Partially Collapsed Gibbs Samplers for Linear Inverse Problems and Applications to X-ray Imaging
Deblurring is one of the most common linear inverse problems in imaging, and most methods for deblurring an image assume that the convolution kernel, the so-called point spread function (PSF), is known or attempt to reconstruct the PSF simultaneously with the image. For many experimental imaging systems, like high energy X-ray radiography systems, the PSF must be computed from calibration imagery. The associated inverse problem is large-scale, so, in order to compute meaningful uncertainties with the reconstruction, it is common to use hierarchical Bayesian models and Markov chain Monte Carlo methods for sampling from the posterior. Classical Gibbs samplers can suffer from long autocorrelation lengths in the Markov chain when the reconstructed quantity in the inverse problem is finely discretized, resulting in poor computational efficiency of the Markov chain. In this work we present partially collapsed Gibbs samplers for linear inverse problems that greatly decrease correlations in the Markov chain, thus increasing the computational efficiency of the algorithms, and we present the results of the enhanced samplers on the problem of reconstructing the point spread function for an X-ray imaging system at the U.S. Department of Energy's Nevada National Security Site.