UMPIRE-Net: Unrolled Magnitude-Phase Regularization Network for Accelerated MRI

Abstract: MRI reconstruction from undersampled k-space measurements is an ill-posed inverse problem. Physics-driven deep learning (PD-DL) methods have shown strong performance for this task by combining the MRI forward model with learned image regularization within algorithm-unrolling frameworks. However, most existing PD-DL methods reconstruct complex-valued images directly, thereby implicitly coupling magnitude and phase within a single learned representation. This coupled regularization may be suboptimal in reconstruction settings where accurate phase modeling plays an important role, such as partial Fourier (PF) imaging, where recovery of the omitted asymmetric k-space measurements depends on the underlying image phase. In such scenarios, explicit modeling of magnitude and phase as separate components may reduce the reliance on externally estimated or predefined phase information. To this end, we propose UMPIRE-Net (Unrolled Magnitude-Phase In REgularization Network), a PD-DL method that introduces separate learned regularizers for magnitude and phase components, together with a novel data-fidelity formulation that enforces measurements consistency. We evaluate UMPIRE-Net for accelerated MRI with PF across different datasets and acceleration factors. Experimental results demonstrate that our proposed method improves reconstruction quality compared with a conventional complex-valued PD-DL baseline, yielding sharper images and reduced artifacts. Code available at: https://github.com/MahdiSaberii/UMPIRE-Net

• The paper introduces UMPIRE-Net, an ADMM-unrolled MRI reconstruction network that separately learns magnitude and sign priors with a smoothed, Nesterov-accelerated data-fidelity unit.
• UMPIRE-Net consistently outperforms a matched complex-valued baseline, achieving up to 1.77 dB higher PSNR, improved SSIM, and approximately 4° lower phase error at acceleration rates of 6 and 8.
• The results show that explicit phase-aware modeling can recover missing partial-Fourier k-space data under self-supervised training, although benefits decrease for noisier fat-saturated knee images.

UMPIRE-Net addresses a structural limitation of physics-driven deep learning (PD-DL) reconstruction for accelerated MRI: conventional unrolled networks regularize the complex-valued image with a single learned prior, implicitly coupling magnitude and phase. In partial Fourier (PF) imaging—where recovery of the omitted asymmetric k-space region depends critically on image phase—this coupling may be suboptimal. The paper, presented at the 2026 IEEE MLSP workshop (2608.14422), proposes an ADMM-unrolled network with separate learned proximal operators for magnitude and sign components, plus a differentiable data-fidelity (DF) unit tailored to this decomposition, and demonstrates consistent gains over a complex-valued PD-DL baseline under self-supervised training.

Motivation and related work

Classical PF methods such as homodyne reconstruction and POCS explicitly exploit phase information to synthesize missing asymmetric k-space samples. Prior separate magnitude/phase approaches either operate in non-DL optimization frameworks (CS with dedicated phase regularization, or phase cycling against phase wrapping) or place neural networks outside a physics-driven unrolled loop. UMPIRE-Net's positioning is to combine both: explicit magnitude/phase decomposition inside an algorithm-unrolling framework. The authors also note a practical training concern—in self-supervised setups such as MM-SSDU, the non-acquired PF region is excluded from the loss mask, giving the network limited direct supervision for recovering those samples; explicit phase modeling is hypothesized to mitigate this.

Method

The reconstruction problem is reformulated with independent regularizers on magnitude and sign:

$$\arg\min_{\mathbf{x}} \|\mathbf{y}_\Omega - \mathbf{E}_\Omega \mathbf{x}\|_2^2 + \mathcal{R}_m(|\mathbf{x}|) + \mathcal{R}_p\left(\tfrac{\mathbf{x}}{|\mathbf{x}|}\right)$$

Regularizing the sign rather than the phase follows prior CS work, since direct phase regularization yields a non-convex objective with spurious local minima from periodicity. ADMM splitting produces three subproblems: magnitude and sign proximal updates (implemented as neural networks) and a DF update that is itself non-convex due to the $|\cdot|$ and sign terms. The authors derive the Wirtinger (CR-calculus) gradient after replacing $|\mathbf{x}|$ with a quadratic smoothing approximation, $(\mathbf{x}\odot\mathbf{x}^H + \epsilon)^{1/2}$, with $\epsilon$ scaled to the zero-filled image energy.

Because the DF subproblem is non-convex, plain gradient descent can stall in local minima. The paper therefore unrolls Nesterov-accelerated GD with learnable step sizes and momentum for the DF unit. All DF parameters ($\beta_1$, $\beta_2$, $\gamma$, steps $\xi$, dual coefficients $\lambda_1$, $|\cdot|$0) are learned and unshared across the $|\cdot|$1 outer unrolls; the DF itself runs 10 inner iterations. Proximal networks use time-embedded U-Nets (TE-UNet) with a ReLU output constraint for magnitude non-negativity.

Experimental setup

Evaluation uses fastMRI coronal proton-density knee data with and without fat saturation (Cor-PD, Cor-PDFS), 300 training slices from 10 subjects and 392 test slices, with retrospective equispaced in-plane undersampling at $|\cdot|$2 (24 ACS lines) combined with PF $|\cdot|$3 along the phase-encoding direction. Training is multi-mask self-supervised (MM-SSDU, loss-mask ratio $|\cdot|$4) with the PF region held out of the training masks, using a normalized $|\cdot|$5-$|\cdot|$6 loss. The conventional PD-DL baseline is a modified TE-UNet in the same ADMM unrolling with 10 CG iterations per DF step, isolating the contribution of the inverse-problem formulation and DF design.

Results

Across both datasets and both acceleration rates, UMPIRE-Net with Nesterov-accelerated DF achieves the best PSNR, SSIM, and phase accuracy. Representative numbers:

Method	Cor-PD R=6 PSNR/SSIM/$|\cdot|$7	Cor-PD R=8	Cor-PDFS R=6	Cor-PDFS R=8
Conv. PD-DL (CG)	35.55 / 0.914 / 19.80°	33.64 / 0.893 / 20.96°	32.54 / 0.750 / 30.80°	31.47 / 0.729 / 31.57°
UMPIRE (GD)	36.42 / 0.914 / 16.60°	35.11 / 0.896 / 16.90°	32.69 / 0.763 / 27.55°	32.18 / 0.746 / 28.79°
UMPIRE (Nesterov)	36.76 / 0.919 / 15.93°	35.41 / 0.902 / 16.82°	32.83 / 0.768 / 26.78°	32.26 / 0.763 / 27.43°

The largest magnitude gains occur on Cor-PD at R=8 (+1.77 dB PSNR), and phase error drops consistently by roughly 4° across all settings. Notably, these gains are achieved under self-supervised training with no fully sampled reference, and the DF region is never directly supervised—supporting the claim that the decomposition itself improves inference of missing PF k-space. Gains on Cor-PDFS are milder, which the authors attribute to lower SNR and noisier phase; this is a candid limitation on how much benefit phase-aware regularization provides when phase estimation is intrinsically difficult. Qualitatively, residual artifacts from conventional PD-DL are partially reduced by UMPIRE with plain GD and eliminated with Nesterov acceleration, indicating the momentum-based DF is a material component of the result rather than an implementation detail.

An ablation of smoothing operators for the DF (Cor-PD, R=8) shows quadratic smoothing (35.11 dB / 0.896 SSIM) outperforming Huber (34.63 / 0.886) and Log-Exp (32.53 / 0.880), suggesting the specific smoothing choice near zero is consequential for the non-convex DF.

Limitations and open questions

The paper concedes several points. Evaluation is restricted to Cartesian knee data at PF $|\cdot|$8 with equispaced undersampling; generalization to other anatomies, trajectories, and PF factors is untested. The benefit shrinks on noisier, fat-saturated data, leaving open how the method behaves in strongly phase-varying regimes such as multi-echo or off-resonance imaging—exactly the settings the authors identify for future evaluation. The DF subproblem remains non-convex even with smoothing and Nesterov momentum, and no convergence guarantees are provided for the unrolled non-convex ADMM. The choice of sign (rather than phase) as the regularized component is inherited from CS practice and is not ablated against alternative decompositions.

Conclusion

UMPIRE-Net shows that decoupling magnitude and phase regularization within a physics-driven unrolled framework, paired with a smoothed, Nesterov-accelerated DF unit, yields measurable improvements—up to +1.77 dB PSNR and ~4° lower phase error—over a matched complex-valued PD-DL baseline for self-supervised partial Fourier MRI reconstruction. The results support the premise that explicit phase-aware modeling reduces reliance on externally estimated phase information, while leaving the extension to spatially varying phase applications and the theoretical behavior of the non-convex DF as open questions.