Redundant sparse representations can significantly improve the MRI image reconstruction with sparsity constraint. An appropriate sparse model is very important to improve image quality even with the same sparsifying transforms and undersampled data. We propose a new fast, stable, compatible and simple iterative thresholding algorithm to solve the analysis sparse models that can obviously improve the image reconstruction for tight-frame-based sparsifying transform in compressed sensing MRI. We theoretically prove the convergence of the proposed projected fast iterative soft-thresholding algorithm (pFISTA). Numerical results show that pFISTA achieves better reconstruction than state-of-art FISTA for synthesis sparse model and more stable and compatible than the state-of-art SFISTA.