Compressed sensing is gradually applied to optical imaging systems because it only needs a very small number of observations to reconstruct the original signal. Aiming to address the problem that the existing reconstruction algorithms have low reconstruction accuracy in video imaging systems with limited sampling resources, a phased reconstruction algorithm is proposed. In the first stage, the algorithm uses the reconstructed keyframes as reference frames to reconstruct the non-key frames through motion compensation and residual reconstruction. After this, it groups the reconstructed non-key frames, enhances them further, and outputs them. In the second stage, the algorithm dynamically selects multi-hypothesis matching blocks from the current frame and the keyframes before and after for the non-key frames reconstructed in the first stage. It then establishes a residual sparse model and completes the reconstruction to output the video reconstruction results. The experimental results show that the average PSNR value of this algorithm is above 41.5 dB, and the average SSIM value is above 0.97. These values represent significant improvements compared to several existing excellent reconstruction algorithms. Compared with the most representative multi-hypothesis algorithm, the average PSNR value and average SSIM value of this algorithm are improved by about 9.1% and 17.3% respectively, leading to better video reconstruction quality.