Abstract: Synthetic Aperture Radar Ground Moving Target Imaging (SAR-GMTIm) is capable of detecting moving target responses in the stationary clutter and forming focused images of the target. There is a widely accepted fact that the moving target response is inherently sparse with respect to the rich clutter. To achieve the recovery of the sparse response of the moving target, algorithms of the first-order are popular, where ISTA and FISTA are typical candidates that can be applied to image for sparse SAR moving target. However, there are some problems such as low computation efficiency and slow convergence. To address the problems, a Greedy Fast Iterative Shrinkage-thresholding Algorithm (Greedy FISTA) is proposed for the representation of the sparse features of SAR-GMTIm. This algorithm improves the FISTA based on the restart framework, and shortens the restart interval and oscillation period of the algorithm, so that a faster convergence rate can be ensured compared with the conventional FISTA. In this paper, both simulated complex SAR data and raw Gotcha data from US Air Force Laboratory are applied to examine the performance of the proposed Greedy FISTA algorithm. Comparisons with the conventional ISTA and FISTA in terms of the required number iteration are performed, and the performance of the sparse recovery are examined by using the phase transition diagram (PTD) analysis. Experimental results show that Greedy FISTA has good imaging performance when applied to the SAR-GMTIm data, and has obvious advantages over other threshold iterative algorithms in terms of convergence speed and sparse recovery accuracy.