More Accurate First Approximation of the MLE

is obtained if the starting potential $\mathbf{V}_0$ in Proposition A.0.1 produces a general-purpose IRF identified for the training image $g^\circ$ rather than the IRF$_0$.

PROPOSITION A.0.3   The potential $\mathbf{V}_0^\mathsf{T} = [V_{\mathrm{irf}:\xi,\eta}(q,s): (q,s)\in\mathcal{Q}^2;(\xi,\eta)\in\mathcal{N}]$ which components $V_{\mathrm{irf}:\xi,\eta}(q,s)=\frac{1}{\rho_{\xi,\eta}\vert\mathcal{N}\vert}V_\mathrm{pix}(q)$ scale the potential values $V_\mathrm{pix}(q)$; $q\in\mathcal{Q}$, of Eq. (A.0.6) reduces the MGRF of Eq. (6.2.5) to the general-purpose IRF with the marginal signal probability distribution $\mathbf{P}_\mathrm{irf} = \mathbf{F}_\mathrm{pix}(g^\circ)$.

Proof. Assuming that $\sum_{s\in\mathcal{Q}}f_{\xi,\eta}(q,s\vert g)=f_\mathrm{pix}(q\vert g)$ for all $(\xi,\eta)\in\mathcal{N}$, the normalised MGRF exponent is as follows:

$$\sum\limits_{(\xi,\eta)\in\mathcal{N}}\rho_{\xi,\eta}\sum\limits_... ...rt g) = \sum\limits_{q\in\mathcal{Q}}V_\mathrm{pix}(q)f_\mathrm{pix}(q\vert g)$$

The assumption holds precisely for the actual marginal signal co-occurrence and signal probability distributions and therefore is asymptotically valid for the empirical distributions, too, providing the lattice $\mathcal{R}$ is sufficiently large to ensure small deviations between the empirical and actual probabilities. $\qedsymbol$

If the potential $\mathbf{V}_0$ from Proposition A.0.3 is used in Proposition A.0.1, the resulting general-case IRF identified from $g^\circ$ has the specific log-likelihood

$$\ell\{\mathbf{V}_0\vert g^\circ\}=\sum\limits_{q\in\mathcal{Q}} f_\mathrm{pix}(q\vert g^\circ) \ln f_\mathrm{pix}(q\vert g^\circ)$$

and the co-occurrence probabilities $p_{\xi,\eta}(q,s)=f_\mathrm{pix}(q\vert g^\circ)f_\mathrm{pix}(s\vert g^\circ)$ for $(q,s)\in\mathcal{Q}^2$ and $(\xi,\eta)\in\mathcal{N}$. Therefore, the $nQ^2$-vector of the expected scaled probabilities is $\E\{\mathbf{F}(g)\vert\mathbf{V}_0\}^\mathsf{T} = \mathbf{P}^\mathsf{T}(g^\cir... ...{\xi,\eta}\boldsymbol{\phi}^\mathsf{T}(g^\circ):(\xi,\eta)\in\mathcal{N}\right]$ where $\boldsymbol{\phi}^\mathsf{T}(g^\circ)= \left[f_\mathrm{pix}(q)g^\circ)f_\mathrm{pix}(s\vert g^\circ): (q,s)\in\mathcal{Q}^2\right]$.

Let $\Delta_{\xi,\eta;q,s}=f_{\xi,\eta}(q,s\vert g^\circ) - f_\mathrm{pix}(q\vert g^\circ)f_\mathrm{pix}(s\vert g^\circ)$ denote the difference between the empirical and actual signal co-occurrence probabilities for the IRF identified from the image $g^\circ$. Let $\mathrm{var}_{q,s}$ be the variance of the latter probability: $\mathrm{var}_{q,s} = f_\mathrm{pix}(q\vert g^\circ)f_\mathrm{pix}(s\vert g^\ci... ...\left( 1 - f_\mathrm{pix}(q\vert g^\circ)f_\mathrm{pix}(s\vert g^\circ) \right)$. Then the gradient $\nabla\ell(\mathbf{V}_0\vert g^\circ) = \mathbf{F}(g^\circ)-\mathbf{P}(g^\circ)\equiv \mathbf{\Delta}(g^\circ)$ of the log-likelihood is the $nQ^2$-vector of the scaled differences: $\mathbf{\Delta}^\mathsf{T}(g^\circ) = \left[ \rho_{\xi,\eta}\Delta_{\xi,\eta;q,s}: (q,s)\in\mathcal{Q}^2 ]: (\xi,\eta)\in\mathcal{N} \right]$ and the covariance matrix $\mathbf{C}\{\mathbf{F}(g)\vert\mathbf{V}_0\}$ is closely approximated by the scaled diagonal matrix $\mathbf{C}_\mathrm{irf} = \mathsf{Diag}\left[\rho_{\xi,\eta}\boldsymbol{\psi}(g^\circ):\xi,\eta)\in\mathcal{N}\right]$ where $\boldsymbol{\psi}(g^\circ)$ is the $Q^2$-vector of the variances: $\boldsymbol{\psi}(g^\circ) = \left[ \mathrm{var}_{q,s}: (q,s)\in\mathcal{Q}^2\right]^\mathsf{T}$.

PROPOSITION A.0.4   The first approximation of the potential MLE in the vicinity of the point $\mathbf{V}_0$ from Proposition A.0.3 in the potential space is $\mathbf{V}^\ast = \mathbf{V}_0 + \lambda^\ast\mathbf{\Delta}(g^\circ)$ with the maximising factor

$$\lambda^\ast = \frac{ \mathbf{\Delta}^\mathsf{T}(g^\circ)\mathbf{... ...m\limits_{(q,s)\in\mathcal{Q}^2} \mathrm{var}_{q,s} \Delta_{\xi,\eta;q,s}^{2} }$$ (A.0.7)

Now for all the signal cardinalities $Q$ the actual MLE for the IRF and its approximation in Proposition A.0.4 completely agree so that the approximation is closer to the actual MLE than the previous one in Proposition A.0.2.