Denoising Score-Matching for Uncertainty Quantification in Inverse Problems

TL;DR

This paper introduces a Bayesian framework using denoising score matching to learn priors for inverse problems, enabling high-quality reconstructions and uncertainty quantification, demonstrated on MRI reconstruction.

Contribution

It combines score matching with Bayesian inference to quantify uncertainty in inverse problems, a novel integration for improved image reconstruction.

Findings

High-quality MRI reconstructions achieved.

Uncertainty quantification for specific image features.

Effective integration of score matching with Bayesian sampling.

Abstract

Deep neural networks have proven extremely efficient at solving a wide rangeof inverse problems, but most often the uncertainty on the solution they provideis hard to quantify. In this work, we propose a generic Bayesian framework forsolving inverse problems, in which we limit the use of deep neural networks tolearning a prior distribution on the signals to recover. We adopt recent denoisingscore matching techniques to learn this prior from data, and subsequently use it aspart of an annealed Hamiltonian Monte-Carlo scheme to sample the full posteriorof image inverse problems. We apply this framework to Magnetic ResonanceImage (MRI) reconstruction and illustrate how this approach not only yields highquality reconstructions but can also be used to assess the uncertainty on particularfeatures of a reconstructed image.