MATH Seminar

Title: Gaussian Markov Random Field Priors and MCMC for Inverse Problems
Seminar: Numerical Analysis and Scientific Computing
Speaker: Johnathan Bardsley of Department of Mathematical Sciences, University of Montana
Contact: Jim Nagy, nagy@mathcs.emory.edu
Date: 2011-11-16 at 12:50PM
Venue: W306
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Abstract: In this talk I will explore the connections between Bayesian statistics and inverse problems. In particular, I will show how familiar quadratic regularization functions can be viewed as prior probability densities arising from Gaussian Markov Random Fields (GMRFs). GMRFs, in turn, correspond to concrete probabilistic assumptions regarding the value of the unknown image at a specific pixel based on the value of its neighbors. With a GMRF prior in hand, I will then show how to perform MCMC sampling of the unknown image and of the noise and prior precision values. The image sampling step is a large-scale structured linear algebra problem that has seen little attention by the numerical linear algebra community. The samples outputted by the MCMC method can be used to compute a reconstructed image, e.g. the sample mean, as well as estimates of the precision parameters, which can in turn be used to estimate the regularization parameter.