# Enhancement of Underwater Images with Retinex Transmission Map and Adaptive Color Correction
## 2 Related Works  
[PERSON]'s study [12] showed that, in the underwater medium, the total radiation reaching the observer consists of three components: Direct Component \(E_{D}\), Forward-scattering Component \(E_{FS}\), and Backscattering Component \(E_{BS}\). The direct component \(E_{D}\) is the portion of the light reflected from the target scene that undergoes attenuation and reaches the observer. At each pixel \(x\) of the image, the direct component can be expressed as:  
\[E_{D}(x)=\int^{c}(x)e^{-p_{\lambda}d(x)}=\int^{c}(x)t^{c}(x), \tag{1}\]  
where \(J^{c}(x)\) is the radiation of the object at the \(x\), \(c\in\{r,g,b\}\) refers to the color channel in the RGB color space, \(d(x)\) is the distance from the object to the observer, and \(p_{\lambda}\) is the attenuation coefficient. \(p_{\lambda}\) represents the absorption and scattering of light when it propagates in the underwater medium, and different wavebands of light have different attenuation coefficients. \(t^{c}(x)=e^{-p_{\lambda}d(x)}\) is called the transmission map of underwater medium. The backscattering component, also known as veiling light, is the prime reason for losing contrast and color cast in underwater images. The backscattering component can be expressed as:  
\[E_{BS}(x)=B^{c}(x)(1-e^{-p_{\lambda}d(x)}), \tag{2}\]  
where \(B^{c}(x)\) is called backscattered light or background light. Because the forward-scattering component has little effect on image distortion, it can be ignored. Therefore, in most underwater image restoration methods [13, 14, 15], the simplified underwater imaging model used is as follows:  
\[I^{c}(x)=J^{c}(x)I^{c}(x)+(1-I^{c}(x))B^{c}(x), \tag{3}\]  
where \(c\in\{r,g,b\}\), and \(I^{c}(x)\) is the corresponding color channel of the observed image \(I(x)\). Image restoration aims to recover the radiation \(J(x)\) of the scene from the observed image \(I(x)\). If \(B(x)\) and \(t(x)\) have been estimated and substituted into Equation (3), the recovery formula of underwater image can be obtained as follows:  
\[J^{c}(x)=\frac{I^{c}(x)-B^{c}(x)}{I^{c}(x)}+B^{c}(x). \tag{4}\]  
The recovery of underwater image in Equation (4) is an ill-posed problem. Thus, some sort of priors could be utilized to estimate the unknown transmission map and backscattered light. Many methods based on the dark channel prior or its variants have emerged in recent years. [PERSON] et al. [5] proposed an efficient and low complexity underwater image enhancement method based on Dark Channel Prior which employed the median filter instead of the soft matting procedure to estimate the transmission map of image. However, since the original DCP was designed for atmospheric image dehazing, the transmission map obtained by DCP was not accurate when there were white objects or artificial light in the underwater scene. The UDCP [6] used the green and blue channel to estimate the transmission map and obtained a more accurate underwater transmission map compared with original DCP. However, such biased estimations of the transmission map lead to the darkening of the overall image in the final restored image. [PERSON] et al. [16] proposed a Generalized Dark Channel Prior (GDCP) for image restoration, which first estimated ambient light using the depth-dependent color change and then estimated the scene transmission. [PERSON] et al. [17] proposed a transmission map optimization method based on statistical model-based background light estimation and the underwater light attenuation prior, which could better recover the details of underwater images. However, this method could not obtain accurate background light value for some close-range images, resulting in an unsatisfactory visual effect. [PERSON] et al. [18] proposed a novel underwater image restoration method based on an adaptive attenuation-curve prior that relied on the statistical distribution of pixel values. Since the scene depth map is very important for the estimation of the transmission map and background light intensity, [PERSON] et al.