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Compressed sensing has shown great potentials in accelerating magneticresonance imaging. Fast image reconstruction and high image quality are twomain issues faced by this new technology. It has been shown that, redundantimage representations, e.g. tight frames, can significantly improve the imagequality. But how to efficiently solve the reconstruction problem with theseredundant representation systems is still challenging. This paper attempts toaddress the problem of applying iterative soft-thresholding algorithm (ISTA) totight frames based magnetic resonance image reconstruction. By introducing thecanonical dual frame to construct the orthogonal projection operator on therange of the analysis sparsity operator, we propose a projected iterativesoft-thresholding algorithm (pISTA) and further accelerate it by incorporatingthe strategy proposed by Beck and Teboulle in 2009. We theoretically prove thatpISTA converges to the minimum of a function with a balanced tight framesparsity. Experimental results demonstrate that the proposed algorithm achievesbetter reconstruction than the widely used synthesis sparse model and theaccelerated pISTA converges faster or comparable to the state-of-art smoothingFISTA. One major advantage of pISTA is that only one extra parameter, the stepsize, is introduced and the numerical solution is stable to it in terms ofimage reconstruction errors, thus allowing easily setting in many fast magneticresonance imaging applications.

Projected Iterative Soft-thresholding Algorithm for Tight Frames in Compressed Sensing Magnetic Resonance Imaging