Auto-binomial Model

assumes the conditional probability $Pr({g}_i \vert {g}^i)$ of a pixel is a binomial distribution. That is, a pixel ${g}_i$, taking values from the set $\mathbf{Q}$, is binomial distributed over $Q$-times independent random trials, with the probability of success, $q$, being,

$$q=\frac{\exp\{\alpha_i+\sum_{j \in \mathcal{N}_i }\beta_{ij}{{g}_j}\}}{1+\exp\{\alpha_i+\sum_{j \in \mathcal{N}_i }\beta_{ij}{{g}_j}\}}$$

.

Hence, the conditional distribution is given by,

$$Pr({g}_i \mid {g}^i)=\mathcal{C}_{Q}^{{g}_i}q^{{g}_i}(1-q)^{Q-{g}_i}$$ (5.2.17)

By Markov-Gibbs equivalence, the Gibbs distribution is specified by the energy function,

$$E({g})=\sum_{i \in \mathbf{R}} \ln \left(\begin{array}{c}{{g}_i}\... ...\in \mathbf{R}} {\alpha}_i{{g}_i}+ \sum_{\{i,j\} \in C}\beta_{ij}{{g}_i}{{g}_j}$$ (5.2.18)

In an auto-binomial model, model parameters are the sets $\{\alpha_i\}$ and $\{\beta_{ij}\}$. The set $\{\alpha_i\}$ has a fixed size, but the size of the set $\{\beta_{ij}\}$ is related to the order of the selected neighbourhood system. To simplify parameter estimation, one could reduce the number of parameters, for instance, by assuming a uniform value ${\alpha}$ for all the sites and using only the simplest neighbourhood system, i.e  4- or 8-nearest neighbours.