In order to remove the image blurring caused by reconstructing video frames independently frame by frame using traditional compressed video sensing method, this paper proposed a new approach to video compressed sensing based on motion estimation and motion compensation by combining the compressed sensing theory with related technology of MPEG standard video coding, so as to remove the spatial and temporal redundancy of video signal. This method took full account of the temporal correlations of video sequences and firstly compensated video frames using forward, backward and bidirectional prediction, then adopted the Backtracking-based Adaptive Orthogonal Matching Pursuit (BAOMP) algorithm to reconstruct the motion prediction residuals and finally reconstructed current frames. The experimental results indicate that the proposed method can gain a better video image quality compared with frame-by-frame reconstruction method and achieve a higher Peak Signal-to-Noise Ratio (PSNR).