## 3 Proposed Method
### Normalized Pixel-Wise Dark Channel Prior  
In the DCP, the spatial resolution of the transmission map \(t(\mathbf{x})\) worsens along object edges because of calculating spatial minimisation in a dark-channel image. Therefore, [PERSON] et al. [5] refined the transmission map via image-matting processing [14]. However, image-matting processing requires a high computational cost and is not acceptable for real-time application. Therefore, they proposed a guided-image filter [7] as a fast image-matting technique. Other researchers also proposed a pixel-wise DCP [15; 16; 17] and a method combining original patch-wise DCP in a flat region and pixel-wise DCP around the edge region [18]. Although pixel-wise DCP can estimate the transmission map \(t(\mathbf{x})\) without selecting a minimum value spatially, the result tends to be darker than the haze image (Figure 2b). The histogram of medium transmission \(t(\mathbf{x})\) in Figure 3 shows that the pixel-wise DCP without normalisation shifts to the left side compared with the histogram of the original patch-wise DCP ([PERSON] et al. [5]). This is why the \(DC(\mathbf{J}(\mathbf{x})/\mathbf{A})\) of Equation (4) cannot be zero by setting the patch size to \(1\times 1\) instead of \(15\times 15\). Therefore, in the proposed method, the \(DC(\mathbf{J}(\mathbf{x})/\mathbf{A})\) of Equation (4) has a small value; the value of \((DC_{f}(\mathbf{x}))\) is defined by multiplying normalised dark channel of haze image \(\mathbf{I}\), which ranges from 0 to 1 and the ratio \(\gamma\) in Equation (8).  
\[DC_{f}\left(\frac{\mathbf{I}(\mathbf{x})}{\mathbf{A}}\right)=\min_{c\in\{r,g, b\}}\left(\frac{I^{c}(\mathbf{x})}{A^{c}}\right), \tag{7}\]  
\[DC_{f}(\mathbf{x})=\gamma\frac{\min_{c\in\{r,g,b\}}\left(\frac{I^{c}(\mathbf{x} )}{A^{c}}\right)-\min_{\mathbf{y}\in\Omega}\left(\min_{c\in\{r,g,b\}}\left( \frac{I^{c}(\mathbf{y})}{A^{c}}\right)\right)}{\max_{\mathbf{y}\in\Omega} \left(\min_{c\in\{r,g,b\}}\left(\frac{I^{c}(\mathbf{y})}{A^{c}}\right)\right)- \min_{\mathbf{y}\in\Omega}\left(\min_{c\in\{r,g,b\}}\left(\frac{I^{c}(\mathbf{ y})}{A^{c}}\right)\right)}, \tag{8}\]  
where \(DC_{p}\) is a pixel-wise dark channel operator, \(\Omega\) is the entire image. The transmission map \(t(\mathbf{x})\) of normalized pixel-wise DCP can be calculated by  
\[t(\mathbf{x})=\frac{1-\omega DC_{p}\left(\frac{\mathbf{I}(\mathbf{x})}{ \mathbf{A}}\right)}{1-DC_{f}(\mathbf{x})}, \tag{9}\]  
The histogram (Figure 3) of the transmission map \(t(\mathbf{x})\) derived by the proposed method shifts towards the right side compared with the histogram of transmission map without normalisation. As a result, the histogram of the proposed method gets close to the original patch-wise DCP. Here, if \(\gamma\) is set to be 0, Equation (9) corresponds to the pixel-wise DCP without normalisation (Figure 2b). Furthermore, setting \(\gamma\) to be a small value (e.g., 0.25) results in a dark image within the yellow dotted rectangle (Figure 2c), but if \(\gamma\) is set to be a large value (e.g., 0.75), the haze-removal effect diminishes within the dashed red rectangle (Figure 2e).  
Figure 2: Differences among haze-removal images with each normalisation parameter \(\gamma\).