# Bayesian Uncertainty Estimation of Learned Variational MRI Reconstruction

arXiv: https://arxiv.org/abs/2102.06665 · HTML: https://mlsanity.com/paper/2102.06665/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 43 references that are in the map.
- First posted on arXiv 12 Feb 2021.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [A Deep Cascade of Convolutional Neural Networks for Dynamic MR Image Reconstruction](https://mlsanity.com/paper/1704.02422/) (2017)
- [End-to-End Variational Networks for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2004.06688/) (2020)
- [An overview of deep learning in medical imaging focusing on MRI](https://mlsanity.com/paper/1811.10052/) (2018)
- [Deep Convolutional Networks as shallow Gaussian Processes](https://mlsanity.com/paper/1808.05587/) (2018)
- [Deep Learning Methods for Parallel Magnetic Resonance Image Reconstruction](https://mlsanity.com/paper/1904.01112/) (2019)
- [A Mean-Field Optimal Control Formulation of Deep Learning](https://mlsanity.com/paper/1807.01083/) (2018)
- [Advancing machine learning for MR image reconstruction with an open competition: Overview of the 2019 fastMRI challenge](https://mlsanity.com/paper/2001.02518/) (2020)
- [Total Deep Variation for Linear Inverse Problems](https://mlsanity.com/paper/2001.05005/) (2020)
- [Uncertainty Quantification in Deep MRI Reconstruction](https://mlsanity.com/paper/1901.11228/) (2019)
- [i-RIM applied to the fastMRI challenge](https://mlsanity.com/paper/1910.08952/) (2019)
- [Simulating single-coil MRI from the responses of multiple coils](https://mlsanity.com/paper/1811.08026/) (2018)
- [Total Deep Variation: A Stable Regularizer for Inverse Problems](https://mlsanity.com/paper/2006.08789/) (2020)
- [Sigma-net: Systematic Evaluation of Iterative Deep Neural Networks for Fast Parallel MR Image Reconstruction](https://mlsanity.com/paper/1912.09278/) (2019)

## Built on by (most cited first, 30 of 30)

- [A Review of Uncertainty Estimation and its Application in Medical Imaging](https://mlsanity.com/paper/2302.08119/) (2023)
- [Physics-Driven Deep Learning for Computational Magnetic Resonance Imaging](https://mlsanity.com/paper/2203.12215/) (2022)
- [Bayesian MRI Reconstruction with Joint Uncertainty Estimation using Diffusion Models](https://mlsanity.com/paper/2202.01479/) (2022)
- [A review of uncertainty quantification in medical image analysis: probabilistic and non-probabilistic methods](https://mlsanity.com/paper/2310.06873/) (2023)
- [Solving Inverse Problems with FLAIR](https://mlsanity.com/paper/2506.02680/) (2025)
- [Learning Regularization Functionals for Inverse Problems: A Comparative Study](https://mlsanity.com/paper/2510.01755/) (2025)
- [A review and experimental evaluation of deep learning methods for MRI reconstruction](https://mlsanity.com/paper/2109.08618/) (2021)
- [Posterior-Variance-Based Error Quantification for Inverse Problems in Imaging](https://mlsanity.com/paper/2212.12499/) (2022)
- [Uncertainty Estimation and Propagation in Accelerated MRI Reconstruction](https://mlsanity.com/paper/2308.02631/) (2023)
- [Task-Driven Uncertainty Quantification in Inverse Problems via Conformal Prediction](https://mlsanity.com/paper/2405.18527/) (2024)
- [Propagation and Attribution of Uncertainty in Medical Imaging Pipelines](https://mlsanity.com/paper/2309.16831/) (2023)
- [NPB-REC: A Non-parametric Bayesian Deep-learning Approach for Undersampled MRI Reconstruction with Uncertainty Estimation](https://mlsanity.com/paper/2404.04550/) (2024)
- [Posterior temperature optimized Bayesian models for inverse problems in medical imaging](https://mlsanity.com/paper/2202.00986/) (2022)
- [PixCUE: Joint Uncertainty Estimation and Image Reconstruction in MRI using Deep Pixel Classification](https://mlsanity.com/paper/2303.00111/) (2023)
- [Uncertainty Estimation and Out-of-Distribution Detection for Deep Learning-Based Image Reconstruction using the Local Lipschitz](https://mlsanity.com/paper/2305.07618/) (2023)
- [Conformal Bounds on Full-Reference Image Quality for Imaging Inverse Problems](https://mlsanity.com/paper/2505.09528/) (2025)
- [Guiding Quantitative MRI Reconstruction with Phase-wise Uncertainty](https://mlsanity.com/paper/2502.20877/) (2025)
- [A Comprehensive Survey on Magnetic Resonance Image Reconstruction](https://mlsanity.com/paper/2503.07097/) (2025)
- [BUFF: Bayesian Uncertainty Guided Diffusion Probabilistic Model for Single Image Super-Resolution](https://mlsanity.com/paper/2504.03490/) (2025)
- [Efficient Noise Calculation in Deep Learning-based MRI Reconstructions](https://mlsanity.com/paper/2505.02007/) (2025)
- [OCSplats: Observation Completeness Quantification and Label Noise Separation in 3DGS](https://mlsanity.com/paper/2508.01239/) (2025)
- [CUTE-MRI: Conformalized Uncertainty-based framework for Time-adaptivE MRI](https://mlsanity.com/paper/2508.14952/) (2025)
- [ReDiff: Reliability-Guided Diffusion for Trustworthy Ultra-Low-Field to High-Field MRI Synthesis](https://mlsanity.com/paper/2603.11325/) (2026)
- [MotionDPS: Motion-Compensated 3D Brain MRI Reconstruction](https://mlsanity.com/paper/2605.22121/) (2026)
- [SoftDropConnect (SDC) -- Effective and Efficient Quantification of the Network Uncertainty in Deep MR Image Analysis](https://mlsanity.com/paper/2201.08418/) (2022)
- [R2D2 image reconstruction with model uncertainty quantification in radio astronomy](https://mlsanity.com/paper/2403.18052/) (2024)
- [Evaluating structural uncertainty in accelerated MRI: are voxelwise measures useful surrogates?](https://mlsanity.com/paper/2503.10527/) (2025)
- [Uncertainty Estimation for Trust Attribution to Speed-of-Sound Reconstruction with Variational Networks](https://mlsanity.com/paper/2504.11307/) (2025)
- [Pixelwise Uncertainty Quantification of Accelerated MRI Reconstruction](https://mlsanity.com/paper/2601.13236/) (2026)
- [Measured-Subspace Consistency: A Plug-and-Play Operator for Diffusion Posterior Sampling in Accelerated MRI Reconstruction](https://mlsanity.com/paper/2606.28448/) (2026)

## Abstract (arXiv)

Recent deep learning approaches focus on improving quantitative scores of dedicated benchmarks, and therefore only reduce the observation-related (aleatoric) uncertainty. However, the model-immanent (epistemic) uncertainty is less frequently systematically analyzed. In this work, we introduce a Bayesian variational framework to quantify the epistemic uncertainty. To this end, we solve the linear inverse problem of undersampled MRI reconstruction in a variational setting. The associated energy functional is composed of a data fidelity term and the total deep variation (TDV) as a learned parametric regularizer. To estimate the epistemic uncertainty we draw the parameters of the TDV regularizer from a multivariate Gaussian distribution, whose mean and covariance matrix are learned in a stochastic optimal control problem. In several numerical experiments, we demonstrate that our approach yields competitive results for undersampled MRI reconstruction. Moreover, we can accurately quantify the pixelwise epistemic uncertainty, which can serve radiologists as an additional resource to visualize reconstruction reliability.