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

TL;DR

This paper introduces a projected iterative soft-thresholding algorithm (pISTA) for magnetic resonance image reconstruction using tight frames, improving convergence speed and stability over existing methods with minimal parameter tuning.

Contribution

It proposes a novel pISTA method leveraging the canonical dual frame for efficient and stable MRI reconstruction with tight frames, and accelerates convergence using Beck and Teboulle's strategy.

Findings

pISTA outperforms synthesis sparse models in image quality

Accelerated pISTA converges faster or comparably to FISTA

The method is stable and easy to parameterize

Abstract

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 iterative soft-thresholding algorithm (ISTA) to tight frames based magnetic resonance image reconstruction. By introducing the canonical dual frame to construct the orthogonal projection operator on the range of the analysis sparsity operator, we propose a projected iterative soft-thresholding algorithm (pISTA) and further accelerate it by incorporating the strategy proposed by Beck and Teboulle in 2009. We theoretically prove that pISTA converges to the minimum of a function with a balanced tight frame sparsity. Experimental results demonstrate that the proposed algorithm achieves better reconstruction than the widely used synthesis sparse model and the accelerated pISTA converges faster or comparable to the state-of-art smoothing FISTA. One major advantage of pISTA 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.