# A Conditional Normalizing Flow for Accelerated Multi-Coil MR Imaging

arXiv: https://arxiv.org/abs/2306.01630 · HTML: https://mlsanity.com/paper/2306.01630/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 54 references that are in the map.
- First posted on arXiv 2 Jun 2023.

## Builds on (most cited first, 30 of 35)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Score-based diffusion models for accelerated MRI](https://mlsanity.com/paper/2110.05243/) (2021)
- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [SRFlow: Learning the Super-Resolution Space with Normalizing Flow](https://mlsanity.com/paper/2006.14200/) (2020)
- [End-to-End Variational Networks for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2004.06688/) (2020)
- [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (2019)
- [Guided Image Generation with Conditional Invertible Neural Networks](https://mlsanity.com/paper/1907.02392/) (2019)
- [Explaining Image Classifiers by Counterfactual Generation](https://mlsanity.com/paper/1807.08024/) (2018)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [Deep Learning Methods for Parallel Magnetic Resonance Image Reconstruction](https://mlsanity.com/paper/1904.01112/) (2019)
- [Solving Linear Inverse Problems Using the Prior Implicit in a Denoiser](https://mlsanity.com/paper/2007.13640/) (2020)
- [Amortised MAP Inference for Image Super-resolution](https://mlsanity.com/paper/1610.04490/) (2016)
- [Deep Bayesian Inversion](https://mlsanity.com/paper/1811.05910/) (2018)
- [Uncertainty Quantification in Deep MRI Reconstruction](https://mlsanity.com/paper/1901.11228/) (2019)
- [Deep Decomposition Learning for Inverse Imaging Problems](https://mlsanity.com/paper/1911.11028/) (2019)
- [Conditional Invertible Neural Networks for Diverse Image-to-Image Translation](https://mlsanity.com/paper/2105.02104/) (2021)
- [Scalable Bayesian uncertainty quantification in imaging inverse problems via convex optimization](https://mlsanity.com/paper/1803.00889/) (2018)

## Built on by (most cited first, 10 of 10)

- [WISE: full-Waveform variational Inference via Subsurface Extensions](https://mlsanity.com/paper/2401.06230/) (2023)
- [Solving multiphysics-based inverse problems with learned surrogates and constraints](https://mlsanity.com/paper/2307.11099/) (2023)
- [Task-Driven Uncertainty Quantification in Inverse Problems via Conformal Prediction](https://mlsanity.com/paper/2405.18527/) (2024)
- [MRI Reconstruction with Regularized 3D Diffusion Model (R3DM)](https://mlsanity.com/paper/2412.18723/) (2024)
- [pcaGAN: Improving Posterior-Sampling cGANs via Principal Component Regularization](https://mlsanity.com/paper/2411.00605/) (2024)
- [Conformal Bounds on Full-Reference Image Quality for Imaging Inverse Problems](https://mlsanity.com/paper/2505.09528/) (2025)
- [Minimax Multi-Target Conformal Prediction with Applications to Imaging Inverse Problems](https://mlsanity.com/paper/2511.13533/) (2025)
- [Advancing MRI Reconstruction: A Systematic Review of Deep Learning and Compressed Sensing Integration](https://mlsanity.com/paper/2501.14158/) (2025)
- [Active MRI Acquisition with Diffusion Guided Bayesian Experimental Design](https://mlsanity.com/paper/2506.16237/) (2025)
- [Separating Intrinsic Ambiguity from Estimation Uncertainty in Deep Generative Models for Linear Inverse Problems](https://mlsanity.com/paper/2605.15050/) (2026)

## Abstract (arXiv)

Accelerated magnetic resonance (MR) imaging attempts to reduce acquisition time by collecting data below the Nyquist rate. As an ill-posed inverse problem, many plausible solutions exist, yet the majority of deep learning approaches generate only a single solution. We instead focus on sampling from the posterior distribution, which provides more comprehensive information for downstream inference tasks. To do this, we design a novel conditional normalizing flow (CNF) that infers the signal component in the measurement operator's nullspace, which is later combined with measured data to form complete images. Using fastMRI brain and knee data, we demonstrate fast inference and accuracy that surpasses recent posterior sampling techniques for MRI. Code is available at https://github.com/jwen307/mri_cnf/