Bayesian inverse problems with neural generative priors

Abstract

Inverse problems are ubiquitous in medical imaging and their solutions are essential for clinical decision making. To allow for quantification of uncertainty a Bayesian approach can be taken, but selecting an informed prior is highly non-trivial. We present a data-driven framework where a generative neural network in the form of a Wasserstein Generative Adversarial Network (WGAN) learns a realistic prior, decoupled from the inverse problem in consideration. The method is tested on a severely undersampled two dimensional MRI analogue. We use two different Markov chain Monte Carlo algorithms for approximating the resulting posterior expectations, one based on piecewise deterministic Markov processes, as well as the preconditioned Crank-Nicolson algorithm. The posterior mean and standard deviation is computed for the reconstructions. Our experiments demonstrate that this approach outperforms classical imaging priors both in the quality of reconstructions and also yielding a realistic posterior, allowing for sharp reconstructions with uncertainty using only a sparse observation.