*3.1. Normalized Pixel-Wise Dark Channel Prior*

In the DCP, the spatial resolution of the transmission map *t*(**x**) worsens along object edges because of calculating spatial minimisation in a dark-channel image. Therefore, He 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 [15–17] 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*(**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*(**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 (He et al. [5]). This is why the *DC*(**J**(**x**)/**A**) of Equation (4) cannot be zero by setting the patch size to 1 × 1 instead of 15 × 15. Therefore, in the proposed method, the *DC*(**J**(**x**)/**A**) of Equation (4) has a small value; the value of (*DCJ*(**x**)) is defined by multiplying normalised dark channel of haze image **I**, which ranges from 0 to 1 and the ratio *γ* in Equation (8).

$$DC\_p\left(\frac{\mathbf{I}(\mathbf{x})}{\mathbf{A}}\right) = \min\_{c \in \{r, \mathbf{g}, b\}} \left(\frac{I^c(\mathbf{x})}{A^c}\right),\tag{7}$$

$$DC\_J(\mathbf{x}) = \gamma \frac{\min\_{\substack{\mathbf{c} \in \{r, \mathbf{g}, b\} \\ \mathbf{y} \in \Omega}} \left( \frac{I^{\mathbf{c}}(\mathbf{x})}{A^{\mathbf{c}}} \right) - \min\_{\mathbf{y} \in \Omega} \left( \min\_{\substack{\mathbf{c} \in \{r, \mathbf{g}, b\} \\ \mathbf{c} \in \{r, \mathbf{g}, b\}}} \left( \frac{I^{\mathbf{c}}(\mathbf{y})}{A^{\mathbf{c}}} \right) \right)}{\max\_{\substack{\mathbf{c} \in \{r, \mathbf{g}, b\} \\ \mathbf{c} \in \{r, \mathbf{g}, b\}}} \left( \frac{I^{\mathbf{c}}(\mathbf{y})}{A^{\mathbf{c}}} \right) \right) - \min\_{\mathbf{y} \in \Omega} \left( \min\_{\mathbf{c} \in \{r, \mathbf{g}, b\}} \left( \frac{I^{\mathbf{c}}(\mathbf{y})}{A^{\mathbf{c}}} \right) \right)},\tag{8}$$

where *DCp* is a pixel-wise dark channel operator, Ω is the entire image. The transmission map *t*(**x**) of normalized pixel-wise DCP can be calculated by

$$t(\mathbf{x}) = \frac{1 - \omega D \mathbf{C}\_p \left(\frac{\mathbf{I}(\mathbf{x})}{\mathbf{A}}\right)}{1 - D \mathbf{C}\_l(\mathbf{x})},\tag{9}$$

The histogram (Figure 3) of the transmission map *t*(**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 *γ* is set to be 0, Equation (9) corresponds to the pixel-wise DCP without normalisation (Figure 2b). Furthermore, setting *γ* to be a small value (e.g., 0.25) results in a dark image within the yellow dotted rectangle (Figure 2c), but if *γ* 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 *γ*.

**Figure 3.** Histogram of medium transmission with each method.

#### *3.2. Acceleration by Down-Sampling*

In the haze-removal method, it is necessary to calculate the transmission map *t*(**x**) for each pixel. Therefore, the calculation time of haze removal depends on the image size, and thus larger image sizes have higher associated calculation costs. Fortunately, observation of the transmission map *t*(**x**) indicates that it is characterised by a relatively low frequency except edges between objects, particularly at different depths. Therefore, we reduced the computation time greatly by down-sampling the input image. It estimated the transmission map *t*(**x**) and atmospheric light **A** using down-sampled image, and then the haze-removal image **J** was estimated by the up-sampled transmission map *t*(**x**) using Equation (6). Figure 4 shows the haze-removal results with different down-sampling ratios. The down-sampling ratio set to 1/4 achieved visually acceptable results, but when it was set to 1/8 or 1/16, halo effects were generated along edges such as along the sides of trees and leaves within the dashed red rectangles (second row of Figure 4e,f). Also, significant aliasing occurred along edges of the bench within the yellow dotted rectangle (third row of Figure 4e,f), and uneven colour occurred in the enhancement results in second row of Figure 4e,f. We therefore set the down-sampling ratio to 1/4 in all further experiments. Also, we used box filtering in down-sampling and bicubic interpolation in up-sampling. Too, acceleration by the down-sampling approach helps with noise suppression by spatial smoothing.

**Figure 4.** Haze image (first row), different haze-removal images (second row), transmission maps (third row) and corresponding down-sampling ratios.
