## 4 Results and Discussion
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 \(\gamma\) is selected. Here, in the case of uniform medium transmission \(t\), \(\min_{\mathbf{y}\in\Omega}(\min_{c\in\{r,g,b\}}\left(I^{c}(\mathbf{y})/A^{c} \right))\) is close to \(1-t\) because \(\  
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 [PERSON] et al. 's method [6] and pixel-wise DCP (\(\gamma=0\)) method in both uniform and nonuniform medium transmission cases. On the other hand, the proposed method outperformed [PERSON] et al. 's method [5] and [PERSON] et al. 's method [10] only in the uniform setting and there are no significant difference in the nonuniform setting. 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**) [PERSON] et al. [6], (**e**) [PERSON] et al. [5], (**f**) [PERSON] et al. [10], (**g**) Pixel-wise DCP without normalisation, (**h**)Proposed method (\(\gamma=0.5\)). 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 [PERSON] 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 [PERSON] et al. 's method [5] (Figure 9c) and our proposed method (Figure 9f). Furthermore, although [PERSON] 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, [PERSON] et al. 's method cannot work well in the underwater image. [PERSON] 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 \(\times\) 680 pixels.