Bayesian MRI Reconstruction with Structured Uncertainty Distributions

Teo Deveney, Ivor J A Simpson, Neill D. F. Campbell

Keywords: Bayesian uncertainty quantification · Generative regularisation · MRI.

Abstract:

We present a Bayesian hierarchical approach to magnetic
resonance imaging (MRI) reconstruction using learned structured uncertainty
distributions. Our method allows for reconstruction of complexvalued
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.

Submission Number: 22