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

Tao Hong, Zhaoyi Xu, Se Young Chun, Luis Hernandez-Garcia, Jeffrey A. Fessler

TL;DR

This work targets fast, provably convergent CS MRI reconstruction by marrying gradient-driven denoisers with a complex quasi-Newton proximal framework. It introduces CQNPM, which leverages a Hermitian positive definite Hessian estimate (MMESHR1) to accelerate convergence in the complex domain and provides rigorous convergence guarantees under nonconvex settings. Empirical results on spiral, radial, and Cartesian acquisitions for brain and knee datasets show that CQNPM outperforms traditional GD/PG/APG and preconditioned PnP methods in both convergence speed and reconstruction quality. The approach offers a theoretically grounded alternative to CNN-only PnP/RED strategies, with potential for extension to 3D and dynamic MRI and to more sophisticated Hessian updating rules.

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.

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

TL;DR

This work targets fast, provably convergent CS MRI reconstruction by marrying gradient-driven denoisers with a complex quasi-Newton proximal framework. It introduces CQNPM, which leverages a Hermitian positive definite Hessian estimate (MMESHR1) to accelerate convergence in the complex domain and provides rigorous convergence guarantees under nonconvex settings. Empirical results on spiral, radial, and Cartesian acquisitions for brain and knee datasets show that CQNPM outperforms traditional GD/PG/APG and preconditioned PnP methods in both convergence speed and reconstruction quality. The approach offers a theoretically grounded alternative to CNN-only PnP/RED strategies, with potential for extension to 3D and dynamic MRI and to more sophisticated Hessian updating rules.

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.
Paper Structure (17 sections, 5 theorems, 48 equations, 12 figures, 3 algorithms)

This paper contains 17 sections, 5 theorems, 48 equations, 12 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Inverse Problems with PnP/RED Priors
  3. Inverse Problems with gradient-driven Denoisers
  4. Contributions and Roadmap
  5. Proposed Method
  6. Convergence Analysis
  7. Numerical Experiments
  8. Spiral Acquisition Reconstruction
  9. Radial Acquisition Reconstruction
  10. Cartesian Acquisition Reconstruction
  11. Convergence Validation
  12. Comparison with the Preconditioned PnP Approach
  13. Conclusion and Future Work
  14. Proof of Lemma 2
  15. Proof of Lemma 3
  16. ...and 2 more sections

Key Result

Lemma 1

Let $f:\,\mathbb C^N\rightarrow (-\infty,\infty]$ be an $L$-smooth function ($L>0$). Then for any $\bm{\mathrm{x}}_1, \bm{\mathrm{x}}_2\in\mathbb C^N$, we have

Figures (12)

  • Figure 1: Magnitude of the complex-valued ground-truth images.
  • Figure 2: The spiral (a) and radial (b) sampling trajectories and the Cartesian sampling mask (c) used in the experiments.
  • Figure 3: The neural network architecture used for the energy function $f_{\bm \theta}(\bm{\mathrm{x}})$, based on cohen2021has. The convolutional kernel size is $3\times 3$ with stride $1$.
  • Figure 4: Comparison of different methods with spiral acquisition on the brain image for $\varepsilon = 4\times10^{-3}$. (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 images of each method at $50$ and $100$th iterations with spiral acquisition. The PSNR values are labeled at the left bottom corner of each image. Second row: the associated error maps ($\times 5$) of the reconstructed images.
  • ...and 7 more figures

Theorems & Definitions (5)

  • Lemma 1: Majorizer of $f$
  • Lemma 2
  • Lemma 3
  • Lemma 4: Bounded Hessian
  • Theorem 1: Convergence results