## 4 Results and Discussion  
In this section, we compare our method with [PERSON] et al. 's method [6], [PERSON] et al. 's method [5] and [PERSON] 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  
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 ‘\(\times\)’ 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**. transmissions. In the uniform medium transmission \(t\), it is set to 0.5 directly. On the other hand, in the nonuniform medium transmission \(t(\mathbf{x})\), 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 \(t(\mathbf{x})\) by applying Equation (2). In the quantitative evaluation, peak-signal-to-noise-ratio (PSNR) and structural similarity (SSIM) [24] are calculated  
\[MSE=\frac{1}{H\mathrm{W}K}\sum_{i=0}^{H-1}\sum_{j=0}^{W-1}\sum_{k =0}^{K-1}\left(G(i,j,k)-J(i,j,k)\right)^{2},\] \[PSNR=20\log_{10}\left(\frac{MAX}{\sqrt{MSE}}\right), \tag{10}\]  
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**) [PERSON] et al. [6], (**d**) [PERSON] et al. [5], (**e**) [PERSON] et al. [10], (**f**) Pixel-wise DCP without normalisation, (**g**) Proposed method (\(\gamma=0.9\)). \[SSIM=\frac{1}{HMK}\sum_{i=0}^{H-1}\sum_{j=0}^{W-1}\sum_{k=0}^{K-1}\frac{\left(2\mu_ {G}(i,j,k)\mu_{J}(i,j,k)+C_{1}\right)}{\left(\mu_{G}(i,j,k)^{2}+\mu_{J}(i,j,k)^{2 }+C_{1}\right)}\frac{\left(2\sigma_{GJ}(i,j,k)+C_{2}\right)}{\left(\sigma_{G}( i,j,k)^{2}+\sigma_{J}(i,j,k)^{2}+C_{2}\right)}, \tag{11}\]  
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; \(\mu_{G}\) and \(\mu_{J}\) are gaussian weighted averages of \(G\) and \(J\), respectively, within local patch; within local patch, \(\sigma_{G}\) and \(\sigma_{J}\) are standard deviations of \(G\) and \(J\), respectively, within local patch; \(\sigma_{GJ}\) is a covariance of \(G\) and \(J\) within local patch; \(C_{1}\) (set to \(0.01^{2}\)) and \(C_{2}\) (set to \(0.03^{2}\)) 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 [PERSON] 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. [PERSON] 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, [PERSON] 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.