## 4 Method
### Transmission Estimation Based on Saturation Line Prior  
The inherent limitations of the DCP have prompted further research into more refined and resilient priors [37; 38; 39]. However, these priors often encounter failure when applied to sandstorm images. This issue can be primarily due to two factors: first, color correction disrupts the pixel consistency in sandstorm images, causing some properties to shift; second, the intrinsic variation between sandstorms and haze conditions limits the applicability of dehazing priors. This subsection aims to offer a feasible approach. The advent of the color attenuation prior (CAP) [36] and saturation-based transmission estimation [40] has validated the role of the saturation component in transmission estimation. However, these methods only consider each pixel's saturation, without accounting for more complex intrinsic relationships among pixels. Recently, the saturation line prior [32] was introduced to address this gap. The saturation line prior describes an approximately linear relationship between the saturation components and the reciprocal of the brightness components of haze images in local blocks. This paper attempts to apply this relationship to sandstorm images. Firstly, by dividing both sides of the atmospheric scattering model equation by the atmospheric light, we obtain a normalized form, as shown in Equation (9):  
\[\frac{I_{cc}^{c}(x)}{A_{adj}^{c}}=\frac{J^{c}(x)}{A_{adj}^{c}}\cdot t(x)+(1-t( x)), \tag{9}\]  
where \(J^{c}(x)\) refers to the color channel \(c\) of the clear image, and \(t(x)\) indicates the transmission map. Supposing that \(I_{N}(x)\) and \(J_{N}(x)\) represent the normalized sand-dust image and the clear image, the above equation can be reformulated as  
Figure 3: Comparison of recovery results from different atmospheric light calculation methods. \[I_{N}(x)=J_{N}(x)\cdot t(x)+(1-t(x)). \tag{10}\]  
Under the premise of a locally monochromatic surface, the saturation component is locally invariant. According to the definition of saturation, the saturation component of \(I_{N}(x)\) can be formulated as  
\[S_{I_{N}}(x)=1-\frac{\underset{ \begin{subarray}{c}\in\{t_{x},\text{g,b}\}\end{subarray} } {\min}J_{N}^{c}(x)\cdot t(x)+(1-t(x))}{\underset{ \begin{subarray}{c}\in\{t_{x}, \text{g,b}\}\end{subarray} }{\max}J_{N}^{c}(x)\cdot t(x)+(1-t(x))}. \tag{11}\]  
Let \(J_{N}^{min}(x)\) and \(J_{N}^{max}(x)\) denote the minimum and maximum color components of \(J_{N}(x)\), respectively. The above equation can be rewritten as  
\[S_{I_{N}}(x)=1-\frac{J_{N}^{min}(x)\cdot t(x)+(1-t(x))}{J_{N}^{max}(x)\cdot t(x )+(1-t(x))}. \tag{12}\]  
Equation (12) can be rewritten as  
\[S_{I_{N}}(x)=1-\frac{J_{N}^{min}(x)\cdot t(x)+(1-S_{J_{N}}(x)\cdot(1-t(x))}{J _{N}^{max}(x)\cdot t(x)+(1-t(x))}-\frac{S_{I_{N}}(x)\cdot(1-t(x))}{J_{N}^{max} (x)\cdot t(x)+(1-t(x))}. \tag{13}\]  
Under the assumption that the RGB channels of the color-corrected sandstorm image share identical transmission, we can derive Equation (14) from Equation (10):  
\[I_{N}^{max}(x)=J_{N}^{max}(x)\cdot t(x)+(1-t(x)). \tag{14}\]  
Given \(1-S_{I_{N}}(x)=\frac{J_{N}^{min}(x)}{J_{N}^{max}(x)}\), substituting (14) into (13) results in  
\[S_{I_{N}}(x)=1-\frac{J_{N}^{min}(x)}{J_{N}^{max}(x)}-\frac{S_{I_{N}}(x)\cdot( 1-t(x))}{I_{N}^{max}(x)}. \tag{15}\]  
By further simplifying the above equation, we obtain  
\[S_{I_{N}}(x)=S_{I_{N}}(x)-S_{I_{N}}(x)\cdot\frac{1-t(x)}{I_{N}^{max}(x)}. \tag{16}\]  
In the HSV color space, suppose that the brightness component is equal to the maximum color channel, expressed as \(V_{I_{N}}(x)=I_{N}^{max}(x)\), with the local saturation and transmission considered as constants. Therefore, we can rewrite Equation (16) as Equation (17):  
\[S_{I_{N}}(x)=k\cdot V_{I_{N}}^{-1}(x)+b, \tag{17}\]  
where \(V_{I_{N}}^{-1}(x)=1/V_{I_{N}}(x)\), \(k=S_{I_{N}}(x)\cdot(t(x)-1)\) denotes the slope, and \(b=S_{I_{N}}(x)\) represents the intercept.