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

Mahdi Saberi; Yaşar Utku Alçalar; Merve Gülle; Chetan Shenoy; Mehmet Akçakaya

Source-linked AI summary

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

Mahdi Saberi, Yaşar Utku Alçalar, Merve Gülle, Chetan Shenoy, Mehmet Akçakaya

arXiv:2608.18036v1Aug 18, 2026eess.IVcs.AIcs.CVcs.LGphysics.med-ph

TL;DR

Highly undersampled MRI reconstruction remains limited by missing information, while auxiliary magnitude measurements have been underexplored. This paper proposes C + Mag, a physics-driven deep learning framework that combines complex and magnitude-only k-space data, achieving improved image quality and phase preservation across dynamic MRI experiments.

Problem
At high acceleration rates, MRI reconstruction remains limited by insufficient acquired information, while practical use of auxiliary magnitude data is largely underexplored.
Method
C + Mag jointly incorporates complex-valued and auxiliary magnitude-only k-space measurements within a unified physics-driven deep learning reconstruction framework.
Results
Across cine, phase-contrast flow, and real-time cine MRI, C + Mag substantially improved image quality and phase preservation over conventional PD-DL methods.
Takeaways & Limitations
Strong temporal consistency of k-space magnitudes supports obtaining informative auxiliary measurements from neighboring steady-state frames without additional acquisition cost.

Abstract

from arXiv · show

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 $\mathbb{C}+\text{Mag}$, 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.

I. INTRODUCTION

The paper addresses severe information limitations in highly undersampled MRI by combining complex-valued and auxiliary magnitude-only measurements within a physics-driven deep learning reconstruction framework. It introduces magnitude-aware optimization and evaluates the approach across retrospective, prospective, and expert-assessed MRI studies.

Motivation: Highly undersampled MRI creates a severely ill-posed inverse problem, and high acceleration can leave residual artifacts, anatomical blurring, and lost fine structural detail.Physics-driven deep learning combines acquisition physics with learned priors but remains constrained by limited acquired information at high acceleration rates.
Motivation: Two magnitude-only measurements can, under suitable conditions, carry information equivalent to one complex linear measurement, motivating mixed-measurement MRI reconstruction.Phase retrieval studies provide the theoretical motivation for combining complex-valued and magnitude-only measurements.
Proposed framework: The proposed C + Mag framework jointly uses complex-valued measurements and auxiliary magnitude constraints from neighboring cardiac phases in a unified reconstruction objective.The framework uses joint complex and magnitude least-squares data-fidelity terms and learned proximal operators through algorithm unrolling.
Optimization: An ADMM-based unrolled algorithm addresses non-differentiable, non-convex magnitude fidelity through quadratically smoothed optimization and momentum-based updates.These design choices are introduced to improve optimization stability for the magnitude-aware data-fidelity formulation.
Evaluation: Evaluations span retrospectively undersampled cine MRI, phase-contrast flow MRI, and prospectively undersampled real-time cine MRI, with improvements over conventional PD-DL methods at high acceleration rates.The framework is additionally assessed through blinded cardiologist evaluations and ablation studies of major design choices.

II. BACKGROUND & MOTIVATION · A. Background on PD-DL Reconstruction in MRI · B. k-space Magnitude Similarity

The background frames MRI reconstruction as regularized recovery from undersampled complex k-space, with PD-DL unrolling data-fidelity and learned-prior operations. The motivation is that highly consistent dynamic-MRI k-space magnitudes can provide auxiliary information without additional scan time.

A. Background on PD-DL Reconstruction in MRI: MRI reconstruction combines complex-valued data fidelity with regularization that incorporates prior information.The encoding operator models undersampled multi-coil measurements, while the regularizer supplies prior information.
A. Background on PD-DL Reconstruction in MRI: PD-DL methods unroll a fixed number of iterative optimization steps, using learnable parameters in data-fidelity and proximal operations.The unrolled framework targets the regularized least-squares reconstruction problem.
A. Background on PD-DL Reconstruction in MRI: The work emphasizes inverse-problem formulation and optimization strategy rather than the training paradigm itself.Both supervised and self-supervised settings are considered in retrospective and prospective undersampling experiments.
B. k-space Magnitude Similarity: Joint sparse phase-retrieval and compressed-sensing theory motivates using accurate magnitude information with incorrect phase alongside complex measurements.The cited theory suggests that two random magnitude measurements can hold the same information as one random complex-valued linear measurement.
B. k-space Magnitude Similarity: Steady-state dynamic MRI offers a natural setting because multi-coil k-space magnitudes remain highly consistent across time without additional scan time.This motivation is tied to large-scale MRI databases and findings on adversarial robustness.
B. k-space Magnitude Similarity: The study quantifies temporal magnitude consistency using cosine similarity across cardiac phases in breath-hold segmented cine acquisitions from the OCMR dataset.Similarities are computed across multiple slices and subjects relative to the phase at the middle of the R-R wave.
B. k-space Magnitude Similarity: c-sim > 0.988 even for distant phases, indicating strong temporal correlation in multi-coil k-space magnitudes.The analysis uses magnitude k-space, discarding phase, and summarizes mean and standard deviation across slices.
B. k-space Magnitude Similarity: Magnitude data from different cardiac phases can therefore improve MRI reconstruction without additional acquisition cost.The auxiliary information is readily available without changes to the cine MRI acquisition.

III. METHODS

The proposed C + Mag framework reconstructs accelerated MRI from jointly enforced complex-valued measurements and auxiliary magnitude-only constraints. It uses ADMM-based unrolling with quadratic smoothing and Nesterov-accelerated updates to address non-differentiability and non-convexity in the magnitude data-fidelity subproblem.

Magnitude-informed formulation: The reconstruction objective jointly enforces consistency with acquired complex measurements and auxiliary magnitude-only measurements from neighboring timeframes.The auxiliary magnitudes ideally cover unsampled k-space locations in the target timeframe, with Δ ⊆ Ω^C and ideally Δ = Ω^C.
ADMM unrolling: The objective is solved using ADMM, with a neural network implementing the regularization-related subproblem and iterative optimization for the data-fidelity subproblem.The parameters μ, λ, and γ are learnable.
Differentiable data fidelity: Quadratic smoothing replaces the non-differentiable absolute-value operator in the magnitude data-fidelity term, using ε > 0.This approach is termed quadratic smoothing (Quad-S), while alternative smoothing formulations are evaluated in an ablation study.
Optimization stability: Because the magnitude fidelity term introduces non-convexity, standard gradient descent may converge to poor local minima.The formulation uses learnable step sizes for gradient-descent iterations within each unrolled step.
Optimization stability: Nesterov acceleration is employed to improve stability in the data-fidelity subproblem, with learnable step sizes and a momentum coefficient.Polyak momentum and Adam are investigated as alternative momentum-based strategies in ablation studies.

IV. IMAGING EXPERIMENTS & IMPLEMENTATION DETAILS · A. Imaging Experiments

The imaging experiments used OCMR data with standardized preprocessing and retrospective time-interleaved undersampling at acceleration factors R ∈{6, 8}. Auxiliary magnitude information was obtained from neighboring cardiac phases, under approved data-sharing and IRB procedures.

IV. IMAGING EXPERIMENTS & IMPLEMENTATION DETAILS: All datasets were acquired under corresponding institutional review board approvals and data-sharing agreements.
IV. IMAGING EXPERIMENTS & IMPLEMENTATION DETAILS: Table I summarizes the datasets and acquisition parameters.
A. Imaging Experiments: The experiments used datasets from the OCMR database.
A. Imaging Experiments: Preprocessing removed readout oversampling, applied phase-encode zeropadding, and standardized spatial and temporal dimensions across subjects.
A. Imaging Experiments: Temporal interpolation standardized the data to 25 cardiac phases.
A. Imaging Experiments: Retrospective undersampling used a time-interleaved shifted equidistant sampling pattern without ACS lines.
A. Imaging Experiments: The retrospective experiments used acceleration factors R ∈{6, 8}.
A. Imaging Experiments: Auxiliary magnitude information was obtained from neighboring cardiac phases.

1) Segmented Breath-Hold Cine MRI: … 1) Network Architecture:

The study evaluates the framework on phase-contrast flow MRI and real-time cine MRI, alongside implementation of a 10-iteration C + Mag PD-DL network. Baselines share the same TE-UNet and ADMM unrolling framework, differing primarily in data-fidelity formulation.

1) Segmented Breath-Hold Cine MRI:: Axial aortic phase-contrast Flow2D MRI from the CMRxRecon2025 challenge dataset were included to evaluate applicability beyond cine imaging.The data used single-slice acquisitions with two velocity encodes per timeframe; preprocessing, retrospective undersampling, and auxiliary magnitude extraction matched the segmented cine experiments.
2) Phase-Contrast Flow MRI:: Free-breathing real-time cine MRI were acquired using balanced steady-state free precession and gradient-echo sequences.These acquisitions form the real-time cine MRI evaluation data.
3) Real-Time (RT) Cine MRI:: Real-time bSSFP cine data were acquired from the NIH Cardiac MRI Raw Data Repository at R = 4 and retrospectively undersampled to R = 8.The data used a time-interleaved shifted equidistant undersampling trajectory, retaining the k-space center line for each t.
3) Real-Time (RT) Cine MRI:: Real-time GRE cine MRI were acquired locally from 16 subjects at 3T with an acceleration factor of R = 8.The acquisitions used a short-axis view under an IRB-approved protocol.
B. Implementation Details: The proposed C + Mag PD-DL network was unrolled for T = 10 iterations using a TE-UNet proximal step and a 10-iteration Nesterov-accelerated GD data-fidelity unit.The TE-UNet used {32, 64, 96} encoder channels with symmetric decoding layers.
1) Network Architecture:: Conventional PD-DL baselines used the same TE-UNet proximal architecture and ADMM-based unrolling framework for fair comparison.They differed primarily in inverse-problem formulation and data-fidelity unit, using GD or conjugate gradient for the standard reconstruction objective.

2) Comparison Methods: … 1) Standard Quantitative Metrics:

The study compares the proposed magnitude-informed reconstruction with conventional methods, using supervised or self-supervised training across retrospective and prospective acquisitions. Evaluation combines image-quality metrics with cardiac function analysis, while experiments isolate auxiliary magnitude constraints through spatial-only regularization.

2) Comparison Methods:: Representative cine MRI reconstructions across acceleration rates and field strengths show that conventional PD-DL methods produce artifacts and blur, whereas the proposed method preserves sharp anatomical structures.The comparison covers R = {6, 8} and 0.55T, 1.5T, and 3.0T scanners.
2) Comparison Methods:: The complex-plus-magnitude formulation is architecture-agnostic and can be incorporated into VSQP, proximal-gradient, or alternative learned proximal-network schemes.ADMM unrolling is used in this work, but the framework is not tied to that algorithm or proximal architecture.
2) Comparison Methods:: Main experiments use spatial-only regularization to isolate the effect of auxiliary magnitude constraints at high acceleration rates.Spatiotemporal regularization is not explored, although the formulation is described as complementary to it.
3) Training:: Retrospectively undersampled segmented cine and phase-contrast flow MRI models were supervised using SENSE-1 coil-combined images as reference data.All training slices and timeframes were used; optimization employed normalized ℓ1-ℓ2 loss and Adam with a learning rate of 5 × 10^-4 for 100 epochs, reserving 10% for validation and tuning.
3) Training:: Prospectively undersampled real-time GRE and bSSFP acquisitions used multi-mask SSDU training with otherwise similar settings.The supplied passage specifies multi-mask SSDU and indicates that the training settings were otherwise similar.
C. Evaluation: For segmented cine and phase-contrast flow datasets with reference images, reconstruction fidelity was assessed using PSNR and SSIM.These metrics were used for quantitative image-quality assessment.
1) Standard Quantitative Metrics:: Cardiac function analysis was performed on segmented breath-hold cine and prospective real-time bSSFP datasets using Segment CMR.The analysis included four test subjects, according to the supplied passage.

2) Cardiac Function Analysis: … A. Retrospectively Accelerated Datasets

Retrospective cine and phase-contrast flow MRI evaluations compare conventional PD-DL with estimated-magnitude and oracle C + Mag reconstructions at accelerations R ∈{6, 8}. The proposed method suppresses undersampling artifacts while preserving cardiac anatomy, magnitude, and phase information, with quantitative and blinded expert assessments defined for these datasets.

A. Retrospectively Accelerated Datasets: Table II reports quantitative comparisons on retrospectively undersampled cine and phase-contrast flow2D MRI data at R ∈{6, 8}.The table includes C + Mag oracle results to show the inverse-problem upper performance bound, not a practical magnitude-estimation setting.
V. RESULTS: In phase-contrast flow MRI at R ∈{6, 8}, C + Mag substantially suppressed residual artifacts and preserved both magnitude and phase information.Conventional PD-DL reconstructions showed residual artifacts and phase inconsistencies in representative results.
2) Cardiac Function Analysis:: For segmented cine data, end-diastolic and end-systolic contours were used to derive EDV, ESV, SV, EF, EDM, and ESM from reconstructed images.Only multiple-slice acquisitions were used for quantification because multiple slices were required.
3) Qualitative Expert Evaluation:: Qualitative assessment used a blinded cardiologist’s 4-point Likert ratings of perceived SNR, blurring, aliasing artifacts, and overall image quality.The former dataset included only four test subjects with multiple slices.
3) Qualitative Expert Evaluation:: Blurring and aliasing were scored on four-level severity scales, and Wilcoxon signed-rank testing used a significance level of P < 0.05.The scoring ranges ran from none or mild through severe, depending on the criterion.
A. Retrospectively Accelerated Datasets: C + Mag was evaluated against conventional PD-DL using adjacent-phase magnitude estimates and an oracle ground-truth magnitude setting.The oracle configuration represents the inverse-problem upper performance bound and is not practical for magnitude estimation.
A. Retrospectively Accelerated Datasets: At R ∈{6, 8}, conventional PD-DL struggled with undersampling artifacts that degraded visualization of cardiac structures.These observations were reported for representative systolic and diastolic cardiac phases.
A. Retrospectively Accelerated Datasets: The proposed C + Mag PD-DL effectively removed artifacts while preserving cardiac anatomy in representative reconstructions.Population-average test-set metrics were summarized in Table II.

1) Segmented Breath-Hold Cine MR: … 2) Real-Time Cine MR (GRE):

Across segmented flow MRI and prospectively undersampled real-time cine MRI, C+Mag suppresses artifacts while preserving anatomical, magnitude, and phase information. In real-time cine, it achieves quality comparable to the clinical R = 4 tGRAPPA baseline despite R = 8 acceleration, with non-significant cardiac-function differences across metrics.

1) Segmented Breath-Hold Cine MR:: C+Mag suppresses residual artifacts while preserving magnitude and phase information in 2D phase-contrast flow MRI at R ∈ {6, 8}.Conventional PD-DL methods show phase inconsistencies, degraded magnitude reconstruction, and inaccurate phase-difference maps in some regions.
B. Prospectively Undersampled Acquisitions: In prospectively undersampled real-time cine acquisitions, fully sampled reference data are unavailable, preventing the oracle C + Mag experiment with ground-truth magnitude information.The experiment therefore evaluates magnitude-only information from nearby time-frames rather than ground-truth magnitudes.
B. Prospectively Undersampled Acquisitions: At R = 8, C + Mag using magnitude-only information from nearby time-frames improves real-time bSSFP cine reconstruction over conventional PD-DL methods.The comparison concerns representative prospectively undersampled real-time bSSFP cine data.
1) Real-Time Cine MR (bSSFP):: At R = 8, C+Mag achieves image quality comparable to clinically used tGRAPPA at R = 4 while using spatial-only regularization.The reconstruction preserves overall structures and suppresses reconstruction artifacts despite the higher acceleration factor.
1) Real-Time Cine MR (bSSFP):: At R = 8, conventional PD-DL improves on tGRAPPA but retains residual artifacts and anatomical-detail loss in systolic and diastolic phases.The passage describes tGRAPPA as exhibiting substantial artifacts at this high acceleration rate.
2) Real-Time Cine MR (GRE):: Cardiac-function differences between the proposed method and baseline were non-significant (P > 0.05) across all bSSFP cine metrics.Metrics include EDV, ESV, EF, SV, EDM, and ESM.
2) Real-Time Cine MR (GRE):: C+Mag preserves cardiac structures while suppressing noise and residual artifacts in prospectively undersampled real-time cine reconstructions.This behavior is reported for both systolic and diastolic phases.

C. Quantitavtive Cardiac Function Analysis … 1) Network architecture for proximal operator:

At R = 8, C + Mag closely matched reference cardiac-function measurements without statistical differences. Blinded readers found reconstructions comparable to references and superior to conventional PD-DL, while ablations identified architecture and optimization choices affecting performance.

C. Quantitavtive Cardiac Function Analysis: At R = 8, C + Mag showed excellent agreement with baseline acquisitions across cardiac-function parameters.The analysis used the most challenging acceleration setting and found volumetric and mass measurements closely matched reference values.
D. Expert Cardiologist Readings: In cine imaging, C + Mag improved cardiac-structure depiction and reduced reconstruction artifacts relative to conventional PD-DL.These improvements resulted in higher overall image-quality assessments.
D. Expert Cardiologist Readings: Blinded readers scored C + Mag comparably to reference acquisitions across perceived SNR, blurring, aliasing, and overall image quality.The evaluation covered segmented cine and phase-contrast flow acquisitions at R = 6 and R = 8, using lower scores to indicate better image quality.
1) Network architecture for proximal operator:: The time-embedded U-Net with unshared data-fidelity parameters achieved the best performance among the tested proximal-operator configurations.Regularizer parameters remained shared, and the experiments used simple GD for solving the data-fidelity subproblem.
D. Expert Cardiologist Readings: For phase-contrast flow imaging, C + Mag provided superior visualization of vascular structures and flow-related features compared with conventional PD-DL.Reader scores remained close to those of reference images despite undersampling, although statistical power was limited by the small number of multi-slice cases.
E. Ablation Studies: Ablations on retrospectively undersampled cine MRI at R = 8 varied proximal-operator architecture, momentum schemes, smoothing operators, and unrolling strategies.Population metrics were reported on the full test set to assess these framework components.
E. Ablation Studies: Among three momentum-based gradient-descent schemes for the data-fidelity subproblem, Nesterov-GD achieved the best performance.The proximal-operator architecture was fixed for this comparison.
E. Ablation Studies: The smoothing ablation compared the proposed quadratic smoothing with Smooth-ℓ1, Pseudo-Huber, and Log-Exp approximations of the absolute-value operator.These alternatives were evaluated for the formulation of Eq. (7).

2) Solving the data fidelity sub-problem: … VI. DISCUSSION AND CONCLUSION

The proposed magnitude-informed PD-DL framework uses an ADMM-based formulation with quadratic smoothing to incorporate auxiliary k-space magnitudes into dynamic MRI reconstruction. Ablations identify Quad-S smoothing and ADMM unrolling as strongest choices, while the framework remains compatible with future architectural and optimization advances.

3) Smoothing for the magnitude term:: Quad-S smoothing consistently achieves the best performance on the full test set despite its simplicity.Smooth-ℓ1, Pseudo-Huber, and Log-Exp alternatives were fine-tuned with δ = 1 for the first two and α = 20 for Log-Exp smoothing.
3) Smoothing for the magnitude term:: The smoothing comparison shows that none of the evaluated alternatives outperforms the proposed Quad-S formulation.The corresponding hyperparameters were fine-tuned on a small validation subset for maximal performance.
4) Unrolling strategies:: The study investigates different unrolling strategies to demonstrate that the proposed formulations are agnostic to the specifics of the algorithm unrolling strategy.
2) Solving the data fidelity sub-problem:: ADMM-based data-fidelity optimization achieves the best performance, outperforming variable splitting with quadratic penalty and proximal gradient descent.PGD substantially degrades because conventional PGD uses only a single data-fidelity update per unroll.
2) Solving the data fidelity sub-problem:: The ADMM formulation outperforms variable splitting with quadratic penalty, although variable splitting retains competitive quantitative performance.
VI. DISCUSSION AND CONCLUSION: The framework incorporates auxiliary magnitude k-space information into MRI reconstruction without additional acquisition cost by using neighboring time-frames in steady-state dynamic MRI.Experiments include retrospectively undersampled cine and phase-contrast flow MRI at R ∈ {6, 8}, together with prospectively undersampled real-time cine MRI.
VI. DISCUSSION AND CONCLUSION: The work’s central contribution is a new inverse-problem formulation for incorporating auxiliary magnitude information into physics-driven deep-learning MRI reconstruction.The formulation is developed particularly for dynamic MRI settings where auxiliary magnitudes require no additional acquisition costs.
VI. DISCUSSION AND CONCLUSION: Network architectures, regularizers, and optimization strategies are complementary to the proposed framework and can be naturally integrated within it.The authors note that ablation studies demonstrate compatibility with such extensions.