Projected 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 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.

💡 Research Summary

The paper tackles the problem of fast and high‑quality magnetic resonance imaging (MRI) reconstruction under the compressed sensing (CS) framework when redundant representations—specifically tight frames—are employed. While tight frames have been shown to improve sparsity and thus image quality, their redundancy makes the standard synthesis‑based iterative soft‑thresholding algorithm (ISTA) unstable and the analysis‑based formulation difficult to solve efficiently.
To overcome these issues, the authors introduce a projected version of ISTA (pISTA). The key insight is that for a tight frame Φ (ΦΦᵀ = I), the canonical dual frame is simply the transpose Φᵀ, and the orthogonal projector onto the range of the analysis operator Ψ = Φᵀ is P = ΦΦᵀ. By inserting this projector after the soft‑thresholding step, the update is forced to stay within the subspace spanned by the frame, thereby preserving the redundancy while avoiding ill‑conditioning. The algorithm proceeds as follows:

1. Gradient step – a standard data‑consistency update x^{k+½}=x^{k}−τAᵀ(Ax^{k}−y), where τ∈(0,2/‖A‖₂²).
2. Analysis and soft‑thresholding – compute z^{k+1}=S_{λτ}(Ψx^{k+½}), with S denoting element‑wise soft‑thresholding.
3. Projection and synthesis – reconstruct x^{k+1}=Φz^{k+1}+(I−ΦΦᵀ)x^{k+½}.

The additional term (I−ΦΦᵀ)x^{k+½} removes components that lie outside the frame’s range, guaranteeing that each iterate belongs to the feasible subspace. The authors prove that, because the gradient is L‑Lipschitz, the soft‑thresholding is non‑expansive, and the projector is also non‑expansive, the composite mapping is a contraction for any τ in the prescribed interval. Consequently, pISTA converges globally to the minimizer of the objective
f(x)=½‖Ax−y‖₂²+λ‖Ψx‖₁,
which corresponds to a balanced sparsity model for tight frames.

To accelerate convergence, the authors adopt the Beck‑Teboulle momentum scheme (FISTA) without modifying its core logic. The accelerated version, pFISTA, updates an auxiliary sequence t_k and uses a linear combination of the two most recent iterates, achieving the optimal O(1/k²) rate for first‑order methods. Importantly, the acceleration does not require any extra parameters beyond the step size τ, preserving the algorithm’s simplicity.

Experimental validation is performed on both 2‑D and 3‑D MRI datasets (brain, knee, etc.) under various acceleration factors (4×, 8×, 12×) and noise levels. The authors compare pISTA and pFISTA against three baselines: (i) synthesis‑based ISTA, (ii) standard FISTA applied to the synthesis model, and (iii) a state‑of‑the‑art smoothing FISTA (SFISTA). Quantitative metrics (PSNR, SSIM) consistently show that pISTA outperforms the synthesis approach by 1.5–2.3 dB in PSNR and improves SSIM by 0.02–0.04, especially in low‑SNR regimes where the redundancy of the tight frame is most beneficial. pFISTA matches or slightly exceeds the convergence speed of SFISTA while retaining comparable reconstruction quality.

A notable practical advantage is the algorithm’s robustness to the choice of τ. The authors demonstrate that a wide range of τ values (approximately 0.5 to 1.5 times the theoretical maximum) yields nearly identical reconstruction errors, which greatly simplifies parameter tuning in clinical settings. Moreover, the computational overhead of pISTA/pFISTA is modest: only forward and adjoint frame transforms (Φ and Φᵀ) are required, with no need for explicit inverse frame calculations or additional regularization terms.

In summary, the paper makes two primary contributions: (1) it formulates a mathematically sound, projection‑based ISTA that leverages the sparsity‑enhancing properties of tight frames while guaranteeing convergence, and (2) it shows that the well‑known FISTA acceleration can be seamlessly integrated, delivering a fast, stable, and easy‑to‑tune reconstruction pipeline. The method’s simplicity, stability, and superior image quality make it a strong candidate for real‑time or ultra‑fast MRI applications. Future work may explore non‑Parseval frames, integration with parallel imaging (SENSE/GRAPPA), and data‑driven frame learning to further push the limits of CS‑MRI.