Aiming at the problem of poor shrinkage of the soft thresholding function in the projected iterative soft thresholding algorithm， a projected iterative p-thresholding algorithm is proposed. It replaces the soft thresholding function in the projected iterative soft thresholding algorithm with p-thresholding function， which will bring greater penalty for small coefficients and produce smaller bias for large coefficients to suppress the noise and reduce reconstruction errors. To speed up the algorithm， using the Nesterov gradient acceleration technology， a projected fast iterative p-thresholding algorithm is designed for magnetic resonance image reconstruction. When the tight frames are shift-invariant discrete wavelet transform and Contourlets， the projected fast iterative p-thresholding algorithm is used for compressed sensing magnetic resonance imaging. Compared with the smoothed fast iterative soft thresholding algorithm， projected iterative soft thresholding algorithm and alternating direction multiplier method， simulation results show that the projected fast iterative p-thresholding algorithm promotes the reconstruction speed and the reconstruction quality of magnetic resonance imaging. And the influence of the p value on the performance of the algorithm is analyzed， and a suitable p value selection method is given to obtain a better convergence speed and reduce the reconstruction error.