Model Parameters and Sufficient Statistics

The partial energy $E_{a}({g}\vert V_{a})$ in Eq. (6.1.1) can be factorised into a product of a potential vector and a vector representing marginal probability distribution of signal co-occurrences $(q,s)\in \mathbf{Q}^2$ at the pixel pairs.

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

$$= \sum_{(q,s) \in \mathbf{Q}^2}V_a{(q,s\vert{g})}H_a{(q,s\vert{g})}$$ (6.2.1)

$$= \mathbf{V}_a\bullet\mathbf{H}_a{({g})}$$ (6.2.2)

$$= \vert\mathbf{C}_a\vert\cdot\mathbf{V}_a\bullet\mathbf{F}_a{({g})}$$ (6.2.3)

Here, $H_a(q,s\mid {g})$ denotes the empirical marginal probability distribution of the pixel pair $(q,s)\in \mathbf{Q}^2$ separated by the displacement vector of the clique family $\mathbf{C}_a$. $\mathbf{H}_a{({g})}=\{H_a{(q,s\mid {g}):(q,s) \in \mathbf{Q}^2}\}$ represents the grey level co-occurrence histogram (GLCH) collected over the image ${g}$ for the clique family $\mathbf{C}_a$, and $\mathbf{F}_a{({g})}=\frac{1}{\vert\mathbf{C}_a\vert}\mathbf{H}_a{({g})}$ is the normalised GLCH. The potential component $V_a(q,s)$ represents the strength of interaction between grey levels $q$ and $s$ over the clique family $\mathbf{C}_a$.

By Eq. (6.2.3), the total energy $E({g})$ in Eq. (6.1.3) can be rewritten as follows,

$$E({g})=\sum_{a \in \mathbf{A}}E_{a}=\mathbf{V}\bullet\mathbf{H}({... ...hbf{V}_a\bullet\mathbf{F}_a({g})}=\mathbf{V}^\mathsf{T} \bullet \mathbf{F}({g})$$ (6.2.4)

Here, $\mathbf{V}=\{\mathbf{V}_a: {a \in \mathbf{A}}\}$, $\mathbf{V}^\mathsf{T}=\{\rho_a\cdot\mathbf{V}_a: a \in \mathbf{A}\}$, $\mathbf{H}{({g})}=\{\mathbf{H}_a{({g})}:a \in \mathbf{A}\}$ and $\mathbf{F}{({g})}=\{\mathbf{F}_a{({g})}:{a \in \mathbf{A}}\}$ denote the potential vector, the weighted potential vector, the GLCH vector and the normalised GLCH vector, respectively, and $\rho_a=\vert\mathbf{C}_a\vert/\vert\mathbf{R}\vert$ denotes the weight or the relative cardinality of the clique family $\mathbf{C}_a$.

After substituting the energy $E({g})$ of Eq. (6.2.4) into Eq. (6.1.3), a generic MGRF model is specified by the following distribution,

$$Pr({g})=\frac{1}{Z_{\mathbf{V},\mathbf{A}}} \exp \left\{\mathbf{V... ...Z_{\mathbf{V},\mathbf{A}}}\exp\{\mathbf{V}^\mathsf{T} \bullet \mathbf{F}({g})\}$$ (6.2.5)

Eq. (6.2.5) reveals a few important statistical properties of a generic MGRF model. First, the Gibbs distribution belongs to the exponential families [2] of probability distributions and therefore to the MaxEnt distributions. The exponential part of the distribution is a linear combination of the potential vector $\mathbf{V}^\mathsf{T}$ and the GLCH vector $\mathbf{H}$. The maximum entropy properties of the exponential families are well known in applied and theoretical statistics [2], well before the MaxEnt principle was reintroduced into texture modelling in [113]. Second, the potential vector $\mathbf{V}^\mathsf{T}$ are the model parameters to be estimated, while the GLCH vector $\mathbf{H}$ are the sufficient statistics to the model.

The GPD function for a generic MGRF model in Eq. (6.2.5) is very similar to that of the FRAME model in Eq. (5.2.25), except that two models involve different sufficient statistics. The FRAME model is derived from GLHs collected from responses of a filter bank, but a generic MGRF model uses the GLCH statistics. If each signal pair $({g}_i,{g}_{i+\alpha}),\: \alpha \in A$ is considered as the output of a local 'filter', then the grey level co-occurrences histogram in a generic MGRF model could be considered a joint distribution of filter responses. Hence, a generic MGRF and the FRAME models are equivalent in this sense.

For both models, identifying an optimal set of characteristic statistics for a training image is a key issue for successful modelling an image. Too big number of statistics involved in a model ultimately lead to intractable complexity of model identification, while insufficient statistics lead to inadequate modelling. In a generic MGRF model, an analytic first approximation of potentials is computed for ranking and selecting the most characteristic clique families, and the GLCHs for selected families act as sufficient statistics of the model. In the FRAME model, identifying sufficient statistics involves a stepwise procedure of filter selection from a predefined filter bank [113]. The procedure is conceptually similar to the method for sequential selection of clique families for an MGRF model [107], i.e. the filter selected at each step maximises the change of the related marginal distribution. A major deficiency of this approach is that the initial filter bank has to be constructed heuristically which might not contain all `important' filters to the image.