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

Tao Hong, Umberto Villa, Jeffrey A. Fessler

TL;DR

This paper addresses CS-MRI reconstruction with gradient-driven denoisers by introducing a generalized Krylov subspace method (GKSM) to efficiently solve the nonconvex optimization. GKSM forms a low-dimensional subspace and solves a proximal problem within it, with closed-form updates for the subspace coefficients and optional convex constraints via projection, while maintaining computational efficiency by limiting A x and A^H x evaluations. A rigorous convergence analysis based on the KL inequality demonstrates monotone descent and eventual convergence to critical points, with explicit rate characterizations depending on the KL exponent. Empirical results on spiral and radial non-Cartesian acquisitions for brain and knee data show GKSM provides substantial speedups over existing methods (e.g., APG, CQNPM) with comparable reconstruction quality, validating its practical impact for fast, reliable MRI reconstruction.

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.

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

TL;DR

This paper addresses CS-MRI reconstruction with gradient-driven denoisers by introducing a generalized Krylov subspace method (GKSM) to efficiently solve the nonconvex optimization. GKSM forms a low-dimensional subspace and solves a proximal problem within it, with closed-form updates for the subspace coefficients and optional convex constraints via projection, while maintaining computational efficiency by limiting A x and A^H x evaluations. A rigorous convergence analysis based on the KL inequality demonstrates monotone descent and eventual convergence to critical points, with explicit rate characterizations depending on the KL exponent. Empirical results on spiral and radial non-Cartesian acquisitions for brain and knee data show GKSM provides substantial speedups over existing methods (e.g., APG, CQNPM) with comparable reconstruction quality, validating its practical impact for fast, reliable MRI reconstruction.

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.

Paper Structure

This paper contains 15 sections, 6 theorems, 38 equations, 9 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries on Krylov Subspace Methods
  3. Proposed Method
  4. Incorporating A Convex Constraint
  5. Discussion
  6. Convergence Analysis
  7. Numerical Experiments
  8. Spiral Acquisition Reconstruction
  9. Radial Acquisition Reconstruction
  10. Effect of TEXT and Convergence Validation
  11. Conclusion
  12. Proof of Lemma 3
  13. Proof of Lemma 4
  14. Proof of Theorem 1
  15. Proof of Theorem 2

Key Result

Lemma 1

Let $f:\,\mathbb C^N\rightarrow (-\infty,\infty]$ be an $L$-smooth function. Then for any $\bm{\mathrm{x}}_1, \bm{\mathrm{x}}_2\in\mathbb C^N$, we have

Figures (9)

  • Figure 1: The neural network architecture used to construct the energy function $f_{\bm\uptheta}(\bm{\mathrm{x}})$ is based on cohen2021has. The convolutional kernels have size $3 \times 3$ with a stride of one.
  • Figure 2: The magnitude of the six brain and knee complex-valued ground truth images.
  • Figure 3: The spiral (a) and radial (b) sampling trajectories.
  • Figure 4: Comparison of different methods with spiral acquisition on the brain $1$ image for $\varepsilon = 5\times10^{-3}$. (a), (b): cost values versus iteration and wall time; (c), (d): PSNR values versus iteration and wall time.
  • Figure 5: First row: the reconstructed brain $1$ images of each method at $50$th and $100$th iterations with spiral acquisition. The PSNR (respectively, SSIM) values are labeled at the left (respectively, right) bottom corner of each image. Second row: the associated error maps ($8 \times$) of the reconstructed images.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Definition 1: Kurdyka–Łojasiewicz inequality attouch2010proximalbolte2014proximal
  • Lemma 1: Majorizer of $f$ hong2025CQNPMCSMRI
  • Lemma 2: Bounded Hessian hong2025CQNPMCSMRI
  • Lemma 3
  • Lemma 4
  • Theorem 1: Descent properties of \ref{['alg:NGKS']}, $K\leq +\infty$
  • Theorem 2: Convergence rates, $K<+\infty$