Harnessing Magnitude-Only and Complex Measurements for Improved Dynamic MRI Reconstruction with Learned Priors

The gist MRI reconstruction methods for undersampled k-space data naturally utilize complex-valued measurements, and this work proposes C + Mag, a magnitude-informed physics-driven deep learning reconstruction method that demonstrates strong consistency of k-space magnitudes across time frames

Motivation

The paper investigates the use of auxiliary k-space magnitude information for accelerated steady-state dynamic MRI reconstruction and demonstrates strong consistency of k-space magnitudes across time-frames. This motivation stems from the observation that two random magnitude measurements can carry information equivalent to one single random complex-valued linear measurement in sparse recovery problems. Prior work has explored simple scenarios of this setting on a limited number of examples with hand-crafted regularizers. The availability of large-scale MRI databases and recent findings in adversarial robustness literature motivated the investigation into whether auxiliary magnitude information can be reliably leveraged for MRI reconstruction without requiring additional scan time.

Proposed Method (C + Mag)

The authors propose a physics-driven deep learning framework for MRI reconstruction using a combination of complex-valued and magnitude-only measurements, referred to as C + Mag reconstruction. This framework formulates a new objective function for MRI reconstruction with joint complex and magnitude least-squares data fidelity terms. The objective is solved using ADMM-based unrolling with learned proximal operators and a novel data-fidelity formulation tailored for mixed complex and magnitude measurements.

Optimization Strategy

The proposed method employs an ADMM-based unrolling framework with a novel magnitude-aware data-fidelity formulation, where quadratically smoothed optimization and momentum-based updates are introduced to address the non-differentiability and non-convexity of the magnitude constraints. The objective function is defined as arg min x yomega − Eomegax2 + λr∆ − E∆x2 + R(x). To handle the non-differentiable term ·, the authors introduce quadratically smoothed optimization (Quad-S) by approximating it as x ≈ p x ⊙ xH + ϵ. Furthermore, to improve stability within the data fidelity subproblem, Nesterov acceleration is employed instead of standard gradient descent.

Experimental Validation

Extensive evaluations were conducted on retrospectively undersampled cine MRI [16] and phase-contrast flow MRI datasets [17], as well as local prospectively undersampled real-time cine MRI databases. The proposed method demonstrates improved artifact suppression, sharper anatomical recovery, and better preservation of phase information compared to conventional PD-DL methods. Quantitative comparisons showed that the C + Mag method achieved superior performance over conventional PD-DL approaches in terms of peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM) for both cine MRI and phase-contrast flow MRI.

Ablation Studies

Key ablation studies were performed to analyze the major design choices of the proposed method. These studies varied the network architecture for the proximal operator, momentum-based GD schemes for solving the DF subproblem, smoothing operators for the · term, and unrolling strategies. The results indicated that NesterovGD achieved the best performance when solving the data fidelity subproblem. Additionally, Quad-S smoothing consistently achieved the best performance for approximating the absolute value operator across all tested configurations. The ADMM-based formulation was found to achieve the best performance compared to variable splitting with quadratic penalty (VSQP) and proximal gradient descent (PGD).

Conclusion

In this work, the focus is on introducing a new inverse problem formulation that enables incorporation of auxiliary magnitude information into PD-DL MRI reconstruction, in particular for dynamic MRI with no additional acquisition costs. The proposed framework can be synergistically combined with different optimization schemes and network architectures, suggesting that further comparisons of different network architectures are not critical for this work. The results demonstrate substantial improvements over conventional PD-DL reconstruction methods in both image quality and phase preservation. The proposed method preserves clinically relevant image features and enables accelerated cardiac MRI without compromising clinical image quality.

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The gist MRI reconstruction methods for undersampled k-space data naturally utilize complex-valued measurements, and this work proposes C + Mag, a magnitude-informed physics-driven deep learning reconstruction method that demonstrates strong consistency of k-space magnitudes across time frames

Motivation

The paper investigates the use of auxiliary k-space magnitude information for accelerated steady-state dynamic MRI reconstruction and demonstrates strong consistency of k-space magnitudes across time frames. This motivation stems from the observation that two random magnitude measurements can carry information equivalent to one single random complex-valued linear measurement in sparse recovery problems. Prior work has explored simple scenarios of this setting on a limited number of examples with hand-crafted regularizers. The availability of large-scale MRI databases and recent findings in adversarial robustness literature motivated the investigation into whether auxiliary magnitude information can be reliably leveraged for MRI reconstruction without requiring additional scan time.

Ablation Studies

Key ablation studies were performed to analyze the major design choices of the proposed method. These studies varied the network architecture for the proximal operator, momentum-based GD schemes for solving the DF subproblem, smoothing operators for the · term, and unrolling strategies. The results indicated that NesterovGD achieved the best performance when solving the data fidelity subproblem. Additionally, Quad-S smoothing consistently achieved the best performance for approximating the absolute value operator across all tested configurations.

1. Improved MRI reconstruction performance in dynamic MRI by jointly leveraging complex-valued measurements and auxiliary magnitude constraints from neighboring timeframes to suppress artifacts and preserve phase information, as demonstrated by C + Mag reconstruction achieving substantial improvements over conventional PD-DL methods in both retrospective and prospective settings.

2. Development of a novel ADMM based unrolled reconstruction algorithm with a magnitude aware data fidelity formulation that includes quadratically smoothed optimization and momentum based updates to improve stability when solving non differentiable and non convex magnitude constraints.

3. Creation of a PD-DL reconstruction framework that jointly leverages complex valued and auxiliary magnitude only measurements through a unified optimization formulation, enabling accelerated MRI reconstruction without requiring additional acquisition cost by utilizing auxiliary magnitude information from neighboring cardiac phases.

4. Enhanced robustness in inverse problem solving by employing Nesterov acceleration for the data fidelity subproblem within the ADMM unrolling framework to improve convergence compared to standard gradient descent methods.

5. Implementation of alternative smooth approximations of the absolute value operator, such as Smooth l1 (Huber) smoothing and Pseudo-Huber (PH-S) smoothing, in ablation studies to investigate their effectiveness in handling non differentiability in magnitude constraints for reconstruction stability.

MRI reconstruction methods for undersampled k-space data naturally utilize complex-valued measurements. Parallel developments in sparse phase retrieval have shown that magnitude-only measurements may provide complementary information for signal recovery. However, their use in MRI reconstruction remains largely unexplored, due to lack of practical settings where informative magnitude measurements can be obtained without additional scan time. In this work, we investigate the use of auxiliary k-space magnitude information for accelerated steady-state dynamic MRI reconstruction, and demonstrate strong consistency of k-space magnitudes across time-frames. Building on this observation, we propose C+ Mag, a magnitude-informed physics-driven deep learning reconstruction method. The proposed method employs an ADMM-based unrolling framework with a novel magnitude-aware data-fidelity formulation, where quadratically smoothed optimization and momentum-based updates are introduced to address the non-differentiability and non-convexity of the magnitude constraints. Experiments on retrospectively undersampled cine MRI and phase-contrast flow MRI datasets, as well as prospectively undersampled real-time cine MRI acquisitions, demonstrate improved artifact suppression, sharper anatomical recovery, and better preservation of phase information compared to conventional PD-DL methods, which is further supported through blinded expert reader evaluations.