Gibbs-Markov Random Field

• A random field Q defined on a lattice L has Gibbs distribution or a Gibbs-Markov random field with respect to a neighborhood system (e.g., a second order neighborhood) iff:
  p(Q) = 1/Z exp(-U(Q))

  Where U(Q) is the energy function defined by summation of the clique potentials and Z is a normalizing constant.

  Random field models, in general, and Gibbs-Markov models in particular provide a tool for introduction of spatial context into pixel labeling problem such as image segmentation and restoration.

  Gibbs-Markov random fields (GMRF) models allow to embody prior information (e.g., local characteristics) about the image in the image model so that Bayes decision theory can be applied to the image segmentation problem.

  Additional prior knowledge such as the sizes, shapes, and orientations of the regions can also be reflected in the parameters of an GMRF model.