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Terahertz video synthetic aperture radar (THz-ViSAR) has tremendous research and application value due to its high resolution and high frame rate imaging benefits. However, it requires more efficient imaging algorithms. Thus, a novel multistage back projection fast imaging algorithm for the THz-ViSAR system is proposed in this paper to enable continuous playback of images like video. The radar echo data of the entire aperture is first divided into multiple sub-apertures, as with the fast-factorized back projection algorithm (FFBP). However, there are two improvements in sub-aperture imaging. On the one hand, the back projection algorithm (BPA) is replaced by the polar format algorithm (PFA) to improve the sub-aperture imaging efficiency. The imaging process, on the other hand, uses the global Cartesian coordinate system rather than the local polar coordinate system, and the wavenumber domain data of the full aperture are obtained step by step through simple splicing and fusion, avoiding the amount of two-dimensional (2D) interpolation operations required for local polar coordinate system transformation in FFBP. Finally, 2D interpolation for full-resolution images is carried out to image the ground object targets in the same coordinate system due to the geometric distortion caused by linear phase error (LPE) and the mismatch of coordinate systems in different imaging frames. The simulation experiments of point targets and surface targets both verify the effectiveness and superiority of the proposed algorithm. Under the same conditions, the running time of the proposed algorithm is only about 6% of FFBP, while the imaging quality is guaranteed.