## 2 Related Works
### Existing Model for Scattering Medium with Spectral Analysis and Integral Imaging  
A scattering model is derived from the relationship between a captured image, a scattering medium, and the original image without the scattering medium. It can be  
Figure 1: Computational reconstruction method in integral imaging. employed to reduce the scattering artifacts in the captured image. The standard model used in dehazing methods is derived from the Bouguer-Lambert-Beer law [39; 40], as follows:  
\[g(x)=t(x)f(x)+(1-t(x))A. \tag{1}\]  
Here, \(g(x)\) represents the image captured by a camera, \(f(x)\) is the original image to be reconstructed, and the intensity of the ambient light is denoted by \(A\). The medium transmission, denoted as \(t(x)\), is the weight function affecting the original image. It can be seen that determining the values of \(A\) and \(t(x)\) is important in order to reconstruct \(f(x)\) from \(g(x)\). However, this model makes it somewhat complicated to separate \(t(x)f(x)\), requiring a complicated dehazing algorithm [41]. Thus, a simpler model is introduced [38], which is written as:  
\[g(x)=\alpha f(x)+(1-\alpha)t(x). \tag{2}\]  
In Equation (2), \(g(x)\) is the image signal affected by the scattering medium, \(f(x)\) is the original signal unaffected by the scattering medium, and \(t(x)\) is the scattering medium. Here, \(\alpha\) is the visibility factor. This model can be easily analyzed in the spectral domain since the model consists of a linear combination of \(f(x)\) and \(t(x)\). Thus, applying the linearity of Fourier transformation to Equation (2), we can easily obtain Equation (3) as follows. \[G(\omega)=\alpha F(\omega)+(1-\alpha)T(\omega), \tag{3}\]  
where \(G(\omega)\), \(F(\omega)\), and \(T(\omega)\) represent the Fourier-transformed signals of \(g(x),f(x)\), and \(t(x)\), respectively. By estimating \(T(\omega)\) and \(\alpha\), we can obtain the reconstructed \(F(\omega)\) from \(G(\omega)\). We utilize the frequency characteristics of the scattering medium to predict \(T(\omega)\). The results presented in Figure 2e demonstrate that the influence of the scattering medium dominates the very-low-frequency components of the image. Therefore, by utilizing the frequency characteristics of the scattering medium, we can determine the low-frequency components of the observed signal \(G(\omega)\) as the components of \(T(\omega)\) and the high-frequency components as the components of \(F(\omega)\). These are then represented by  
\[T(\omega)\simeq\frac{1}{(1-\alpha)}G(\omega),\ \ \omega\leq f_{c}, \tag{4}\]  
\[F(\omega)\simeq\frac{1}{\alpha}G(\omega),\ \ \omega>f_{c}. \tag{5}\]  
The spectrum-based method [38] is illustrated in Figure 3. To estimate the effect of the scattering medium, \(f_{c}\) is determined to separate lower-band and higher-band signals. The cut-off frequency \(f_{c}\) is used for a low-pass filter to remove the effect of the scattering medium. The remaining high-frequency signal is then utilized to restore the high-frequency components of the original signal using the visibility coefficient \(\alpha\). The cut-off frequency \(f_{c}\) is determined through Equation (6). \[f_{c}=\int_{0}^{1}\lvert G_{N}\left(\omega\right)\rvert d\omega\rvert, \tag{6}\]  
where \(G_{N}(\omega)\) is the normalized value of \(G(\omega)\). The visibility coefficient \(\alpha\) is determined by applying the Beer-Lambert-Bouguer law. According to this law, the amount of transmitted light \(I(z)=I_{0}\cdot\exp(-kz)\) is defined, where \(I_{0}\) is the amount of light from the original source and \(z\) is the distance from the sensor. Here, \(k\) denotes the total attenuation coefficient depending on the wavelength of the illumination and the medium. This coefficient primarily determines the amount of attenuation, absorption, and scattering of light within the scattered environment. Since visibility decreases due to the attenuation of the original signal, the total attenuation coefficient and the visibility coefficient \(\alpha\) are related.