Unadjusted Langevin Sampling for Uncertainty Estimation in MRI Reconstruction - Theory and Numerical Validation

Motivation: Langevin sampling can provide uncertainty estimation for reconstructions$$$\,$$$from undersampled k-space data by computing$$$\,$$$samples of the posterior distribution. This is of particular$$$\,$$$interest when used with$$$\,$$$machine-learning priors.$$$\,$$$However, to$$$\,$$$sample the true data$$$\,$$$distribution sufficiently well, the parameters of the$$$\,$$$algorithm must be$$$\,$$$chosen correctly. Goal(s): Establish Langevin sampling as a$$$\,$$$reliable tool for uncertainty estimation. Approach: Using a theoretical analysis of Langevin sampling one can predict the$$$\,$$$difference of the sampled distribution$$$\,$$$to the true posterior.$$$\,$$$These results are validated numerically$$$\,$$$and compared to the pseudo-replica method for a basic SENSE$$$\,$$$reconstruction. Results: The theoretical predictions$$$\,$$$are confirmed in the numerical results.$$$\,$$$For correctly$$$\,$$$chosen parameters, the predicted$$$\,$$$variance maps$$$\,$$$can be related to the results obtained with the$$$\,$$$pseudo-replica method. Impact: Uncertainty estimation via posterior sampling is an important tool to understand the reliability of reconstruction using generative machine-learning models.