The ability to reconstruct high-quality images from undersampled MRI data is vital in improving MRI temporal resolution and reducing acquisition times. Deep learning methods have been proposed for this task, but the lack of verified methods to quantify the uncertainty in the reconstructed images hampered clinical applicability. We introduce "NPB-REC", a non-parametric fully Bayesian framework, for MRI reconstruction from undersampled data with uncertainty estimation. We use Stochastic Gradient Langevin Dynamics during training to characterize the posterior distribution of the network parameters. This enables us to both improve the quality of the reconstructed images and quantify the uncertainty in the reconstructed images. We demonstrate the efficacy of our approach on a multi-coil MRI dataset from the fastMRI challenge and compare it to the baseline End-to-End Variational Network (E2E-VarNet). Our approach outperforms the baseline in terms of reconstruction accuracy by means of PSNR and SSIM ($34.55$, $0.908$ vs. $33.08$, $0.897$, $p<0.01$, acceleration rate $R=8$) and provides uncertainty measures that correlate better with the reconstruction error (Pearson correlation, $R=0.94$ vs. $R=0.91$). Additionally, our approach exhibits better generalization capabilities against anatomical distribution shifts (PSNR and SSIM of $32.38$, $0.849$ vs. $31.63$, $0.836$, $p<0.01$, training on brain data, inference on knee data, acceleration rate $R=8$). NPB-REC has the potential to facilitate the safe utilization of deep learning-based methods for MRI reconstruction from undersampled data. Code and trained models are available at \url{https://github.com/samahkh/NPB-REC}.

通过非参数全贝叶斯框架NPB-REC，可以从欠采样数据中重建MRI图像，并对重建图像的不确定性进行估计。此方法改进了重建图像的质量，并提供了更好的与重建误差相关的不确定性度量，对MRI重建的深度学习方法的安全应用具有潜力。

NPB-REC：一种非参数贝叶斯深度学习方法用于不完全采样磁共振重构及不确定性估计