Bayesian Inference and Uncertainty Quantification for Medical Image Reconstruction with Poisson Data

We provide a complete framework for performing infinite dimensional Bayesian inference and uncertainty quantification for image reconstruction with Poisson data. In particular, we address the following issues to make the Bayesian framework applicable in practice. We first introduce a positivity-preserving reparametrization, and we prove that under the reparametrization and a hybrid prior, the posterior distribution is well-posed in the infinite dimensional setting. Second, we provide a dimension-independent Markov chain Monte Carlo algorithm, based on the preconditioned Crank--Nicolson Langevin method, in which we use a primal-dual scheme to compute the offset direction. Third, we give a method combining the model discrepancy method and maximum likelihood estimation to determine the regularization parameter in the hybrid prior. Finally we propose to use the obtained posterior distribution to detect artifacts in a recovered image. We provide an example to demonstrate the effectiveness of the proposed method.