A Convergent Generalized Krylov Subspace Method for Compressed Sensing MRI Reconstruction with Gradient-Driven Denoisers

TL;DR

This paper introduces a generalized Krylov subspace method for efficient MRI reconstruction using gradient-driven denoisers, providing theoretical convergence guarantees and demonstrating superior computational performance.

Contribution

It develops a novel Krylov subspace algorithm with convergence guarantees for nonconvex problems in MRI reconstruction, bridging theory and practical efficiency.

Findings

GKSM achieves faster MRI reconstruction compared to existing methods.

Theoretical analysis confirms convergence in nonconvex settings.

Numerical experiments validate accuracy and efficiency.

Abstract

Model-based reconstruction plays a key role in compressed sensing (CS) MRI, as it incorporates effective image regularizers to improve the quality of reconstruction. The Plug-and-Play and Regularization-by-Denoising frameworks leverage advanced denoisers (e.g., convolutional neural network (CNN)-based denoisers) and have demonstrated strong empirical performance. However, their theoretical guarantees remain limited, as practical CNNs often violate key assumptions. In contrast, gradient-driven denoisers achieve competitive performance, and the required assumptions for theoretical analysis are easily satisfied. However, solving the associated optimization problem remains computationally demanding. To address this challenge, we propose a generalized Krylov subspace method (GKSM) to solve the optimization problem efficiently. Moreover, we also establish rigorous convergence guarantees for GKSM in nonconvex settings. Numerical experiments on CS MRI reconstruction with spiral and radial acquisitions validate both the computational efficiency of GKSM and the accuracy of the theoretical predictions. The proposed optimization method is applicable to any linear inverse problem.