Unsupervised 4D Flow MRI Velocity Enhancement and Unwrapping Using Divergence-Free Neural Networks

Abstract: This work introduces an unsupervised Divergence and Aliasing-Free neural network (DAF-FlowNet) for 4D Flow Magnetic Resonance Imaging (4D Flow MRI) that jointly enhances noisy velocity fields and corrects phase wrapping artifacts. DAF-FlowNet parameterizes velocities as the curl of a vector potential, enforcing mass conservation by construction and avoiding explicit divergence-penalty tuning. A cosine data-consistency loss enables simultaneous denoising and unwrapping from wrapped phase images. On synthetic aortic 4D Flow MRI generated from computational fluid dynamics, DAF-FlowNet achieved lower errors than existing techniques (up to 11% lower velocity normalized root mean square error, 11% lower directional error, and 44% lower divergence relative to the best-performing alternative across noise levels), with robustness to moderate segmentation perturbations. For unwrapping, at peak velocity/velocity-encoding ratios of 1.4 and 2.1, DAF-FlowNet achieved 0.18% and 5.2% residual wrapped voxels, representing reductions of 72% and 18% relative to the best alternative method, respectively. In scenarios with both noise and aliasing, the proposed single-stage formulation outperformed a state-of-the-art sequential pipeline (up to 15% lower velocity normalized root mean square error, 11% lower directional error, and 28% lower divergence). Across 10 hypertrophic cardiomyopathy patient datasets, DAF-FlowNet preserved fine-scale flow features, corrected aliased regions, and improved internal flow consistency, as indicated by reduced inter-plane flow bias in aortic and pulmonary mass-conservation analyses recommended by the 4D Flow MRI consensus guidelines. These results support DAF-FlowNet as a framework that unifies velocity enhancement and phase unwrapping to improve the reliability of cardiovascular 4D Flow MRI.

• The paper introduces DAF-FlowNet, an unsupervised framework that parameterizes velocities as the curl of a learned vector potential to guarantee divergence-free fields.
• It leverages cosine-based data consistency loss and Fourier feature encoding to simultaneously denoise and unwrap phase-wrapped 4D Flow MRI data.
• Validated on synthetic and in-vivo datasets, it achieves lower RMSE, reduced divergence, and improved preservation of hemodynamic details over traditional methods.

Unsupervised Divergence- and Aliasing-Free Neural Networks for 4D Flow MRI Velocity Enhancement and Unwrapping

Introduction and Problem Scope

4D Flow MRI facilitates time-resolved, volumetric measurements of cardiovascular hemodynamics, offering crucial insight into global and regional flow parameters. Major barriers to clinical adoption stem from the inherently low velocity-to-noise ratio (VNR), suboptimal spatial-temporal resolution, and phase aliasing artifacts, chiefly phase wraps occurring when velocities surpass the VENC threshold. Accurate downstream physics-based biomarkers such as wall shear stress and pressure gradients demand highly reliable, divergence-free velocity fields with minimal aliasing and noise. Traditional supervised and physics-guided denoising methods balance data fidelity with physical constraints but require careful hyperparameter tuning, are sensitive to segmentation errors, or fail to address phase unwrapping robustly in low VENC or noisy settings.

The presented work targets these limitations via a novel, unified unsupervised approach, DAF-FlowNet, which parameterizes 4D Flow MRI velocities as the curl of a learned vector potential, thus guaranteeing mass conservation (divergence-free) by construction. Simultaneous denoising and unwrapping are achieved via a cosine-based data consistency loss, and the entire architecture sidesteps explicit divergence-penalty weighting. This representation yields robust velocity enhancements, phase unwrapping, and superior internal consistency in mass conservation metrics, as demonstrated on both synthetic and in-vivo patient cohorts.

Methodological Innovation

DAF-FlowNet encodes spatial coordinates through random Fourier features, enabling the full multilayer perceptron to synthesize a vector potential field $\mathbf{\Phi}$ over the masked fluid domain. The velocity $\mathbf{V}$ at each point is analytically computed as $\mathbf{V} = \nabla \times \mathbf{\Phi}$, inherently satisfying $\nabla \cdot \mathbf{V} = 0$. This physics-inspired, coordinate-based representation obviates the pitfalls of explicit regularization and improves adaptability across varying image domains via a domain transfer scaling factor on the Fourier bandwidth.

For restoration, a composite loss is minimized: (1) a cosine similarity term between predicted and measured (noisy, possibly wrapped) velocities—enabling unwrapping by leveraging periodicity; (2) a no-slip boundary loss enforcing zero velocity at the segmented vessel wall. Network optimization uses large coordinate sample batches to stabilize learning. Direct Fourier feature encoding allows tuning of spectral bias: a single scale parameter, $\sigma$, controls the trade-off between denoising and preservation of high-frequency flow structures.

Figure 1: The DAF-FlowNet pipeline with Fourier-encoded coordinates, MLP-based vector potential inference, and automatic differentiation for divergence-free velocities.

Synthetic Validation: Denoising, Divergence Minimization, and Unwrapping

Performance benchmarking (Figure 2) spans denoising, divergence minimization, and phase unwrapping on synthetically generated 4D Flow MRI datasets (physical CFD-constrained forward model with variable VENC and Gaussian noise). Across all tested noise levels, DAF-FlowNet yielded up to 11% lower normalized RMSE, up to 11% lower directional error, and 44% lower RMS divergence than any baseline (divergence-corrected, projection, and CNN-based).

Figure 2: Quantitative metrics (VelNRMSE, directional error, RMS divergence) for denoising/divergence minimization under varying noise on synthetic data.

Qualitative inspection (Figure 3) illustrates preserved core flow patterns and effective noise suppression; any residual error is primarily confined to vessel boundaries. DAF-FlowNet remains highly robust to moderate segmentation perturbations during enhancement, outperforming RBF and DFW which display sensitivity to mask dilation.

In unwrapping, DAF-FlowNet consistently achieves the lowest residual folded voxels (0.18% and 5.2% for VENC 150 and 100 cm/s, improvements of 72% and 18% over best comparators), though is less effective for very low VENC (70 cm/s), where classical graph-based approaches outperform.

Figure 3: Velocity magnitude, error maps, and wrapped/denoised outputs for synthetic experiments, demonstrating artifact removal and error localization.

Combined Enhancement and Unwrapping

A key result is the superior performance of DAF-FlowNet in simultaneous denoising and de-aliasing of phase-wrapped, noisy 4D Flow MRI. For challenging cases (VENC 100 cm/s with varying noise), DAF-FlowNet maintains lower velocity RMSE, reduced divergence, and fewer residual wraps than best sequential pipelines (GC3D + DFW), with gains up to 15% in velocity RMSE and 28% in divergence. Crucially, this is achieved in a single-stage, end-to-end unsupervised framework, avoiding the cumulative error of sequential post-processing.

Architectural Sensitivity and Spectral Bias

Hyperparameter importance analysis (Figure 4) underscores Fourier scale $\sigma$ as the dominant factor influencing denoising and feature retention. Insufficient $\sigma$ (too low) leads to underfitting and poor frequency recovery, while excessive $\sigma$ leads to noise amplification. Embedding size, network depth, and width are secondary. For domain transfer (e.g., patient data with different resolution/extents), optimal $\sigma$ is scaled accordingly.

Figure 4: Hyperparameter importance, highlighting the preeminent role of Fourier scale $\sigma$ over other architectural parameters.

In-Vivo Application and Mass Conservation Consistency

Clinical validation in hypertrophic cardiomyopathy (HCM) patients demonstrates generalizability. DAF-FlowNet effectively denoises, removes aliasing, and crucially preserves fine-grained hemodynamics essential for clinical measures. Unlike DFW, which sometimes oversmooths and degrades mass conservation, DAF-FlowNet yields consistent reductions in inter-plane flow biases: e.g., for HNCM aortic bias decreases from $\mathbf{V}$0 to $\mathbf{V}$1 L/min (relative bias from $\mathbf{V}$2 to $\mathbf{V}$3). Pulmonary flow bias is similarly improved, supporting higher fidelity in derived hemodynamic diagnostics.

Figure 5: Clinical in-vivo validation with (A) denoised diverged flows, (B) correction of aliasing in wrapped regions, and (C) preservation of aortic/pulmonary flow waveforms across processing methods.

Computational Efficiency and Practicality

A principal limitation is increased computational expense compared to traditional projection-based or sequential pipelines. DAF-FlowNet per-timeframe runtime exceeds 90 seconds for synthetic, and several minutes for in-vivo data. However, the parallelizable nature of coordinate-based MLPs, and possible meta-learning/fine-tuning initialization, may reduce this burden in future implementations.

Implications, Theoretical and Practical

• Physics-Constrained Representations: Embedding divergence-free structure directly into the parameterization, instead of penalizing divergence, achieves robust, domain-adaptive denoising and de-aliasing with less need for hyperparameter re-tuning and segmentation precision.
• Unified Denoising/Unwrapping: The cosine-based loss naturally facilitates aliasing correction in heterogeneous VENC and SNR regimes, enabling quality improvements even in challenging clinical acquisition scenarios.
• Generalizability: Fourier feature domain transfer and coordinate-based parameterization suit variable clinical and simulated imaging domains.

Future avenues include explicit multi-VENC integration to enhance global optimum attainment in extreme aliasing scenarios, adaptation to spatiotemporal super-resolution, and meta-learned initializations to decrease per-scan fitting times. Integration with more advanced Fourier embedding or spectral bias adaptation mechanisms (e.g., spatially adaptive/progressive encoding) could further enhance robustness.

Conclusion

The DAF-FlowNet framework establishes the practical and theoretical value of physics-constrained, coordinate-based neural representations for the restoration of 4D Flow MRI velocity fields. By parameterizing velocities as the curl of a learned vector potential and directly embedding mass conservation, it achieves superior denoising, aliasing correction, and mass-conservation consistency against a broad range of baselines. The approach sets a new methodological foundation for MRI-based hemodynamics, particularly for quantitative cardiovascular analysis, with natural extensions to demanding clinical and research imaging scenarios.