Gaussian Markov Random Field Priors for

Abstract. In this paper, our focus is on the connections between the methods of (quadratic) regularization for inverse problems and Gaussian Markov ran-dom field (GMRF) priors for problems in spatial statistics. We begin with the most standard GMRFs defined on a uniform computational grid, which corre-spond to the oft-used discrete negative-Laplacian regularization matrix. Next, we present a class of GMRFs that allow for the formation of edges in recon-structed images, and then draw concrete connections between these GMRFs and numerical discretizations of more general diffusion operators. The bene-fit of the GMRF interpretation of quadratic regularization is that a GMRF is built-up from concrete statistical assumptions about the values of the unknown at each pixel given the values of its neighbors. Thus the regularization term corresponds to a concrete spatial statistical model for the unknown, encap-sulated in the prior. Throughout our discussion, strong ties between specific GMRFs, numerical discretizations of diffusion operators, and corresponding regularization matrices, are established. We then show how such GMRF pri-ors can be used for edge-preserving reconstruction of images, in both image deblurring and medical imaging test cases. Moreover, we demonstrate the ef-fectiveness of GMRF priors for data arising from both Gaussian and Poisson noise models. 1