Argonauts:Markov Random Fields
From Wasteland
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Sun 6 November 2005
Spatio-temporal coordinates
WMVL, 11.00am to 1.00pm (exactly) + 15 min for conclusions.
Attendees (in alphabetical order)
- Ramón Casero Cañas.
- Mike Kadour.
- Jeong-Gyoo Kim.
- Etienne von Lavante.
- Rohan Loveland.
- Olivier Noterdaeme.
Minutes
The argonauts have now a mailing list at googlegroups. The WMVL photocopier has been dubbed Science-unfriendly, as it tried to eat part of the research material. It was fun to take it apart and put it back together, but Science is not supposed to be fun, anyway, and we have no time to waste. Brainstorming for improving the whiteboard wipping system would be great.
Some argonauts are coming late to the meetings. May God have mercy on their souls. Some argonauts are timid and don't participate aggresively. Introducing physical punishment is under consideration in both cases.
Etienne provided Appendix C of Guofang Xiao's thesis, which is a brief summary of Markov Random Field (MRF) Theory. We went to hell and back to understand the Appendix, but as proper scientists we patted ourselves on the back, agreed that this was due to confusing notation rather than our own shortcomings and in the end I think that we got a pretty good basic idea of what MRFs are about.
Basically, a MRF is a vector of random variables X = (X_1, ... X_N) on a lattice S with a neighbourhood system N. On images, each X_i corresponds to one pixel. The lattice S is the 2D layout of the pixels. The neighbourhood is defined on terms of cliques, or tetris-like pieces formed by pixels, such that all pixels in a clique touch each other.
What is useful of a MRF is its local characteristic: The probability that a pixel has a certain value can be computed from the neighbourhood. So you don't need all the pixels in the image.
The probability cannot be computed directly though, so you need the Hammersley-Clifford theorem to tell you that X is a MRF on S with respect to N, if and only if X is a Gibbs Random Field (GRF) on S with respect to N.
Of course we have no clue about the theorem, but let's just show some faith here. In the end, what matters is that you can compute the probability of a pixel having a certain value as a normalized exp(-U/T), where T is a constant and U is the energy of your neighbourhood.
Energy here is just a sum of potencials. In particular, for each clique in your neighbourhood you have a potencial, which is just a way of evaluating how far is the value you want for your central pixel from the value of the pixels in the neighbourhood.
So if the potencial goes up, probability goes down, i.e. a pixel will probably have similar values to other pixels in the same neighbourhood.
The final 15 min presentation of conclusions was useful. We'll do it again in the future. Jeong-Gyoo kindly volunteered to lead it at gunpoint.
It turns out that MRF are often used in combination with the Bayes theorem (the former provide an estimate of the a priori probability to the latter). We did this in our meeting of the 3 March 2006 Markov Random Fields and Bayes.
Most of the crew went to Taylor's for a sandwich, in what could be considered a submarine mission (lovely weather in Oxford). We ate at the Engineering Cafe.
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