Scalable Block Gibbs Sampling for Image Deblurring in X-Ray Radiography

Abstract

Quantitative image analysis in the security sciences formulates an image deblurring problem as a Bayesian inverse problem to reduce and quantify noise and blur. We consider images of size 16 megapixels and, since each pixel represents an unknown, the dimension of the Bayesian inverse problem is on the order of 107. The large dimension poses numerical and computational difficulties for two reasons. First, Markov chain Monte Carlo (MCMC), typically used to solve a Bayesian inverse problem, is generally slow to converge in high dimensions. Second, even generating one step in a Markov chain is challenging at this size. We present a Gibbs sampler that is scalable to the large dimension required in the security sciences and its scalability is achieved in two steps. We (i) accelerate MCMC convergence by exploring banded structure in the posterior precision matrix; and (ii) use a matrix-free implementation, because constructing and storing even sparse matrices is infeasible in our target application.