International Journal for Uncertainty Quantification, 1(1):xxx–xxx, 2015 A METROPOLIS-HASTINGS METHOD FOR LINEAR INVERSE PROBLEMS WITH POISSON LIKELIHOOD AND GAUSSIAN PRIOR

Poisson noise models arise in a wide range of linear inverse problems in imaging. In the Bayesian setting, the Poisson likelihood function together with a Gaussian prior yields a posterior density function that is not of a well known form and is thus difficult to sample from, especially for large-scale problems. In this work, we present a method for computing samples from posterior density functions with Poisson likelihood and Gaussian prior, using a Gaussian approximation of the posterior as an independence proposal within a Metropolis-Hastings framework. To define our priors, we use Gaussian and Laplace-distributedMarkov random fields, which are the Bayesian analogues of smoothness and total variation regularization, respectively. For the Laplace prior, a Gaussian approximation is used, whereas for the Gaussian prior, the scaling (or regularization) parameter is sampled using a hierarchical Gibbs sampler, eliminating the need to choose a regularization parameter a priori. The results are demonstrated on synthetic data–including a synthetic X-ray radiograph generated from a radiation transport code–and on real images used to calibrate a pulsed power high-energy X-ray source at a U.S. Department of Energy X-ray radiography facility