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A Complex Quasi-Newton Proximal Method for Image Reconstruction in Compressed Sensing MRI

Tao Hong, Luis Hernandez-Garcia, Jeffrey A. Fessler

TL;DR

This work addresses speeding up compressed sensing MRI reconstruction by introducing a complex-domain quasi-Newton proximal framework (CQNPM) that uses a symmetric rank-1 (SR1) Hessian approximation to form a weighted proximal step. The method handles a composite objective with a data fidelity term and a wavelet+TV regularizer via an efficiently computed weighted proximal mapping, leveraging a dual Chambolle/FISTA approach and, when advantageous, partial smoothing. Numerically, CQNPM achieves faster convergence in both iterations and CPU time than traditional accelerated proximal methods and primal-dual schemes on non-Cartesian radial and spiral trajectories, with robust performance across gamma and maxiter settings and improvements at higher data input SNR. The results suggest practical impact for large-scale CS-MRI reconstructions and indicate directions for future work, including fixed-weight Hessian approximations to further speed up convergence.

Abstract

Model-based methods are widely used for reconstruction in compressed sensing (CS) magnetic resonance imaging (MRI), using regularizers to describe the images of interest. The reconstruction process is equivalent to solving a composite optimization problem. Accelerated proximal methods (APMs) are very popular approaches for such problems. This paper proposes a complex quasi-Newton proximal method (CQNPM) for the wavelet and total variation based CS MRI reconstruction. Compared with APMs, CQNPM requires fewer iterations to converge but needs to compute a more challenging proximal mapping called weighted proximal mapping (WPM). To make CQNPM more practical, we propose efficient methods to solve the related WPM. Numerical experiments on reconstructing non-Cartesian MRI data demonstrate the effectiveness and efficiency of CQNPM.

A Complex Quasi-Newton Proximal Method for Image Reconstruction in Compressed Sensing MRI

TL;DR

This work addresses speeding up compressed sensing MRI reconstruction by introducing a complex-domain quasi-Newton proximal framework (CQNPM) that uses a symmetric rank-1 (SR1) Hessian approximation to form a weighted proximal step. The method handles a composite objective with a data fidelity term and a wavelet+TV regularizer via an efficiently computed weighted proximal mapping, leveraging a dual Chambolle/FISTA approach and, when advantageous, partial smoothing. Numerically, CQNPM achieves faster convergence in both iterations and CPU time than traditional accelerated proximal methods and primal-dual schemes on non-Cartesian radial and spiral trajectories, with robust performance across gamma and maxiter settings and improvements at higher data input SNR. The results suggest practical impact for large-scale CS-MRI reconstructions and indicate directions for future work, including fixed-weight Hessian approximations to further speed up convergence.

Abstract

Model-based methods are widely used for reconstruction in compressed sensing (CS) magnetic resonance imaging (MRI), using regularizers to describe the images of interest. The reconstruction process is equivalent to solving a composite optimization problem. Accelerated proximal methods (APMs) are very popular approaches for such problems. This paper proposes a complex quasi-Newton proximal method (CQNPM) for the wavelet and total variation based CS MRI reconstruction. Compared with APMs, CQNPM requires fewer iterations to converge but needs to compute a more challenging proximal mapping called weighted proximal mapping (WPM). To make CQNPM more practical, we propose efficient methods to solve the related WPM. Numerical experiments on reconstructing non-Cartesian MRI data demonstrate the effectiveness and efficiency of CQNPM.
Paper Structure (22 sections, 3 theorems, 30 equations, 26 figures, 3 algorithms)

This paper contains 22 sections, 3 theorems, 30 equations, 26 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Discretized Total Variation
  5. Weighted Proximal Mapping
  6. Complex Quasi-Newton Proximal Methods
  7. Compute Weighted Proximal Mapping
  8. Compute the Weighted Proximal Mapping when alpha
  9. Partial Smoothing
  10. Numerical Experiments
  11. Radial Acquisition MRI Reconstruction
  12. Spiral Acquisition MRI Reconstruction
  13. The Choice of gamma
  14. The Choice of maxiter
  15. Reconstruction with High Data Input SNR
  16. ...and 7 more sections

Key Result

Proposition 1

Let where $\bm w_k(\bm z,\bm P,\bm Q)=\bm v_k-\bar{\lambda}{\bm B}_k^{-1} \left(\alpha\bm T^\mathcal{H}\bm z+(1-\alpha)\mathrm{vec}\left(\mathcal{L}(\bm P,\bm Q)\right)\right)$ and $\mathcal{P} = \mathcal{P}_1~\text{or}~\mathcal{P}_2$ depending on which TV is used. Then the optimal solution of eq:TVWavR

Figures (26)

  • Figure 1: The magnitude of the complex-valued ground truth images.
  • Figure 2: The non-Cartesian MRI trajectories used in this paper.
  • Figure 3: Cost values versus iteration (top) and CPU time (bottom) of the brain image with regularizer $h(\bm x) = \|\bm T \bm x\|_1$ and $\lambda=5\times10^{-4}$ for a left invertible wavelet transform $\bm T$ with $5$ levels. Acquisition: radial trajectory with $96$ projections, $512$ readout points, and $12$ coils.
  • Figure 4: First row: the ground truth image and PSNR values versus CPU time; second to third row: the reconstructed brain images at $3$, $10$, $13$, and $16$th iteration with \ref{['fig:brain:radial:cost']} setting; fourth row: the zoomed-in regions and the corresponding error maps ($\times 5$) of the $16$th iteration reconstructed images.
  • Figure 5: Cost values versus iteration (top) and CPU time (bottom) of the brain image with regularizer $h(\bm x)=\alpha \|\bm T\bm x\|_1+(1-\alpha)\mathrm{TV}(\bm x)$ and same acquisition as \ref{['fig:brain:radial:cost']}. The parameters were $\lambda = 6\times 10^{-4}$ and $\alpha = \frac{1}{6}$.
  • ...and 21 more figures

Theorems & Definitions (7)

  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Remark 1
  • Theorem 1: Theorem 3.4, becker2019quasi
  • proof