Compressed sensing (CS) breaks Shannon/Nyquist sampling theorem bottleneck. It captures and represents signals at a sampling rate significantly below the Nyquist rate, and then original signals can be accurately or high precisely recovered by solving sparse optimization problems based on signal sparsity or compressibility. CS has a good application prospect in remote sensing imagery, especially in Infrared Remote Sensing Video. The CS-based remote sensing includes two stages: onboard encoding imaging and offline decoding recovery. Video offline decoding recovery is one of the core questions in CS-Based Infrared Remote Sensing Video systems. In this paper, firstly, we introduce a coupled optimization models which is composed of a single model and an error model for video offline decoding recovery. This paper shows that the coupled models can easily improve speed and accuracy to recover video of Infrared Remote Sensing. Secondly, we review the Bregman method and linearized Bregman method. Furthermore, a linearized Bregman error iteration algorithm is proposed for solving the coupled models, thus lead to better convergence rates and error performances. In numerical experiments, we compare the convergence rates of the original Bregman method and the linearized Bregman method from the first frame picture, and test the performance of the linearized Bregman method for video recovery with the single model and coupled models. Numerical experiments demonstrate the effectiveness of the proposed algorithm. The comparison with the single model is included.