## 3 Analysis of Image Transmission Channel and Quality Evaluation  
Like wireless signals, as illustrated in Figure 1, attenuation of optical signals also occurs in other transmission channels such as water, air, etc. The attenuation effects of the optical beam in the medium involves two completely different processes, i.e., absorption and scattering effects [17; 18; 19]. The absorption is an irreversible process in which the molecules in the medium absorb photons and convert them into thermal or chemical energy before they are radiated again. The scattering effect refers to the process in which photons are incident into the medium without energy conversion, and then are emitted again in a random direction. Two important parameters can reflect the absorption and scattering effects. The absorption coefficient \(a(\lambda)\) and scattering coefficient \(b(\lambda)\) indicate the absorptivity and scattering rate per unit length in the medium, respectively. The channel attenuation coefficient \(c(\lambda)\) of light can be expressed by the following formula:  
\[c(\lambda)=a_{\omega}(\lambda)+a_{\gamma}(\lambda)+a_{\varphi}(\lambda)+b_{ \omega}(\lambda)+b_{\varphi}(\lambda), \tag{1}\]where \(a_{\omega}(\lambda)\), \(a_{\gamma}(\lambda)\), and \(a_{\varphi}(\lambda)\) represent the absorption coefficient of main medium particles (e.g., water, air), metal pipe wall, and other suspended particles, respectively, \(b_{w}(\lambda)\) and \(b_{\varphi}(\lambda)\) denote the scattering coefficient of main medium and other suspended particles, and \(\lambda\) indicates the wavelength of visible light [20; 21; 22]. Here, in Figure 1, the image transmission channel based on VLRC in the pipe is composed of a transmitter, a receiver, and several relay nodes. For each transmitter of the nodes, the light transmission will be influenced by factors such as the optical design of the convex lens and the design of the transmitter circuit [23]. Also, for each receiver of the nodes, the sensitivity of the light detector, the selection of filter and optical design of the convex lens can affect the transmission of the light. After considering the above factors, the image transmission channel based on VLRC can be described as the following equation:  
\[P_{n}(r)=P_{n-1}(t)\psi_{n-1}(t)\psi_{n}(r)exp[-c(\lambda)D]\times\frac{\pi R ^{2}}{\pi\big{(}Dtan\frac{\theta}{2}\big{)}^{2}}, \tag{2}\]  
where \(P_{n}(r)\) indicates the \(n\)\(th\) received optical power, \(P_{n-1}(t)\) represents the ( \(n-1\)) \(th\) transmitted optical power, \(\psi_{n-1}(t)\) is the optical efficiency of ( \(n-1\)) \(th\) transmitter, \(\psi_{n}(r)\) indicates the optical efficiency of \(n\)\(th\) receiver, \(R\) denotes the convex lens radius of the receiver, and \(D\) and \(\theta\) represent the transmission distance and divergence angle of the light, respectively. In this research, to evaluate the VLRC-based image transmission performance within the pipe, an image quality evaluation framework is built based on the comparison between the final received images and original images. This framework focuses on evaluating image transmission performance through some typical image quality assessment algorithms such as mean square error (MSE), peak signal-to-noise ratio (PSNR), mean absolute error (MAE), and structural similarity index measure (SSIM), etc [24]. Here, we assume that the original two-dimensional gray image is \(I=F(i,j)\), the size of image is \(M\times N\) pixels, the received gray image is \(I^{\prime}=F^{\prime}(i,j)\), and then the MSE and MAE can be described as follows:  
\[MSE=\frac{1}{M\times N}\times\sum_{i=1}^{M}\sum_{j=1}^{N}[f(i,j)-f^{\prime}(i, j)]^{2}, \tag{3}\]  
\[MAE=\frac{1}{M\times N}\times\sum_{i=1}^{M}\sum_{j=1}^{N}\big{|}f(i,j)-f^{ \prime}(i,j)\big{|}, \tag{4}\]  
where \(1\leq i\leq M,\ 1\leq j\leq N\), \(f(i,j)\) and \(f^{\prime}(i,j)\) represent the pixel value of the original image and the final image at the pixel point of \((i,j)\), respectively, \(M\) and \(N\) denote the width and height, respectively.