## 1 Introduction  
Underwater optical wireless communication (UOWC) plays an important role in military, environmental detection, marine exploration and disaster prevention, and has attracted more and more attention in recent years [1; 2; 3]. The communication distance is limited due to serious absorption and scattering of water [4]. To achieve long-distance underwater communication, a single photon detector with single photon limit sensitivity is used to detect very weak light signal at the receiver [5; 6; 7; 8; 9; 10; 11]. The communication signals are extracted from the discrete single photon pulses that output from single photon detector. Video communication is an important application aspect of underwater optical wireless communication because of its vivid and picturesque characteristic. Due to fluctuation of photon flux and quantum efficiency of photon detection, long-distance photon counting underwater optical wireless communication has the characteristics that the link is easily interrupted, the bit error rate is high, and the burst error is large. It is necessary to improve the transmission reliability through error correction coding. Traditional video compression standards, such as MPEG or H.26X, use a hybrid coding method combining predictive coding and transform coding to compress video sequences, which make video frames depend heavily on each other [12; 13]. Thus, if a data frame were lost, other data frames would not be decoded. In a photon-counting underwater optical wireless communication system, the signal in a time slot is a discrete single-photon pulse sequence, and even there no photon is detected in many time slots, causing a lot of symbol deletion and packet loss. Therefore, the traditional video compression standards and coding have great difficulties to restore video [14; 15]. In 2006, [PERSON] et al. proposed a compressed sensing (CS) theory, which broke through the limitation of traditional Nyquist sampling theorem [16; 17]. According to the CS theory, signal can be sampled and compressed at the same time; as a result, it is widely used in the field of video compression [18; 19; 20; 21; 22; 23]. In 2008, [PERSON] et al. [18] firstly applied compressed sensing theory to video coding, and proposed compressive video sampling (CVS). On the other hand, this paper proposes to divide video frames into key frames and non-key frames, where key frames use traditional video coding, while non-key frames use compressed sensing coding. The experimental results show that, on conditions of a better reconstruction, the method saves nearly half of the video collection. In 2009, [PERSON] et al. proposed distributed compressive video sensing (DCVS) based on inter-frame correlation [19; 20]. The encoding end allocates different measurement rates to key frames and non-key frames to reduce the amount of data transmitted, and each frame is compressed by CS measurement. The decoding end uses joint decoding to reconstruct, while the amount of data at the transmitting end is reduced and the complexity is transmitted to the decoding end. [PERSON] et al. [21] investigated dynamic measurement rate allocation in block-based DCVS, which can adaptively adjust measurement rates by estimating the sparsity of each block via feedback information. [PERSON] et al. [22] proposed residual distributed compressive video sensing based on double side information (RDCVS-DSI). In RDCVS-DSI, by taking advantage of the frequency domain characteristics of the image and the correlation between consecutive frames, the low-quality video frames are regarded as the first side information in the encoding process, and the second side information is generated from the motion estimation of non-key frames. Performance analysis and simulation results show that the RDCVS-DSI model can reconstruct the high-fidelity video sequence with lower complexity.