A Simple and Fast Iterative Soft-thresholding Algorithm for Tight Frames in Compressed Sensing Magnetic Resonance Imaging

Compressed sensing has shown great potentials in accelerating magnetic resonance imaging. Fast image reconstruction and high image quality are two main issues faced by this new technology. It has been shown that, redundant image representations, e.g. tight frames, can significantly improve the image quality. But how to efficiently solve the reconstruction problem with these redundant representation systems is still challenging. This paper attempts to address the problem of applying fast iterative soft-thresholding algorithm (FISTA) to tight frames based magnetic resonance image reconstruction. By introducing the canonical dual frame, we construct an orthogonal projection operator on the range of the analysis sparsity operator and propose a new algorithm, called the projected FISTA (pFISTA). We theoretically prove that pFISTA converges to the minimum of a function with a balanced tight frame sparsity. One major advantage of pFISTA is that only one extra parameter, the step size, is introduced and the numerical solution is stable to it in terms of image reconstruction errors, thus allowing easily setting in many fast magnetic resonance imaging applications.