Characteristic Neighbourhood

The characteristic neighbourhood represents an interaction structure constituted by the most energetic clique families. Most of clique families in a texture have rather low partial energy, and therefore related pixel interactions have little or no impact on texture patterns. In contrast, only a small number of most energetic clique families are major contributors to a texture. This observation suggests only the most energetic clique families should be included in the interaction structure.

The selection of clique families is based on partial energy which measures the probabilistic strength of related pixel interaction. Given the first approximation of potential $\mathbf{V}_a^{\circ}(q,s)$ in Eq. (6.3.2) and the partial energy $E_{a}$ in Eq. (6.1.3), an approximation of partial energy for a clique family $\mathbf{C}_a$ can be computed from a training sample ${g}^{\circ}$ by:

$$E_{a}^{\circ}({g}\vert V_{a})=\vert\mathbf{C}_a\vert \cdot \lambd... ...mathbf{Q}^2}(F_a(q,s \mid {g}^{\circ})-M{(q,s)})\cdot F_a(q,s \mid {g}^{\circ})$$ (6.3.3)

Since the dimension of the image lattice $\vert\mathbf{R}\vert$ and the factor $\lambda_{0}$ are constant, the approximation of partial energy can be simplified further to a relative partial energy, $\mathsf{\varepsilon}_{a}^{\circ}({g}\vert V_{a})$, for simplicity.

$$\mathsf{\varepsilon}_{a}^{\circ}({g}\vert V_{a})={\vert\mathbf{C}... ...athbf{Q}^2} (F_a(q,s \mid {g}^{\circ})-M{(q,s)})\cdot F_a(q,s \mid {g}^{\circ})$$ (6.3.4)

The relative partial energy $\mathsf{\varepsilon}_{a}^{\circ}({g}\vert V_{a})$ provides a relative measure of contribution and allows to rank all the clique families accordingly. The most characteristic clique families can be found by using a threshold in the simplest case. To a good approximation, the relative partial energies of all clique families are assumed to follow a Gaussian distribution. Therefore, the threshold $\theta$ is heuristically decided by a function of the mean $\bar{\mathsf{\varepsilon}}$ and the standard deviation $\sigma_{\mathsf{\varepsilon}}$ of the Gaussian [37].

$$\theta=\bar{\mathsf{\varepsilon}}+c\sigma_{\mathsf{\varepsilon}}$$ (6.3.5)

However, this over-simplified method of determining the interaction structure by using a threshold does not take statistical interplay among clique families into account, which might result in neglecting some important clique families. In other words, only statistical independent clique families should be included into the interaction structure [39]. In another approach, clique families are selected iteratively based on their statistical impact to the probability distribution [107]. In this method, the interaction structure grows from a single clique family with maximum energy and only one clique family is selected at each iteration. The statistical impact of a clique family is defined by the change to the probability distribution after adding the family into the structure. This method involves computational expensive operations of re-sampling distribution and re-collecting GLCH statistics at each step, but the recovered structure is more adequate and considerably smaller than the structure obtained by thresholding.