## 2 Joint Source-Channel Coding Scheme Based on Residual Distributed Compressive Video Sensing  
According to the CS theory, accurate reconstruction of a signal can be obtained by sampling the signal in a small amount, when the signal is sparse or the signal can be sparse under certain conditions. Compressed sensing theory can be abstracted into the mathematical expression \(y~{}=~{}\Phi x\), where \(x\) is sparse signal of \(N\) dimensions, \(\Phi\) is measurement matrix with dimension \(M\times N\), and \(y\) is measured value. If the signal \(x\) has a basic base in a certain domain, e.g., \(x=\Psi\alpha\), where \(\alpha\) is a sparse signal and \(\Psi\) is a sparse matrix with dimension \(N\times N\), the mathematical expression becomes \(y=\Phi\Psi\alpha\). In reconstruction process, the value of \(\alpha\) is obtained by solving the underdetermined equations, and the signal \(x\) is obtained by \(\alpha\). The algorithm is usually solved by Orthogonal Matching Pursuit (OMP) [24], Total variation Augmented Lagrangian Alternating Direction Algorithm (TVAL3) [25], etc. Joint source-channel coding scheme based on residual distributed compressive video sensing is shown in Figure 1. A video sequence consists of several GOPs (group of pictures), where a GOP consists of a key frame and non-key frame; each frame can be seen as a sparse signal of \(N\) dimensions [20]. To ensure the reconstruction performance and reduce the amount of data, the key frame adopts a higher measurement rate for the compressed sensing measurement. Besides, an extremely sparse residual matrix is obtained by subtracting key frames from non-key frames, and a lower measurement rate for the compressed sensing measurement is adopted. Suppose that \(\Phi_{k}\) is measurement matrix of key frame with dimension \(M_{k}\times N\), \(\Phi_{cs}\) is measurement matrix of residual matrix with dimension \(M_{k}\times N\), the measurement rate of key frames is \(r_{k}=M_{b}/N\), and the measurement rate of residuals matrix is \(r_{cs}=M_{cs}/N\). Assuming that the number of key frames is \(n_{k}\), the number of residuals matrix is \(n_{cs}\), respectively. Then, the average measurement rate (AMR) is:  
\[r_{ave}=\frac{\left(r_{k}\times n_{k}+r_{cs}\times n_{cs}\right)}{\left(n_{k}+n _{cs}\right)} \tag{1}\]  
Each measured value is converted into a data frame for transmission in the underwater channel, and the data frame format is shown in Figure 2. The data frame consists of the frame header \"FF FF FF\", the sequence number of the measured value in the matrix, the positive and negative sign, integer and decimal part of the measured value, and CRC (Cyclic Redundancy Check) checksum value. In this way, a measured value is packed. Transmitter and receiver follow this predetermined data format for transmission and reception. Receiver decodes the received packets and restores the measured value. According to the CRC, the correct measured value is placed in the measurement matrix by the serial number, and the wrong measured value is discarded (assignment 0). The measurement matrix is generated by setting the same random number seed as the sender; it does not need to be transmitted. The measured value matrix and the corresponding measurement matrix can be used to reconstruct the image frame by compression sensing algorithm. The key frames are reconstructed directly. The non-key frames are obtained by  
Figure 1: Joint source-channel coding scheme based on residual distributed compressive video sensing. Figure 2: Data frame format. summing the reconstructed residual matrix and the key frames. Finally, the reconstructed video frame is restored to the video. Light propagation in water suffers from attenuation through both absorption and scattering, which may cause data loss or bit error. The measured values determined by the distributed compressive video sensing theory are of equal importance.