed method (*γ* = 0.9).

In the quantitative evaluation, peak-signal-to-noise-ratio (PSNR) and structural similarity (SSIM) [24] are calculated

$$MSE = \frac{1}{HWN} \sum\_{i=0}^{H-1} \sum\_{j=0}^{W-1} \sum\_{k=0}^{K-1} \left( G(i, j, k) - J(i, j, k) \right)^2,$$

$$PSNR = 20 \log\_{10} \left( \frac{MAX}{\sqrt{MSE}} \right), \tag{10}$$

$$SSIM = \frac{1}{HWN} \sum\_{i=0}^{H-1} \sum\_{j=0}^{W-1} \sum\_{k=0}^{K-1} \frac{\left(2\mu\_{\mathbb{G}}(i,j,k)\mu\_{\mathbb{J}}(i,j,k) + \mathbb{C}\_1\right)}{\left(\mu\_{\mathbb{G}}(i,j,k)^2 + \mu\_{\mathbb{J}}(i,j,k)^2 + \mathbb{C}\_1\right)} \frac{\left(2\sigma\_{\mathbb{G}\mathbb{J}}(i,j,k) + \mathbb{C}\_2\right)}{\left(\sigma\_{\mathbb{G}}(i,j,k)^2 + \sigma\_{\mathbb{J}}(i,j,k)^2 + \mathbb{C}\_2\right)'},\tag{11}$$

where *H*, *W* and *K* are image size as height and width and number of colour channel respectively; *G* and *J* are the ground-truth image and haze-removal result, respectively; *MAX* is maximum possible value of ground-truth image; *μ<sup>G</sup>* and *μ<sup>J</sup>* are gaussian weighted averages of *G* and *J*, respectively, within local patch; within local patch, *σ<sup>G</sup>* and *σ<sup>J</sup>* are standard deviations of *G* and *J*, respectively, within local patch; *σG J* is a covariance of *G* and *J* within local patch; *C*<sup>1</sup> (set to 0.012) and *C*<sup>2</sup> (set to 0.032) are small constants. Secondly, we compared with proposed method and conventional method to actual haze image as qualitative visual evaluation. Finally, we show the comparison of computation time by each method and image size.

In common of qualitative evaluation in results with setting the uniform or the nonuniform medium transmission, the results of Figures 7 and 8 show that both our proposed method (Figures 7g and 8h) and He et al.'s method [5] (Figures 7d and 8e) can obtain highly accurate haze-removal images that are indistinguishable from the original haze-free image. Cai et al.'s method [10] can also obtain highly accurate haze-removal images in outdoor scene such as cityscape and landscape images (Figures 7e and 8f). However, Cai et al.'s method [10] cannot remove the haze in underwater scene. The reason is that underwater images are not included in the training data. The results of pixel-wise DCP without normalisation (Figures 7f and 8g) are darker than their original haze-free images (Figure 7a).

In the quantitative evaluation with uniform setting in Table 1, it is apparent that our proposed method can obtain the highest PSNR and SSIM values compared with conventional methods if the appropriate value for *γ* is selected. Here, in the case of uniform medium transmission *t*, min**y**∈Ω(min*c*∈{*r*,*g*,*b*}(*Ic*(**y**)/*Ac*)) is close to 1 <sup>−</sup> *<sup>t</sup>* because min**y**∈Ω(min*c*∈{*r*,*g*,*b*}(*Jc*(**y**)/*Ac*)) is close to 0, and max**y**∈Ω(min*c*∈{*r*,*g*,*b*}(*Ic*(**y**)/*Ac*)) is close to 1 because max**y**∈Ω(min*c*∈{*r*,*g*,*b*}(*Jc*(**y**)/*Ac*)) is close to 1 in Equation (8). As the result, the appropriate value of *γ* is close to 1 when the *ω* equals to 1. From Table 1, the proposed method can obtain the best results when *γ* is set to a large value. On the other hand, in the quantitative evaluation with nonuniform setting shown in Table 2, some results from He et al.'s method [5] achieved better performance than the proposed method. The main reason is that it is not easy to estimate an appropriate *γ* in the case of nonuniform medium transmission *t*(**x**) because it depends on the haze scene. How to automatically determine an appropriate value from the distribution of haze in the scene is our future work.


**Table 1.** Quantitative evaluation with PSNR and SSIM [24] for simulated haze images generated using uniform transmission map (*t* = 0.5). First row is PSNR value, and second row is SSIM value in each cell.

**Figure 8.** Comparison of proposed method with conventional method using simulated haze image generated with nonuniform transmission map. (**a**) Original haze-free image, (**b**) Manual segmented image, (**c**) Simulated haze image, (**d**) Tarel et al. [6], (**e**) He et al. [5], (**f**) Cai et al. [10], (**g**) Pixel-wise DCP without normalisation, (**h**)Proposed method (*γ* = 0.5).

We used the paired t-test to verify whether any performance differences between the proposed method and state-of-the-art methods are statistically significant. The test results are summarized in Table 3. The statistically significant methods (p < 0.05) are indicated by "Yes" and others are indicated by "No". As shown in Table 3, the proposed method outperformed Tarel et al.'s method [6] and pixel-wise DCP (*γ* = 0) method in both uniform and nonuniform medium transmission cases. On the other hand, the proposed method outperformed He et al.'s method [5] and Cai et al.'s method [10] only in the uniform setting and there are no significant difference in the nonuniform setting.


**Table 2.** Quantitative evaluation with PSNR and SSIM [24] for simulated haze images generated using nonuniform transmission map. First row is PSNR value, and second row is SSIM value in each cell.

**Table 3.** Paired *t*-test results between proposed method and conventional methods. The statistically significant methods (p < 0.05) are indicated by "Yes" and others are indicated by "No".


Figure 9 shows that our haze-removal method produced good results for processing actual haze images. Closer qualitative evaluation confirms that the images processed by our proposed method (Figure 9f) are visually similar to those obtained by He et al.'s method [5] (Figure 9c). We can see that the results of pixel-wise DCP without normalisation (Figure 9e) are also unnaturally darker than those obtained by He et al.'s method [5] (Figure 9c) and our proposed method (Figure 9f). Furthermore, although Tarel et al.'s method [6] obtained clearer haze-removal results in the pumpkin, bridge and townscape images compared with our proposed method results, our evaluation confirmed that the colours of the park, bridge and townscape images changed from those of the original haze images. We also noted the occurrence of halo effects in the train image. In addition, Tarel et al.'s method cannot work well in the underwater image. Cai et al.'s method [10] (Figure 9d) can remove haze more naturally than other methods. In particular, it can remove haze uniformly in the sky region and the colour is more natural. On the other hand, it cannot work well in the underwater image.

Figure 10 shows computation time for each image size for each method, assuming a i7-5557U (3.1 GHz, 2 cores, 4 threads) without GPU acceleration and main memory size is 16 GB. All methods are implemented in MATLAB. Using conventional methods, it takes several tens of seconds, and they cannot achieve real-time calculation. However, our proposed method can achieve real-time calculation until image size exceeds 1024 × 680 pixels.

**Figure 9.** Comparison of haze-removal results by our proposed method with conventional methods applied to actual haze images. (**a**) Haze image, (**b**) Tarel et al. [6], (**c**) He et al. [5], (**d**) Cai et al. [10], (**e**) Pixel-wise DCP without normalisation, (**f**) Proposed method (*γ* = 0.5).

**Figure 10.** Comparison of computation time for each image size and each method.
