Converting a Gaussian Markov random field

[mort.motes]

Hi I am currently reading:

http://www.cs.cornell.edu/~asaxena/learningdepth/saxena_ijcv07_learningdepth.pdf

which deals with reconstructing depth from a single still image.

A gaussian multiscale markov random field is trained in a supervised context where the model is shown below:

http://img534.imageshack.us/img534/1259/combineda.jpg

now this model is converted into a standard multivariate gaussian (indicated by the arrow) but how is that conversion possible? I have read that it basically is a matter of completing the square but is there some more detailed explanation for this somewhere besides:

http://en.wikipedia.org/wiki/Completing_the_square

which don't really describe the techniques used on markov random fields.

 

[SW VandeCarr]

I'm not really familiar with image generation using Markov random fields. The following paper discusses approaches using a Bayesian approach. Perhaps it will be useful to you.

http://www.scss.tcd.ie/JiWon.Yoon/papers/MRF/Image%20segmentation%20image.pdf