Analytic First Approximation of Potentials

Model parameters $\mathbf{V}^\mathsf{T}$ in Eq. (6.2.5) are estimated by computing MLE  $\mathbf{V}^*$. Given a neighbourhood $\mathcal{N}$ and a training image ${g}^{\circ}$, the log-likelihood function $\ell(\mathbf{V}\vert{g}^{\circ})=\log P({g}^{\circ}\vert\mathbf{V})$ of the potential is:

$$\ell(\mathbf{V}\vert{g}^{\circ})=\mathbf{V}^\mathsf{T}\mathbf{F}(... ...\sum_{{g}\in\mathcal{S}} \exp\left( \mathbf{V}^\mathsf{T}\mathbf{F}({g})\right)$$ (6.3.1)

The MLE  $\mathbf{V}^*$ can not be obtained analytically because the second item in Eq. (6.3.1) involves the entire configuration space.

In a generic MGRF model, a quadratic approximation to the log-likelihood function $\ell(\mathbf{V}\vert{g}^{\circ})$ based on Taylor series is applied to simplify the log likelihood function and allows to analytically compute an first approximation of potentials, $\mathbf{V}^{\circ} \approx \mathbf{V}$. For a clique family $\mathbf{C}_a$, the first approximation of potential $\mathbf{V}_a^{\circ}$ is given by,

$$\begin{array}{lll} \mathbf{V}_a^{\circ}&=&\{V_a^{\circ}(q,s),... ...dot \left(F_a(q,s \mid {g}^{\circ})-M{(q,s)}\right) \end{array}$$ (6.3.2)

where ${g}^{\circ}$ is the training image, $M(q,s)=\frac{1}{{Q}^2}$ is the centred marginal probabilities of signal co-occurrences for the IRF, and $\rho_{a}=\frac{\vert\mathbf{C}_a\vert}{\vert\mathbf{R}\vert}$ gives the relative size of the clique family $\mathbf{C}_a$ with respect to the lattice size. The scaling factor $\lambda_{0}$ is calculated using the same GLCHs and centred grey level histogram (GLH). A more detailed derivation of the first approximation of potentials can be found in [37,36,38]. More accurate model identifications are given in Appendix A.

The first approximation of potentials might not be adequate for computing an accurate posterior distribution $Pr({g}^{\circ})$, but it gives a sub-optimal estimate of partial energy for each clique family and allows to identify the characteristic neighbourhood of the model.