[Paper Review] Denoising Score-Matching for Uncertainty Quantification in Inverse Problems

Zaccharie Ramzi, Benjamin Rémy+3 more|arXiv (Cornell University)|Nov 16, 2020

Medical Imaging Techniques and Applications29 references4 citations

TL;DR

This paper proposes a Bayesian framework for MRI reconstruction that uses deep denoising score matching to learn a prior from data, enabling uncertainty quantification via annealed Hamiltonian Monte Carlo sampling. The method achieves high-quality reconstructions while explicitly quantifying uncertainty through posterior sampling, revealing data-informative regions and unstable features not captured by deterministic networks.

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.

Motivation & Objective

• To address the lack of uncertainty quantification in deep learning-based inverse problems, particularly in MRI reconstruction.
• To develop a scalable, data-driven Bayesian framework that leverages score-based generative modeling for prior learning.
• To enable full posterior sampling in high-dimensional image spaces using score networks trained via denoising score matching.
• To demonstrate that uncertainty can be quantified and visualized through posterior samples, revealing regions poorly constrained by data.

Proposed method

• A residual U-Net architecture is trained via denoising score matching to estimate the score of the data distribution, learning a prior on MRI images.
• The score network is regularized via spectral norm constraints to improve training stability and score quality.
• Annealed Hamiltonian Monte Carlo (HMC) is used to sample from the posterior distribution p(x|y), combining the learned prior with the likelihood from k-space measurements.
• A final denoising step is applied to the last HMC sample using the Expected Denoised Sample (EDS) scheme to improve reconstruction quality.
• The framework is applied to retrospective under-sampling in single-coil MRI, using the fastMRI dataset with acceleration factor 4.
• The method avoids explicit density estimation by relying solely on score gradients, enabling scalable inference in high-dimensional image spaces.

Experimental results

Research questions

• RQ1 Can denoising score matching be effectively used to learn a prior for Bayesian MRI reconstruction?
• RQ2 How well can posterior sampling via annealed HMC quantify uncertainty in reconstructed MRI features?
• RQ3 Can this score-based approach outperform deterministic deep learning methods in terms of uncertainty awareness while maintaining reconstruction quality?
• RQ4 What is the impact of spectral normalization on score network training and posterior sampling stability?
• RQ5 How does the framework handle data with varying noise levels or under-sampling patterns?

Key findings

• The method achieved a mean PSNR of 30.04 dB across posterior samples, significantly outperforming zero-filled reconstruction (25.55 dB) and approaching the state-of-the-art UPDNet (32.15 dB).
• Individual posterior samples had an average PSNR of 27.63 dB, indicating that the sampling process preserves image quality while enabling uncertainty assessment.
• Posterior samples revealed high variability in the red region of interest, indicating low data constraint—a feature not detectable via deterministic networks.
• The framework successfully scaled to high-dimensional MRI data (320×320×2), demonstrating the scalability of score-based models over GANs or VAEs.
• Despite challenges in HMC convergence, the final EDS denoising step significantly improved sample quality, suggesting a viable path for practical deployment.
• The approach enables visual uncertainty quantification: regions with high posterior variance correspond to features poorly supported by the measurements, aiding clinical interpretation.