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Plug and play methods for magnetic resonance imaging (long version)

Rizwan Ahmad, Charles A. Bouman, Gregery T. Buzzard, Stanley Chan, Sizhou Liu, Edward T. Reehorst, Philip Schniter

arXiv:1903.08616v5cs.CV

TL;DR

MRI acquisition is slow, creating a need for accurate reconstruction from undersampled data while accommodating scan-specific forward models and limited training data. The paper reviews PnP methods that alternate denoising with forward-model recovery, interprets them through consensus equilibrium, and demonstrates them on cardiac and knee MRI. Across these examples, application-specific PnP CNNs outperform traditional CS methods and perform on par with modern deep-learning methods using considerably less training data.

  • Problem

    MRI’s slow acquisition motivates undersampling, while learning-based reconstruction can require large fully sampled multi-coil datasets and may generalize poorly across dataset-specific forward models.

  • Method

    The paper reviews PnP algorithms that iteratively alternate sophisticated image denoising with forward-model signal recovery and analyzes their equilibrium interpretation.

  • Results

    PnP with application-specific CNN denoisers significantly outperformed traditional CS methods and performed on par with modern deep-learning methods using considerably less training data.

  • Takeaways & Limitations

    PnP methods can exploit state-of-the-art denoisers while decoupling image modeling from MRI forward modeling, supporting recovery across scans with changing acquisition operators.

Abstract

from arXiv · show

Magnetic Resonance Imaging (MRI) is a non-invasive diagnostic tool that provides excellent soft-tissue contrast without the use of ionizing radiation. Compared to other clinical imaging modalities (e.g., CT or ultrasound), however, the data acquisition process for MRI is inherently slow, which motivates undersampling and thus drives the need for accurate, efficient reconstruction methods from undersampled datasets. In this article, we describe the use of "plug-and-play" (PnP) algorithms for MRI image recovery. We first describe the linearly approximated inverse problem encountered in MRI. Then we review several PnP methods, where the unifying commonality is to iteratively call a denoising subroutine as one step of a larger optimization-inspired algorithm. Next, we describe how the result of the PnP method can be interpreted as a solution to an equilibrium equation, allowing convergence analysis from the equilibrium perspective. Finally, we present illustrative examples of PnP methods applied to MRI image recovery.

I. INTRODUCTION

MRI provides valuable soft-tissue imaging but acquires data slowly, motivating undersampling and reconstruction methods that balance fidelity, efficiency, robustness, and generalization. The article frames PnP methods as a way to combine sophisticated denoisers with MRI forward models.

  • MRI offers non-invasive soft-tissue imaging but typically requires slow acquisition, with clinical exams potentially lasting more than an hour.
  • Undersampling reduces MRI acquisition time but makes accurate recovery from incomplete measurements necessary.
  • Learning-based MRI methods can outperform compressive-sensing methods, but often require large fully sampled multi-coil datasets and may generalize poorly across forward models.
  • PnP algorithms alternate denoising with forward-model signal recovery, separating image modeling from scan-specific coil sensitivities, sampling patterns, and resolution.
  • The standard MRI model represents undersampled Fourier measurements from multiple coils, with acceleration rate R ≜ N/M and noise commonly modeled as AWGN after pre-whitening.
  • Traditional regularization encodes prior knowledge through φ(x), but designing regularizers that capture MRI structure while remaining tractable is difficult, especially with multiple transforms and weights.

IV. PLUG-AND-PLAY METHODS

PnP methods replace proximal denoisers within optimization-inspired MRI recovery algorithms while retaining separate data-fidelity updates. Their empirical success motivates equilibrium-based analysis because generic denoisers need not correspond to an explicit regularizer.

  • PnP reconstruction iteratively calls a sophisticated denoiser while recovering x from measurements y within a larger optimization or inference algorithm.
  • Prox-based PnP: Variable splitting introduces v to decouple regularization from data fidelity, while an external constraint asymptotically enforces x = v.
  • ADMM’s penalty parameter η affects convergence speed but not the final solution, whereas in PnP ADMM the ratio σ²/η affects the fixed point.
  • Prox-based PnP: PnP ADMM plugs a powerful image denoiser such as BM3D into ADMM in place of the proximal denoiser associated with a regularizer.
  • Because generic denoisers may not be proximal maps, PnP iterations need not minimize an explicit regularized cost, leaving convergence and limit questions open.
  • PnP ADMM, FISTA, and PDS differ mainly in data-fidelity handling: ADMM uses a proximal update, whereas FISTA and PDS use gradient-descent steps.
  • PnP FISTA constrains η, restricting reachable fixed points relative to PnP ADMM and PnP PDS; proximal updates may also be more costly than gradient steps.

B. The balanced FISTA approach

The balanced FISTA approach extends transform-domain regularization and uses a finite-parameter unconstrained formulation. With β = 1/η, FISTA with transform thresholding yields a related balanced algorithm.

  • The analysis-form regularizer uses an ℓ1 penalty on transform coefficients, with proximal denoising derived for the associated optimization problem.
  • For orthogonal Ψ, the proximal denoiser reduces to transform thresholding, which is efficient because it requires forward and reverse transforms.
  • Non-orthogonal tight frames can improve results, but their proximal denoisers generally lack closed-form solutions.
  • The synthesis formulation introduces transform coefficients α and a parameter β, with the original constrained problem recovered only in the limiting case β → ∞.
  • With finite β, the balanced problem is not equivalent to the constrained formulation; choosing β = 1/η in FISTA produces the balanced approach.

C. Regularization by denoising

RED recovers MRI images by solving a denoising-based optimality condition, connecting plug-in denoisers to KDE-MAP estimation and score matching. Its interpretation as explicit regularization depends on restrictive denoiser properties, while practical variants can still yield state-of-the-art recoveries.

  • RED formulation: RED recovers x from measurements by solving an optimality condition using an arbitrary plug-in denoiser f and tuning parameter η.The approach was proposed as a denoiser-based signal reconstruction method rather than as a conventional proximal algorithm.
  • Empirical performance and solvers: Empirical studies report that sophisticated denoisers such as BM3D can produce better recoveries with PnP than regularization-based methods, while RED solutions can be state-of-the-art when η is well tuned.The paper also describes GD, inexact ADMM, fixed-point, and accelerated proximal-gradient algorithms for solving RED’s condition.
  • Conditions and limitations: Explicit regularization interpretations of RED require local homogeneity and a symmetric Jacobian, properties not satisfied by several popular denoisers.If the Jacobian is non-symmetric, the RED solutions cannot be represented as minimizers of a regularized loss of the stated form.
  • KDE-MAP interpretation: When f is the MMSE denoiser, RED exactly computes the MAP estimate under a kernel-density-estimation prior.The equivalence follows by combining Tweedie’s formula with the KDE-MAP optimality condition.
  • Score matching: With a practical non-MMSE denoiser, RED instead provides a score-matching approximation to KDE-MAP.Training the denoiser by mean-squared error makes its scaled residual approximate the KDE prior score.
  • Comparison with prox-based PnP: RED and prox-based PnP begin from different MAP formulations: KDE-based MAP for RED and φ-based MAP for PnP ADMM.The two approaches therefore replace different denoising operators, even though both use plug-in denoisers.
  • Comparison with prox-based PnP: Under symmetric linear denoisers, RED and prox-based PnP share a regularizer but use different losses and additional post-processing.This comparison clarifies that the two methods are not generally equivalent despite their related denoising structure.

D. Approximate Message Passing

Approximate message passing inserts denoisers into an iterative reconstruction scheme with an Onsager correction. Under suitable random-matrix assumptions, its denoiser inputs and error can be characterized by scalar Gaussian noise and state evolution, but MRI’s Fourier structure requires adaptations.

  • AMP formulation: PnP AMP estimates the signal iteratively with a possibly changing denoiser, and its Jacobian trace can be approximated numerically.The PnP version extends AMP to denoising-based reconstruction, including complex-valued MRI settings.
  • AMP mechanism: The Onsager correction distinguishes AMP from unaccelerated FISTA and provides AMP’s special large-system properties.Removing this term yields an ISTA-like recursion with a particular step size.
  • Large-system theory: Under i.i.d. Gaussian A and Lipschitz denoisers, PnP AMP has rigorous large-system guarantees including an AWGN denoiser-input property and exact state-evolution prediction of mean-squared error.These results hold as M,N →∞ with fixed M/N.
  • Large-system theory: With an MMSE denoiser and a unique state-evolution fixed point, AMP’s error converges to the minimum MSE.With a proximal denoiser under the same fixed-point condition, AMP converges to the MAP solution even when φ is non-convex.
  • VAMP: VAMP extends rigorous state-evolution and denoiser-input results from i.i.d. Gaussian matrices to right-rotationally invariant matrices.PnP VAMP was later analyzed under Lipschitz denoisers and can be viewed as a symmetric ADMM variant with adaptive penalty.
  • MRI adaptation: MRI violates the random-matrix assumptions underlying AMP and VAMP because its Fourier-structured sensing matrix and images do not randomize coefficients as required.A PnP VAMP method addresses this by reconstructing wavelet coefficients, whose energy decreases exponentially across wavelet bands.

V. UNDERSTANDING PNP THROUGH CONSENSUS EQUILIBRIUM

The paper frames consensus equilibrium as a way to analyze what plug-and-play methods compute and whether they converge when denoisers are not proximal maps. This addresses unresolved questions for PnP ADMM and RED.

  • Motivation: Consensus equilibrium is introduced to determine what PnP ADMM and RED solve when their plug-in denoisers are not the corresponding proximal or MMSE denoisers.The framework also targets convergence analysis for these algorithms.

A. Consensus equilibrium for prox-based PnP

Consensus equilibrium reframes prox-based PnP algorithms as seeking agreement between denoising and data-fitting operators. This viewpoint supports convergence analysis and extends beyond ADMM to related solvers.

  • A. Consensus equilibrium for prox-based PnP: Prox-based PnP algorithms seek a consensus equilibrium in which denoising and data-fitting operators agree on the recovered image.The opposing correction terms balance the two operators at equilibrium.
  • A. Consensus equilibrium for prox-based PnP: At a fixed point, the data-fitting correction accounts for measurement-inconsistent components through the k-space residual projected into the image domain.The correction does not address errors outside the row-space of A.
  • A. Consensus equilibrium for prox-based PnP: The PnP error combines with measurement noise so that the corresponding image-domain correction is canceled by the denoiser.This relation connects reconstruction error, noise, and the equilibrium correction term.
  • A. Consensus equilibrium for prox-based PnP: The same equilibrium condition can be reached with solvers beyond ADMM, including fixed-point methods based on Mann iteration.The framework therefore separates the equilibrium target from the particular numerical solver.
  • A. Consensus equilibrium for prox-based PnP: When the denoiser is nonexpansive and is a sub-gradient of a convex function, PnP ADMM converges to a global solution.A decreasing-η variant can guarantee convergence under appropriate conditions when practical denoisers violate these assumptions.
  • A. Consensus equilibrium for prox-based PnP: Under technical conditions including non-expansiveness, PnP FISTA converges to the same fixed point as PnP ADMM.The result extends equilibrium-based convergence analysis to another prox-based PnP method.

B. Consensus equilibrium for RED

RED algorithms can also be expressed through consensus equilibrium, enabling direct comparison with prox-based PnP and convergence analysis under denoiser conditions. Their equilibrium differs mainly in where the correction is applied.

  • B. Consensus equilibrium for RED: Proximal-gradient and ADMM-based RED algorithms seek a consensus equilibrium between the RED denoiser and data-fitting operators.The equilibrium is represented by the pair (b̂x_red, b̂u_red).
  • B. Consensus equilibrium for RED: The RED and prox-based PnP equilibrium conditions have the same form when L = 1, but apply the correction on opposite sides of denoising.RED adds the correction after denoising, whereas prox-based PnP applies it before denoising.
  • B. Consensus equilibrium for RED: RED’s fixed-point equation requires the data-fitting correction to balance the denoiser correction.This provides an image-domain interpretation of the RED equilibrium.
  • B. Consensus equilibrium for RED: Using Mann iteration, PG RED converges to a fixed point when the denoiser is nonexpansive and L > 1.The convergence statement is conditional on these denoiser and algorithmic-parameter assumptions.

A. Parallel cardiac MRI

The paper applies PnP methods to parallel cardiac cine MRI using a CNN denoiser designed for spatiotemporal data. Across retrospectively undersampled test datasets, PnP-CNN achieved the highest reported reconstruction quality, while its preliminary results leave room for architectural improvement.

  • A. Parallel cardiac MRI: A CNN denoiser was trained specifically for cardiac cine data because available sophisticated denoisers are limited and BM4D is slow for this application.The CNN operates on spatiotemporal volumetric patches.
  • A. Parallel cardiac MRI: The study trained on 50 fully sampled, high-SNR cine datasets from eight healthy volunteers collected at 3 T and 1.5 T.The datasets covered short-axis, two-chamber, three-chamber, and four-chamber views.
  • A. Parallel cardiac MRI: Testing used four fully sampled cine datasets from two healthy volunteers, retrospectively undersampled at acceleration rates R = 6, 8, and 10.The measurements were compressed to C = 12 virtual coils and modeled using ESPIRiT-estimated sensitivity maps.
  • A. Parallel cardiac MRI: PnP-CNN used PnP ADMM with 100 iterations and was compared against CS-UWT, CS-TV, L+S, and PnP-UWT.Each ADMM iteration used four conjugate-gradient steps to approximate the data-fitting update.
  • A. Parallel cardiac MRI: Across all four datasets and three acceleration rates, PnP-CNN achieved the highest rSNR relative to the fully sampled reference.It also better preserved anatomical heart details than the CS methods and PnP-UWT.
  • A. Parallel cardiac MRI: The cardiac cine results are preliminary, and optimizing the denoiser architecture may further improve PnP-CNN performance.

B. Single-coil fastMRI knee data

The evaluation uses retrospectively undersampled MRI data and compares PnP-CNN with compressed-sensing and U-Net-based recovery methods. On the reported fastMRI validation set, PnP-CNN achieved the highest rSNR while remaining competitive in SSIM.

  • Methods: PnP-CNN used PnP ADMM with a two-channel DnCNN trained on only 267 central training slices.The training-noise variance and σ2/η were manually adjusted to maximize rSNR.
  • Methods: PnP-CNN was compared with TV and U-Net baselines under random and fixed sampling masks.The U-Nets included models trained with full or substantially smaller datasets and with fixed or random masks.
  • Results: PnP-CNN achieved the highest rSNR for both random and fixed testing masks, while U-Net achieved slightly higher SSIM in both tests.PnP-CNN performed similarly to the U-Nets and significantly better than TV.
  • Results: The smaller-data U-Net performed significantly worse than the other U-Nets and was not competitive with PnP-CNN despite using the same training data.The fixed-mask full-data U-Net performed best when tested with its matching mask but dropped considerably with random masks.

VII. CONCLUSION

The paper presents PnP methods as an avenue for compressive MRI recovery, applying them to cardiac cine and knee images from highly undersampled data. Application-specific CNN denoisers significantly outperformed traditional CS methods, matched modern deep-learning methods, and required considerably less training data.

  • Conclusion: PnP recovery exploited richer image structure through state-of-the-art denoisers and was evaluated on cardiac cine and knee MRI.The conclusion frames these results as demonstrating the potential of PnP methods for MRI reconstruction.
  • Conclusion: Application-specific CNN-based PnP methods significantly outperformed traditional CS methods and performed on par with modern deep-learning methods.These findings were reported for highly undersampled cardiac cine and knee datasets.
  • Conclusion: PnP methods achieved this deep-learning-level performance with considerably less training data.
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