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

TL;DR

This paper introduces a convergent complex quasi-Newton proximal method for faster MRI reconstruction in compressed sensing, leveraging gradient-driven denoisers with theoretical guarantees and improved convergence speed.

Contribution

It proposes a novel complex quasi-Newton proximal algorithm with a modified Hessian estimation for complex domains, providing convergence guarantees for nonconvex MRI reconstruction problems.

Findings

Faster convergence compared to existing methods

Effective reconstruction on Cartesian and non-Cartesian data

Theoretical convergence guarantees for nonconvex settings

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 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.