Introduction

A generic MGRF model involves an arbitrary structure of multiple pairwise pixel interactions for describing spatially homogeneous and piecewise-homogeneous textures with translation-invariant pixel interactions.

Considering a random field model of textures, signal value at each pixel (e.g. grey level) probabilistically depends on values at the neighbouring pixels. By analogy with an interacting particle system in physics, statistical dependence among pixels is referred to pixel interaction, and a texture, as an interacting pixel system, is the direct result of various pixel interactions. In a generic MGRF model, pixel interaction has ``no physical meaning'', reflecting only ``the fact that some spatial signal combinations in a particular texture are more frequent than others'' [38].

A generic MGRF model provides a framework of computing the strength of pixel interactions for recovering a characteristic neighbourhood to identify a texture. Only pairwise pixel interactions in second-order cliques are involved in an explicit, arbitrary geometric structure of multiple pixel interactions. Compared to the neighbourhood systems of other MGRF models, e.g., a rectangular window for the auto-models, the arbitrary structure of multiple pixel interactions is more flexible and efficient, because it covers both short- and long-range pixel interactions at a low computational cost.

Pixel interaction is assumed to be translational-invariant, so that a pairwise pixel interaction has the same strength at every pixel of an image. Therefore, for simplicity, pairwise cliques in an image are grouped into clique families. A pairwise clique of pixels $i$ and $j$ is denoted by $C=\{(i,j):i,j \in \mathbf{R}\}$. Thus a clique family $\mathbf{C}_a$, representing the group of all the pairwise cliques having a same relative displacement vector $(\delta{x_a},\delta{y_a})$, is denoted by $\mathbf{C}_a=\mathbf{C}_{\delta{x_a},\delta{y_a}}=\{(i,j):i,j \in \mathbf{R}; i-j=(\delta{x_a},\delta{y_a})\}$. The set of all the clique families in an image is denoted by $\mathbf{C}=\{\mathbf{C}_a: a\in \mathbf{A}\}$. Here, $\mathbf{A}$ is a set of all possible displacement vectors in an image. The interaction structure of a generic MGRF model is a subset of $\mathbf{C}$, which consists of clique families representing the most characteristic long- and short- range pixel interactions.

To quantitatively measure pixel interactions, a generic MGRF model uses notions of Gibbs potential  $V_a({g}_i,{g}_j:(i,j)=\mathbf{C}_{a})$, partial energy  $E_{a}({g}\vert V_{a})$, and total energy $E({g})$. The potential represents the probabilistic strength of pixel interaction within a clique $C \in \mathbf{C}_a$. The partial energy sums the potentials over the family $\mathbf{C}_a$, and the total energy sums the partial ones over the set $\mathbf{C}$ of all the families involved in the image. Eqs. (6.1.1) and (6.1.2) show relations among three measures of pixel interactions.

$$E_{a}({g}\vert V_{a})=\sum_{C\in \mathbf{C}_a}V_a({g}_i,{g}_j:(i,j)=C).$$ (6.1.1)

$$E({g})=\sum_{a\in\mathbf{A}}{E_{a}({g}\vert V_{a})}$$ (6.1.2)

A generic MGRF model is specified by a Gibbs distribution of partial energies over all clique families, with the Gibbs energy being the total energy of the image. Combining Eqs. (6.1.1) and (6.1.2), the Gibbs distribution function could be in one of the following forms,

$$\begin{array}{lll} Pr({g})&=&\frac{1}{Z} \displaystyle\exp \l... ...bf{C}_a}V_a\left({g}_i,{g}_j:(i,j)=C\right)\right\} \end{array}$$ (6.1.3)

The partial energy represents the significance of impact of a clique family and the related pixel interaction to the entire texture. The higher the partial energy, the more important a clique family is in forming texture features. A generic MGRF model selects a characteristic neighbourhood $\mathcal{N}$, consisting of only the most energetic clique families, for a compact texture description. In doing this, the model function in Eq. (6.1.3) is largely simplified. Meanwhile, the model quality is not compromised, because only less important trivial clique families are neglected.