Multi-dimensional multivariate Gaussian Markov random fields with application to image processing

Image Deblurring, Gaussian Markov Random Fields, and Neumann Boundary Conditions

In this paper we consider the inverse problem of image deblurring with Neumann boundary conditions. Regularization is incorporated by using Gaussian Markov random fields (GMRFs) to model an appropriate prior on the image pixel values. We provide a linear algebraic framework for GMRFs, and we establish an important connection between GMRFs studied in the statistical literature, and negative-Lapl...

Approximating Hidden Gaussian Markov Random Fields

This paper discusses how to construct approximations to a unimodal hidden Gaussian Markov random field on a graph of dimensionnwhen the likelihood consists of mutually independent data. We demonstrate that a class of non-Gaussian approximations can be constructed for a wide range of likelihood models. They have the appealing properties that exact samples can be drawn from them, the normalisatio...

Subset Selection for Gaussian Markov Random Fields

Given a Gaussian Markov random field, we consider the problem of selecting a subset of variables to observe which minimizes the total expected squared prediction error of the unobserved variables. We first show that finding an exact solution is NP-hard even for a restricted class of Gaussian Markov random fields, called Gaussian free fields, which arise in semi-supervised learning and computer ...

Gaussian Markov Random Fields: Theory and Applications

3D Bayesian Regularization of Diffusion Tensor MRI Using Multivariate Gaussian Markov Random Fields

3D Bayesian regularization applied to diffusion tensor MRI is presented here. The approach uses Markov Random Field ideas and is based upon the definition of a 3D neighborhood system in which the spatial interactions of the tensors are modeled. As for the prior, we model the behavior of the tensor fields by means of a 6D multivariate Gaussian local characteristic. As for the likelihood, we mode...