# Haze Removal Based on Refined Transmission Map for Aerial Image Matching
## 2 Physical-Based Image Dehazing and Matching
### Scale Invariant Feature Transform with Embedded Color  
The SIFT [49] operator is broadly used in image matching due to its robustness to the rotation of the image, scaling, and variations in luminance. This algorithm consists of the following five steps: scale-space extrema detection, key-point localization, orientation assignment, key-point descriptor, and key-point matching. Moreover, SIFT with embedded color (C-SIFT) [50] was used in this study. The C-SIFT algorithm is robust because the local features include color information for image dehazing. In general, the camera response, as shown in (17), is given by [51]:  
\[\rho_{i}{}^{xy}=\int_{w}\{L_{p}(\lambda)+E(\lambda)R(\lambda)\}S_{i}(\lambda)d \lambda,\ \ (i=R,G,B) \tag{21}\]  
where \(\rho_{i}\) is the response of the camera, \(\lambda\) is the wave length of the color, \(L_{p}(\lambda)\) is the path radiance, \(E(\lambda)\) is the illumination, \(R(\lambda)\) is the surface reflectance, and \(S(\lambda)\) is the sensitivity of the camera. In addition, the \(xy\)-coordinates are allocated in the array to every point. \[\rho_{i}{}^{xy}=\int_{w}L_{p}(\lambda)E(\lambda)R(\lambda)S_{i}(\lambda)d \lambda+\ \zeta_{i}{}^{xy},\ \ (i=R,G,B) \tag{22}\]  
here,  
\[\zeta_{i}{}^{xy}=\int_{w}L_{p}(\lambda)S_{i}(\lambda)d\lambda,(i=R,G,B)\]  
Although haze cannot be entirely removed by dehazing processing, it is assumed that \(\zeta_{i}{}^{xy}\) is not a dominant component. Here, if it is assumed that the camera sensitivities are delta functions, Equation (21) can be revised for any wavelength \(k\). Note that realistic camera sensitivities are not delta functions, which is an issue that is discussed later. \[ \begin{array}{c}\rho_{k}{}^{xy}=\int_{w}E(\lambda)R(\lambda)\delta(\lambda- \lambda_{k})d\lambda\\ =\ E(\lambda_{k})R(\lambda_{k})\end{array}  \tag{23}\]  
The color response at the Planckian locus of an illuminant for any pixel is defined as follows:  
\[\rho_{k}=c_{1}\lambda_{k}{}^{-5}exp\bigg{(}\frac{-c_{2}}{T\lambda_{k}}\bigg{)} R(\lambda_{k}) \tag{24}\]  
where  
\[E(\lambda,T)=c_{1}\lambda^{-5}\bigg{[}exp\bigg{(}\frac{-c_{2}}{T\lambda}\bigg{)} -1\bigg{]}^{-1} \tag{25}\]  
Here, \(c_{1}=3.74183\times 10^{-16}\ \mathrm{Wm}^{2}\) and \(c_{2}=1.4388\times 10^{-2}\ \mathrm{mK}\). The natural logarithm is used to separate the two components, namely, the surface and illuminant, using Equation (24) as follows:  
\[ln\rho_{k}=ln\Big{[}c_{1}\lambda_{k}{}^{-5}R(\lambda_{k})\Big{]}-\frac{c_{2} }{T\lambda_{k}} \tag{26}\]  
Following the determination of the color responses for each RGB channel, the ratio for each channel is calculated using the same process as follows:  
\[\begin{array}{c}ln\frac{R}{G}=ln\rho_{R}-ln\rho_{G}=R_{R}-R_{G}+\frac{1}{T}( E_{R}-E_{G})\\  
Equation (27) shows that the relationship between the coordinates of the log-chromaticity and the altered level in the illuminant is linear, whereas \(R_{k}\) is not dependent on the illuminant color. In addition, the relationship between the log-chromaticity and change in the surface reflectance is also linear, as shown in Equation (29).  
\[\left(ln\frac{R}{G}-\frac{R_{R}-R_{G}}{R_{B}-R_{G}}ln\frac{B}{G}\right)T=E_{R} -E_{G}-\frac{R_{R}-R_{G}}{R_{B}-R_{G}}(E_{B}-E_{G}) \tag{29}\]  
When using a real camera, because it is impossible to implement the abovementioned sensitivities using a delta function, the proposed method uses the channel filter function of Worldview-2 to consider the characteristics of aerial images, rather than the mathematical approach presented in Equations (22)-(29). Next, the transformation of a real camera response into a new response is explained. A linear regression model [52] is used. Finally, 1485 Munsell patches for which the color-invariant values are normalized at the minimum and maximum values are used as the input values of the C-SIFT as shown in Algorithm 2. ```
Input: Dehazed Image Output: Valid key-point matches
1: Detect the scale-space extrema. 2: Localize the key points. 3: Perform orientation assignment on each key point to achieve image rotation.