Markov Random Field Theory

The spatial property can be modelled through different aspects, among which, the contextual constraint is a general and powerful one. Markov random field (MRF) theory provides a convenient and consistent way to model context-dependent entities such as image pixels and correlated features. This is achieved by characterizing mutual influences among such entities using conditional MRF distributions. In an MRF, the sites in $\mathcal S$ are related to one another via a neighbourhood system, which is defined as ${\mathcal N}=\{{\mathcal N}_i, i \in \mathcal S\}$, where ${\mathcal N}_i$ is the set of sites neighbouring i, $i \notin {\mathcal N}_i$ and $i \in {\mathcal N}_j \Longleftrightarrow j \in {\mathcal N}_i$. A random field X said to be an MRF on $\mathcal S$ with respect to a neighbourhood system $\mathcal N$ if and only if

$$\begin{eqnarray*}&& P({\mathbf x})>0, \forall \mathbf x \in \mathcal X \\ && P(x_i\vert x_{{\mathcal S}-\{i\}})=P(x_i\vert x_{{\mathcal N}_i}) \end{eqnarray*}$$

Note, the neighbourhood system can be multi-dimensional. According to the Hammersley-Clifford theorem [1], an MRF can equivalently be characterized by a Gibbs distribution. Thus,

$$P({\mathbf x})=Z^{-1}\exp(-U(\mathbf x)),$$ (5)

where

$$Z=\sum_{{\mathbf x} \in \mathcal X}\exp(-U({\mathbf x}))$$ (6)

is a normalizing constant called the partition function, and U(x) is an energy function of the form

$$U({\mathbf x})=\sum_{c \in \mathcal C}V_c({\mathbf x}),$$ (7)

which is a sum of clique potentials V_c(x) over all possible cliques $\mathcal C$. A clique c is defined as a subset of sites in $\mathcal S$ in which every pair of distinct sites are neighbours, except for single-site cliques. The value of V_c(x) depends on the local configuration on clique c. For more detail on MRF and Gibbs distribution see [12].