# Physics-informed compressed sensing for PC-MRI: an inverse Navier-Stokes problem

arXiv: https://arxiv.org/abs/2207.01466 · HTML: https://mlsanity.com/paper/2207.01466/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 58 references that are in the map.
- First posted on arXiv 4 Jul 2022.

## Builds on (most cited first, 4 of 4)

- [Can stable and accurate neural networks be computed? -- On the barriers of deep learning and Smale's 18th problem](https://mlsanity.com/paper/2101.08286/) (2021)
- [4DFlowNet: Super-Resolution 4D Flow MRI using Deep Learning and Computational Fluid Dynamics](https://mlsanity.com/paper/2004.07035/) (2020)
- [Variational regularisation for inverse problems with imperfect forward operators and general noise models](https://mlsanity.com/paper/2005.14131/) (2020)
- [Joint phase reconstruction and magnitude segmentation from velocity-encoded MRI data](https://mlsanity.com/paper/1908.05285/) (2019)

## Built on by (most cited first, 6 of 6)

- [Implicit neural representations for unsupervised super-resolution and denoising of 4D flow MRI](https://mlsanity.com/paper/2302.12835/) (2023)
- [Physics-informed graph neural networks for flow field estimation in carotid arteries](https://mlsanity.com/paper/2408.07110/) (2024)
- [Bayesian inverse Navier-Stokes problems: joint flow field reconstruction and parameter learning](https://mlsanity.com/paper/2406.18464/) (2024)
- [Image Reconstruction for Accelerated MR Scan with Faster Fourier Convolutional Neural Networks](https://mlsanity.com/paper/2306.02886/) (2023)
- [Using Powerful Prior Knowledge of Diffusion Model in Deep Unfolding Networks for Image Compressive Sensing](https://mlsanity.com/paper/2503.08429/) (2025)
- [Sampling Innovation-Based Adaptive Compressive Sensing](https://mlsanity.com/paper/2503.13241/) (2025)

## Abstract (arXiv)

We formulate a physics-informed compressed sensing (PICS) method for the reconstruction of velocity fields from noisy and sparse phase-contrast magnetic resonance signals. The method solves an inverse Navier-Stokes boundary value problem, which permits us to jointly reconstruct and segment the velocity field, and at the same time infer hidden quantities such as the hydrodynamic pressure and the wall shear stress. Using a Bayesian framework, we regularize the problem by introducing a priori information about the unknown parameters in the form of Gaussian random fields. This prior information is updated using the Navier-Stokes problem, an energy-based segmentation functional, and by requiring that the reconstruction is consistent with the k-space signals. We create an algorithm that solves this reconstruction problem, and test it for noisy and sparse k-space signals of the flow through a converging nozzle. We find that the method is capable of reconstructing and segmenting the velocity fields from sparsely-sampled (15% k-space coverage), low (~10) signal-to-noise ratio (SNR) signals, and that the reconstructed velocity field compares well with that derived from fully-sampled (100% k-space coverage) high (>40) SNR signals of the same flow.