# Multi-Module Combination for Underwater Image Enhancement
## 3 Proposed Method
### Visibility Restoration Module  
Due to the presence of underwater microorganisms and suspended particles, light is both scattered and absorbed as it propagates through the water. This phenomenon can result in images that are characterized by blurriness and darkness, ultimately leading to reduced visibility. This section addresses this issue by designing an underwater image visibility restoration module that employs an image defogging method based on the rank-one transmission prior [18]. This method assumes that the imaged scene is predominantly illuminated by spatially homogeneous light, except in the vicinity of the light source. The thickness of the scattered light is influenced by the depth of the scene and the degree of light transmission. Based on the theory of underwater optical imaging [23], the influence of the water body on light propagation is primarily manifested through three physical effects: the forward scattering effect caused by suspended particles, the backward scattering effect resulting from the water itself, and the direct attenuation effect that occurs during light transmission. The experimental data indicate that when the shooting distance is less than 2 m, the contribution of the forward scattering component to imaging quality decreases to less than 5%, rendering it negligible. \[I_{c}\approx D_{c}+B_{c} \tag{10}\]  
where \(I_{c}\) is the image taken directly under water, \(D_{c}\) is the direct attenuation component and \(B_{c}\) is the backscattering component, Equation (10) can be further subdivided as follows:  
\[I_{c}(x)=J_{c}(x)t_{c}(x)+A_{c}(1-t_{c}(x)) \tag{11}\]  
In Equation (11), \(x\) is the coordinate of any pixel in the image. c is the R, G, and B color channels of an image. \(J_{c}(x)\) represents the ideal clear image. \(A_{c}\in\mathbb{R}^{1\times 1\times 3}\) represents the ambient background light intensity in the R, G, and B channels, \(t_{c}\in\mathbb{R}^{m\times n\times 3}\) and \(t_{c}(x)\) represent the transmittance of each pixel. The value of this is related to the absorption and scattering of the light waves by seawater in the scene, and the expression is as follows:  
\[t_{c}(x)=e^{-\beta(\lambda)d(x)} \tag{12}\]  
\(\beta(\lambda)\) is the attenuation coefficient, including the effect of absorption of the light wave scattering effect. Since the attenuation coefficients of underwater light are different for different frequencies, they vary with the color channel. \(d(x)\) is the scene depth. The transmittance of the portion of ambient scattered light that affects imaging is described in Equation (11) by \(1-t_{c}(x)\), where \(\overset{\sim}{t}\) is used to denote the transmission map. Transmittance \(\overset{\sim}{t}\) is strongly correlated with the scattered light, and if the spectrum of the scattered light is given, then the transmittance can be expressed as their correlation coefficient. Assuming that L\({}_{1}\) denotes an image of an ideal outdoor scene, image I is the image to be processed of size m \(\times\)\(n\), and the transmittance map \(\overset{\sim}{t}(x)\in\mathbb{R}^{1\times 3}\) forms the rank-one prior matrix \(\widetilde{T}\in\mathbb{R}^{r\times 3}(r=m\times n)\). \(\widetilde{t}(x)+t_{c}(x)=(1,1,1)\). The relation between the transmittance map \(\widetilde{\mathfrak{t}}(x)\) and the spectrum is  
\[\widetilde{t}(x)=C(x)\mathbf{b} \tag{13}\]  
where \(C\in\mathbb{R}^{r}\) is the vector of coefficients, \(\mathbf{b}\in\mathbb{R}^{1\times 3}\) is the transport basis of \(\widetilde{t}\), and \(\widetilde{t}\in\mathbb{R}^{m\times n\times 3}\). The modulo expansion of \(\widetilde{\mathfrak{t}}\) is the matrix \(\widetilde{T}\in\mathbb{R}^{3\times r}\) with the expression:  
\[\widetilde{T}=(\mathbf{c}\mathbf{b})^{\top} \tag{14}\]  
where \(\mathbf{c}\) is the column main order vector of \(\mathcal{C}\). \(\widetilde{T}\) is the rank-one matrix and the first-order prior on the scattering map.