Compressed sensing has shown great potentials in accelerating magnetic resonance imaging¹. Fast image reconstruction and high image quality are two main issues faced by this technology. It has been shown that, redundant image representations, e.g. tight frames, can significantly improve the image quality^2,3. But how to efficiently solve the reconstruction problem with these redundant representation systems is still challenging. In this paper, we propose a projected iterative soft-thresholding algorithm (pISTA) and its acceleration (pFISTA) for MRI image reconstruction via sparse representation under tight frames. We theoretically prove that pISTA converges to a minimizer of an objective function with a balanced tight frame sparsity^4,5. Our algorithms use much less memory than typical algorithms for both the synthesis and analysis sparse models. Besides, compared with the synthesis sparse model, the proposed algorithms achieve better image quality. Also, both pISTA and pFISTA have only one parameter to tune and the numerical solution is stable to it in terms of image reconstruction errors, which allows easily setting in many fast MRI applications.

With the canonical dual frame, we rewrite the analysis model to be a constrained synthesis-like one. This inspires us to apply algorithms that are usually fit for synthesis models, e.g. iterative soft-thresholding algorithm (ISTA), to analysis models. In order to keep the simplicity of ISTA, we propose to replace a constrained proximal map by an unconstrained proximal map plus the orthogonal projection onto the constrained subspace. Therefore, the proposed algorithm is called projected ISTA (pISTA). Furthermore, the same accelerating strategy as FISTA⁶ is introduced, resulting in the projected FISTA (pFISTA).

We proved that the the proposed algorithm converges to a minimizer of an objective function with a balanced tight frame sparsity. The derivation of pFISTA is shown in Fig.1.

Our pFISTA works in image domain, and there is no need to store any tight frame coefficients. Therefore, the pFISTA can significantly reduce memory consumption for highly redundant systems. Besides, this algorithm allows compatible programming for various tight-frames with different redundancy since one only need the forward and inverse tight-frame and thresholding.

Fig.2 shows the empirical convergence of FISTA⁶, SFISTA⁷ and pFISTA using shift-invariant discrete wavelet transform (SIDWT). It implies that both pFISTA and SFISTA, solving approximately analysis sparse models, produce much lower reconstruction error, than the original FISTA, solving synthesis sparse model. Advantage of analysis sparse model is also found on the reconstructed images as shown in Fig. 4, where wavelet basis-like artifacts are observed for FISTA method. Fig.3 shows that our pFISTA leads to similar reconstruction errors with only one free parameter $gama$ while the state-of-the-art SFISTA is sensitive to an free parameter $miu$.

We propose a projected iterative soft-threshoding algorithm (pISTA) and futher accelerate it with the same strategy as FISTA, namely pFISTA, to solve sparse image reconstruction from undersampled measurements in fast magnetic resonance imaging. We theoretically prove that the proposed algorithm converges to the balanced sparse model. Numerical results show that pFISTA achieves better reconstruction than FISTA for synthesis sparse model and converges faster or comparable to the state-of-art SFISTA⁷ for analysis sparse model. One main advantage of pFISTA is that reconstructed errors are stable to the step size, thus allowing widely usage for different tight frames in magnetic resonance image reconstructions. In the future, the convergence of pFISTA for general frames/dictionaries will be analyzed and this algorithm will be used for other advanced adaptively sparse representations in compressed sensing MRI. More information can be found at the full-length paper shared at arXiv ⁸.