Convergent Complex Quasi-Newton Proximal Methods
for Gradient-Driven Denoisers
in Compressed Sensing MRI Reconstruction

arXiv Code Supp

Abstract

In compressed sensing (CS) MRI, model-based methods are pivotal to achieving accurate reconstruction. One of the main challenges in model-based methods is finding an effective prior to describe the statistical distribution of the target image. Plug-and-Play (PnP) and REgularization by Denoising (RED) are two general frameworks that use denoisers as the prior. While PnP/RED methods with convolutional neural networks (CNNs) based denoisers outperform classical hand-crafted priors in CS MRI, their convergence theory relies on assumptions that do not hold for practical CNNs. The recently developed gradient-driven denoisers offer a framework that bridges the gap between practical performance and theoretical guarantees. However, the numerical solvers for the associated minimization problem remain slow for CS MRI reconstruction. This paper proposes a complex quasi-Newton proximal method (CQNPM) that achieves faster convergence than existing approaches. To address the complex domain in CS MRI, we propose a modified Hessian estimation method that guarantees Hermitian positive definiteness. Furthermore, we provide a rigorous convergence analysis of the proposed method for nonconvex settings. Numerical experiments on both Cartesian and non-Cartesian sampling trajectories demonstrate the effectiveness and efficiency of our approach.

Problem

$$ \text{Denoiser:} \,\, {\umD}_{\boldsymbol \theta } \equiv \uvx - \nabla_{\uvx} f_{\boldsymbol \theta} (\uvx) $$ $$ {\uvx}^* = \arg\min_{\uvx\in\mathcal C} F(\uvx) \equiv f_{\boldsymbol \theta}(\uvx) + \underbrace{\frac{1}{2}\|\umA\uvx - \uvy\|_2^2}_{h(\uvx)} $$

CQNPM: Reconstruction Algorithm

Algorithm Initialization: $\uvx_1$ and stepsize $\alpha_k > 0$ For $k = 1, 2, \ldots$ until convergence: Estimate $\umH_k \succ 0$ and $\umB_k\succ 0$ using Algorithm 2 (see paper). Update: $$ \uvx_{k+1} \leftarrow \operatorname{prox}^{\umB_k}_{\alpha_k \, h + \iota_{\mathcal{C}}} \left( \uvx_k - \alpha_k \, \umH_k \, \nabla_{\uvx} f_{\boldsymbol \theta}(\uvx_k) \right), $$ where $$ \iota_{\mathcal C}(\uvx) = \begin{cases} 0, & \text{if } \uvx \in \mathcal{C}, \\ +\infty, & \text{otherwise}, \end{cases} $$ and $$ \operatorname{prox}^{\umB_k}_{\alpha_k \, h + \iota_{\mathcal{C}}}(\cdot) \triangleq \arg\min_{\uvx\in\mathcal C}\frac{1}{2}\|\uvx-\cdot \|_{\umB_k}^2+ \alpha_k h(\uvx). $$

Results

Figure 1: Cost and PSNR versus Iter. and Wall Time. GD: projected gradient descent; PG: proximal gradient; APG: accelerated proximal gradient.

Figure 2: Reconstructed Images.

Figure 3: Convergence results. (a) $F(\uvx_k)$: cost value; (b) $E(\uvx_k)=\|\uvx_k-\uvx_{k+1}\|_2^2$.

BibTeX citation

@article{hong2025CQNPMCSMRI,
title={Convergent Complex Quasi-Newton Proximal Methods for Gradient-Driven Denoisers in Compressed Sensing MRI Reconstruction},
author={Hong, Tao and Xu, Zhaoyi and Chun, Se Young and Hernandez-Garcia, Luis and Fessler, Jeffrey A},
year={2025},
journal={arXiv:2505.04820},
url={https://arxiv.org/abs/2505.04820},
}