*2.1. Estimation of Transmission Map*

Medium transmission *t*(**x**) [1,5] is expressed by

$$t(\mathbf{x}) = \mathbf{e}^{-\beta d(\mathbf{x})},\tag{2}$$

where *β* is the scattering coefficient of the atmosphere and *d*(**x**) is the depth at coordinate **x**. He et al. [5] used DCP, indicating that at least one RGB colour channel within most local patch has a low-intensity value

$$D\mathbb{C}\left(\mathbf{J}(\mathbf{x})\right) = \min\_{\mathbf{y}\in\Omega(\mathbf{x})} \left(\min\_{c\in\{r,\mathbf{g},b\}} \left(J^{c}(\mathbf{y})\right)\right),\tag{3}$$

where *Jc*(**y**) is a colour channel of haze-free image **J**(**y**) at coordinate **y** and *DC* is the dark-channel operator which extracts a mimimum RGB colour channel in a local patch Ω(**x**) centered at coordinate **x**. From Equations (1) and (3) can be rewritten as

$$DC\left(\frac{\mathbf{I}(\mathbf{x})}{\mathbf{A}}\right) = \tilde{t}(\mathbf{x})DC\left(\frac{\mathbf{J}(\mathbf{x})}{\mathbf{A}}\right) + (1 - \tilde{t}(\mathbf{x})) ,\tag{4}$$

where ˜*t*(**x**) is the coarse transmission map based on patch and the argument **I**(**x**)/**A** and **J**(**x**)/**A** are to be element-wise division. If Ω(**x**) is set to large patch size (e.g., 15 × 15), *DC*(**J**(**x**)/**A**) should tend to be zero. Finally, the transmission ˜*t*(**x**) can be estimated by Equation (5).

$$f(\mathbf{x}) = 1 - \omega D \mathbf{C} \left(\frac{\mathbf{I}(\mathbf{x})}{\mathbf{A}}\right). \tag{5}$$

where *ω* is the haze removal rate which is considered to the human perception for depth scene (0.95 in the He et al. [5]). Since ˜*t*(**x**) is calculated by each large patch to satisfy the DCP, ˜*t*(**x**) is not smooth in edge region and the spatial resolution is lost. To solve the problem, He et al. [5] refined the transmission map ˜*t*(**x**) using image-matting processing [14] as post-processing. However, such processing requires high computational cost and several tens of seconds to execute the haze-removal method.
