PriorLEARN
Prior learning
This subproject deals with the solution of linear and nonlinear inverse problems that will arise in the context of dynamic MRI reconstruction problems. The main innovation of this subproject is to largely replace the deterministic optimization-based methods that have dominated in recent years by probabilistic methods based on statistical modeling and Bayesian inference techniques. This opens up a much broader view of the solution of inverse problems than was previously possible with classical variational methods. The probabilistic approach allows to compute a complete distribution of solutions, which in turn can be used to compute different point estimates such as the posterior expectation. Furthermore, the probabilistic approach allows to compute posterior variances which in turn can be used for uncertainty quantification.
For inverse problems such as those studied here, it is crucial to have access to strong priors (or regularizers), as these can lead to reliable image reconstructions even with noisy and incomplete data. This is of even greater importance for the dynamic and active image reconstruction problems we address in this project. The main goal of this subproject is therefore to develop new techniques for learning such strong priors from data.
In preliminary work on static MRI reconstruction, we have already obtained promising results showing that the learned priors can lead to high quality reconstructions even in cases where established regularizers such as total variation fail. We will employ different parametric models, starting from the celebrated FRAME model (also know as the Fields-of-Experts model) over Gaussian Mixture Models to full neural networks) and use different unsupervised (maximum likelihood learning, score-matching) and supervised (bilevel optimization) learning techniques. For computing posterior expectations and variances, it will also be important to develop faster sampling techniques that are needed for maximum likelihood learning and Bayesian inference. To provide theoretical performance guarantees, we will study uncertainty quantification in the framework of conformal prediction, for example based on the posterior variance.