Bayesian MRI Reconstruction with Structured Uncertainty Distributions

We present a Bayesian hierarchical approach to magnetic resonance imaging (MRI) reconstruction using learned structured uncertainty distributions. Our method allows for reconstruction of complex-valued MRI images in a probabilistic manner that goes beyond standard pixelwise uncertainty. We use a variational autoencoder architecture (VAE) prior with an expressive correlated Gaussian decoding distribution obtained via a sparse parameterisation of the precision matrix, and model the posterior uncertainty in the latent and image space using a similarly correlated variational approximation. The resulting posterior is fully marginalisable over the VAE latent, and provides interpretable insights into the spatial structure of the reconstruction distribution that are not seen in existing methods. Diagnostic posterior pixelwise correlations and residual structure show a principled decay of prior correlation influence with increasing data, and we demonstrate that these modelled posterior statistics are representative of the true reconstruction error. This allows us to answer questions like “how much data is required to resolve a local region to a specific spatial accuracy”. We also provide numerical experiments demonstrating that our method maintains excellent pixelwise reconstruction performance and well-calibrated posterior coverage even in extremely sparse data scenarios.