Bayesian Uncertainty-Aware MRI Reconstruction

Ahmed Karam Eldaly; Matteo Figini; Daniel C. Alexander

eess.IVMar 13, 2026

Bayesian Uncertainty-Aware MRI Reconstruction

Authors: Ahmed Karam Eldaly, Matteo Figini, Daniel C. Alexander

Organizations: Department of Computer Science, University of Exeter, UK · UCL Hawkes Institute, Department of Computer Science, University College London, UK

Abstract

We propose a novel framework for joint magnetic resonance image reconstruction and uncertainty quantification using under-sampled k-space measurements. The problem is formulated as a Bayesian linear inverse problem, where prior distributions are assigned to the unknown model parameters. Specifically, we assume the target image is sparse in its spatial gradient and impose a total variation prior model. A Markov chain Monte Carlo (MCMC) method, based on a split-and-augmented Gibbs sampler, is then used to sample from the resulting joint posterior distribution of the unknown parameters. Experiments conducted using single- and multi-coil datasets demonstrate the superior performance of the proposed framework over optimisation-based compressed sensing algorithms. Additionally, our framework effectively quantifies uncertainty, showing strong correlation with error maps computed from reconstructed and ground-truth images.

Figures & tables

	Under-sampling Ratio
Method	5%	10%	20%	30%	40%
IFFT	19.85 $\pm$ 2.8	15.32 $\pm$ 2.5	12.08 $\pm$ 2.2	9.72 $\pm$ 1.8	7.67 $\pm$ 1.5
ADMM-Wav	5.62 $\pm$ 1.5	3.71 $\pm$ 1.2	2.26 $\pm$ 1.0	1.45 $\pm$ 0.8	0.98 $\pm$ 0.6
ADMM-TV	5.56 $\pm$ 1.5	3.68 $\pm$ 1.2	2.25 $\pm$ 1.0	1.42 $\pm$ 0.8	0.98 $\pm$ 0.6
MCMC-TV	5.65 $\pm$ 1.6	3.60 $\pm$ 1.2	2.11 $\pm$ 1.0	1.30 $\pm$ 0.8	0.90 $\pm$ 0.6

Table 1: Average and standard deviation of RMSE using different under-sampling ratios. Bold indicates best, and underlined indicates second best results.

SR	5%	10%	20%	30%	40%
CC	0.80 $\pm$ .068	0.79 $\pm$ .059	0.79 $\pm$ .051	0.75 $\pm$ .047	0.74 $\pm$ .045

Table 2: Average correlation coefficient (CC) computed between marginal standard deviation and error maps.

Figure 1: Results of (a) image reconstruction of a brain test image from HCP data set using the four tested methods at increasing under-sampling ratio.

Figure 2: Error map images computed between ground truth and image estimates in Fig. 1 .

Figure 3: Marginal standard deviation of the MCMC-TV estimates in Fig. 1 .

Figure 4: Results of reconstruction of three brain test images from the real low-field MRI M4Raw using multicoil data at 20% under-sampling ratio.

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Trustworthy MRI Reconstruction via Bayesian Uncertainty Quantification with Sparsity Prior Models

Ahmed Karam Eldaly, Matteo Figini, Daniel C. Alexander

Department of Computer Science, University of Exeter, Exeter EX4 4QF, U.K. · UCL Hawkes Institute, Department of Computer Science, University College London, London, U.K.

Sparse Recovery Uncertainty Quantification Bayesian Inverse Problems Accelerated MRI Reconstruction Magnetic Resonance Imaging

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We propose a novel Bayesian framework for joint image reconstruction and uncertainty quantification from compressed sensing magnetic resonance imaging data. The problem is formulated as a linear inverse problem, where prior distributions are assigned to the unknown image parameters. Specifically, the image is assumed to be sparse in a given transform domain. We develop a general framework applicable to any sparsifying transform and demonstrate its performance using (1) a total variation transform based on image spatial gradients and (2) a wavelet-domain transform. Bayesian inference is performed using a split-and-augmented Gibbs sampler, while the resulting non-differentiable conditional distributions are efficiently sampled using a proximal Markov chain Monte Carlo method. The proposed algorithms are validated on both single-coil and multi-coil datasets using various k-space sampling patterns and acceleration factors. The results demonstrate that the proposed Bayesian methods consistently outperform their optimisation-based counterparts in image reconstruction while providing uncertainty estimates for the reconstructed images. Furthermore, the estimated uncertainty maps show a strong correlation with the true reconstruction errors and substantially outperformed deep learning-based uncertainty estimation methods.

Abstract

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