Structure-Preserving Correction Learning for Sparse Bayesian Inference in Brain Source Imaging
Authors: Marco Morik, Xiao Ruiting, Shinichi Nakajima, Stefan Haufe, Ismail Huseynov
Abstract
Classical sparse Type-II Bayesian methods for M/EEG brain imaging support joint estimation of source and noise hyperparameters, but rely on fixed iterative update rules. Although these updates are principled and interpretable, their dynamics cannot be adapted from data. We propose to learn the update mechanism itself while preserving the underlying Bayesian structure by unfolding a classical joint hyperparameter-learning solver into a trainable neural architecture whose layers mirror the original iterations. The resulting framework is initialized to recover the classical solver exactly before training and is enriched through progressively more expressive correction-learning mechanisms, ranging from learnable biases to adaptive MLP and attention-based contextual refinements. In this way, training does not replace Bayesian inference with a black-box predictor, but instead learns structured correction terms while retaining the interpretability and model-based character of the original update dynamics. Structured correction learning therefore aims to improve empirical reconstruction performance without replacing the original model-based inference mechanism. Experimental results show that the learned correction variants improve reconstruction performance and convergence behavior over the baseline unfolded solver while preserving its algorithmic transparency.
Objective: Sparse Bayesian learning provides an effective framework to solve high-dimensional problems in brain signal decoding. However, conventional likelihoods regarding data distributions, such as Gaussian or Bernoulli, are potentially inadequate for handling the noisy recordings of brain activity. Hence, this work aims to formulate a robust sparse Bayesian learning framework to address noisy high-dimensional brain activity decoding. Methods: Motivated by the commendable robustness of the minimum error entropy learning criterion for addressing non-Gaussian signals, this study reformulated the sparse Bayesian learning framework under a generalized Bayesian paradigm, in which the model parameter is regulated with the minimum error entropy loss rather than a conventional likelihood function. Results: Our developed SBL-MEE algorithm was evaluated with two real-world brain decoding tasks of regression and classification scenarios, respectively. Experimental results demonstrated that our approach not only realizes superior brain decoding performance than existing methods, but also presents more physiologically interpretable decoder patterns. Conclusion: Although minimum error entropy is not constructed from an arbitrary probabilistic distribution, it is effective to establish noise-robust inference in sparse Bayesian learning method. Significance: This work provides a powerful tool to improve brain activity decoding capability, particularly regarding the noisy high-dimensional setting, thus promoting biomedical engineering applications such as brain-computer interface.
Magnetic resonance imaging (MRI) is highly susceptible to patient motion due to its relatively long acquisition times and the fact that data are acquired sequentially in k-space. Even small patient movements introduce phase inconsistencies across measurements, leading to severe artifacts such as blurring, ghosting, and geometric distortions that can compromise diagnostic quality. Retrospective motion compensation remains challenging, particularly in accelerated acquisitions, due to the ill-posed nature of the joint reconstruction and motion estimation problem. In this work, we propose a unified Bayesian framework for motion-compensated 3D MRI that jointly estimates the anatomical image, rigid-body motion parameters, and coil sensitivity maps directly from motion-corrupted k-space data. Our approach integrates pretrained 3D complex-valued score-based diffusion models as expressive anatomical image priors within a physics-based forward model. Inference is performed by alternating diffusion posterior image updates with efficient proximal optimization steps for motion and coil sensitivity estimation, enabling fully unsupervised reconstruction without the need for paired motion-free training data. Experiments on simulated and real-motion brain MRI datasets demonstrate that the proposed method achieves improved image quality and motion robustness compared to state-of-the-art classical and learning-based motion correction techniques, particularly in the presence of severe motion and high acceleration.
Antonio Ortiz-Gonzalez, Erich Kobler, Lukas Schletter +1
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.
Ahmed Karam Eldaly, Matteo Figini, Daniel C. Alexander