# Bayesian inference and uncertainty quantification for image reconstruction with Poisson data

## TL;DR

This paper develops a comprehensive Bayesian framework for image reconstruction from Poisson data, including a positivity-preserving reparametrization, a dimension-independent MCMC algorithm, and methods for regularization parameter estimation and artifact detection.

## Contribution

It introduces a novel infinite-dimensional Bayesian approach with a hybrid prior, a new MCMC algorithm, and techniques for regularization and artifact detection in image reconstruction.

## Key findings

- The posterior distribution is well-posed in infinite dimensions.
- The dimension-independent MCMC algorithm effectively samples the posterior.
- The method successfully detects artifacts in reconstructed images.

## Abstract

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 MCMC 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.