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Undirected Graphical Models: Markov Random Field

Mesfin Dema March 30,2011. Undirected Graphical Models: Markov Random Field. Contents. Importance of Context Where Can We Use Context? Context Modeling in Baye’s Rule Images as Random Fields Markov Random Field(MRF) Gibbs Random Field(GRF) Equivalence of MRF and GRF MAP for MRF.

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Undirected Graphical Models: Markov Random Field

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  1. Mesfin Dema March 30,2011 Undirected Graphical Models:Markov Random Field

  2. Contents • Importance of Context • Where Can We Use Context? • Context Modeling in Baye’s Rule • Images as Random Fields • Markov Random Field(MRF) • Gibbs Random Field(GRF) • Equivalence of MRF and GRF • MAP for MRF

  3. Importance of Context a) Natural Image b) Image obtained by shuffling intensities of image in (a)

  4. Importance of Context Histogram of a) Natural Image b) Image obtained by shuffling intensities of image in (a)

  5. Importance of Context “Show me your friends and I will tell you who you are.”- MRF

  6. Where Can We Use Context?

  7. Context Modeling in Baye’s Rule • Joint Probability • Conditional Probability

  8. Images as Random Fields

  9. Markov Random Fields(MRF) • A random field is a MRF if • The joint probability is strictly positive • Markovianity

  10. Markov Random Fields(MRF) First -order 4-neighbors How can we formulate the MRF mathematically?

  11. Gibbs Random Field(GRF) • A random field is a GRF if the joint distribution has the form • If the GRF is defined in terms of

  12. Gibbs Random Field(GRF)

  13. GRF and MRF Equivalence • Hammersley-Clifford Theorem: x is an MRF on if and only if it is a GRF on • This theorem shows how local interaction can result in global optimization.

  14. Example: Image De-noising

  15. Example: Image De-noising 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

  16. Example: Image De-noising For the previous example

  17. Monte Carlo Markov Chain (MCMC) • Allows to simulate a probability distribution by constructing a Markov chain.

  18. MAP for MRF • Most of the application are Inference which requires posterior probability. Maximum A Posteriori(MAP) is used to select the best configuration. • The likelihood term can be modeled to fit the data. Mostly Gaussian distribution is used.

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