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

arXiv: https://arxiv.org/abs/2604.00205 · HTML: https://mlsanity.com/paper/2604.00205/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 51 references that are in the map.
- First posted on arXiv 31 Mar 2026.

## Builds on (most cited first, 9 of 9)

- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Learned Initializations for Optimizing Coordinate-Based Neural Representations](https://mlsanity.com/paper/2012.02189/) (2020)
- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [SAPE: Spatially-Adaptive Progressive Encoding for Neural Optimization](https://mlsanity.com/paper/2104.09125/) (2021)
- [4DFlowNet: Super-Resolution 4D Flow MRI using Deep Learning and Computational Fluid Dynamics](https://mlsanity.com/paper/2004.07035/) (2020)
- [FreSh: Frequency Shifting for Accelerated Neural Representation Learning](https://mlsanity.com/paper/2410.05050/) (2024)
- [Implicit neural representations for unsupervised super-resolution and denoising of 4D flow MRI](https://mlsanity.com/paper/2302.12835/) (2023)
- [VAST: Vascular Flow Analysis and Segmentation for Intracranial 4D Flow MRI](https://mlsanity.com/paper/2601.13393/) (2026)

## Built on by (most cited first, 1 of 1)

- [Learning in the Transverse Subspace: A Minimal Representation for Divergence-Free Operator Learning](https://mlsanity.com/paper/2609.35884/) (2026)

## Abstract (arXiv)

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.