Actual vs. Estimated Potentials for a General-case IRF.

In contrast to the $\mathrm{IRF}_\mathrm{eq}$, a general-case IRF on $\mathcal{R}$ has an arbitrary marginal probability distribution of signals $\mathbf{P}_\mathrm{irf}=(p_\mathrm{pix}(q): q\in\mathcal{Q})$. This IRF can be represented similarly to Eq. (6.2.5) by using the pixel-wise potential:

$$P_\mathrm{irf}(g^\circ)= \frac{1}{Z_\mathrm{irf}}\exp \left(\sum\... ...\vert \mathbf{V}^\mathsf{T}_\mathrm{pix}\mathbf{F}_\mathrm{pix}(g^\circ)\right)$$ (A.0.5)

where $\mathbf{V}_\mathrm{pix}^\mathsf{T} = [V_\mathrm{pix}(q): q\in\mathcal{Q}]$ is the potential $Q$-vector, $\mathbf{F}^\mathsf{T}_\mathrm{pix}(g^\circ) = [f_\mathrm{pix}(q\vert g^\circ): q\in\mathcal{Q}]$ is the corresponding vector of empirical marginal probabilities of signals, and $Z_\mathrm{irf}$ is the partition function. The latter has in this case the analytical form: $Z_\mathrm{irf}=\left( \sum_{q\in\mathcal{Q}}\exp V_\mathrm{pix}(q) \right )^{\vert\mathcal{R}\vert}$.

Assuming the centred potential, $\sum_{q\in\mathcal{Q}}V_\mathrm{pix}(q)=0$, it is easily shown that the actual potential MLE and its first approximation obtained for a training image $g^\circ$ much as in Proposition A.0.2 but for the IRF in Eq. (A.0.5) are, respectively,

$$\begin{array}{lll} V_\mathrm{pix}(q) & = & \ln f_\mathrm{pix}... ... f_\mathrm{pix}(q\vert g^\circ)-\frac{1}{Q} \right) \end{array}$$ (A.0.6)

Table A.1 presents both the estimates in a special case when one intensity, $q_\mathrm{sp}$, has the empirical probability $f_\mathrm{pix}(q_\mathrm{sp}\vert g^\circ) = f$ and all remaining intensities are equiprobable, $f_\mathrm{pix}(q\vert g^\circ) =\frac{1- f}{Q-1}$; $q\in\mathcal{Q}\backslash q_\mathrm{sp}$. The estimates are given in function of $Q$ and the relative probability $\beta=\frac{f(Q-1)}{1-f}$. For small $Q$, both the estimates are close to each other except for $f \approx 1$. But for larger $Q$, the approximate MLE of Eq. (A.0.4) exceeds considerably the actual one so that the approximation may be intolerably inaccurate for the MGRFs, too.

Table: Approximate (``e"), $V_\mathrm{pix}(q_\mathrm{sp})$, and actual (``a"), $V^\ast_\mathrm{pix}(q_\mathrm{sp})$, MLE of the centred potentials specifying the generic IRFs for the relative probability $\beta=\frac{f(Q-1)}{1-f}$ if $f_\mathrm{pix}(q_\mathrm{sp}\vert g^\circ) = f$ and $f_\mathrm{pix}(q\vert g^\circ) =\frac{1- f}{Q-1}$ for $q\in\mathcal{Q}\backslash q_\mathrm{sp}$.

		Relative probabilities $\beta$
$Q$		1.0	2.0	5.0	10	20	50	100	200	500	$10^3$	$10^4$	$10^5$	$\infty$
2	e	0.00	0.67	1.33	1.64	1.81	1.92	1.96	1.98	1.99	2.00	2.00	2.00	2.00
	a	0.00	0.35	0.80	1.15	1.50	1.96	2.30	2.65	3.11	3.45	4.61	5.76	$\infty$
	$f$	0.50	0.67	0.83	0.91	0.95	0.98	0.99	1.00	1.00	1.00	1.00	1.00	1.00
$2^2$	e	0.00	0.80	2.00	2.77	3.30	3.70	3.84	3.92	3.97	3.98	4.00	4.00	4.00
	a	0.00	0.52	1.21	1.73	2.25	2.93	3.45	3.97	4.66	5.18	6.91	8.63	$\infty$
	$f$	0.25	0.40	0.63	0.77	0.87	0.94	0.97	0.98	0.99	1.00	1.00	1.00	1.00
$2^3$	e	0.00	0.89	2.67	4.24	5.63	6.88	7.40	7.69	7.87	7.94	7.99	8.00	8.00
	a	0.00	0.61	1.41	2.01	2.62	3.42	4.03	4.64	5.44	6.04	8.06	10.1	$\infty$
	$f$	0.13	0.22	0.42	0.59	0.74	0.88	0.93	0.97	0.99	0.99	1.00	1.00	1.00
$2^4$	e	0.00	0.94	3.20	5.76	8.69	12.1	13.8	14.8	15.5	15.8	16.0	16.0	16.0
	a	0.00	0.65	1.51	2.16	2.81	3.67	4.32	4.97	5.83	6.48	8.63	10.8	$\infty$
	$f$	0.06	0.12	0.25	0.40	0.57	0.77	0.87	0.93	0.97	0.99	1.00	1.00	1.00
$2^5$	e	0.00	0.97	3.56	7.02	11.9	19.4	24.2	27.6	30.1	31.0	31.9	32.0	32.0
	a	0.00	0.67	1.56	2.23	2.90	3.79	4.46	5.13	6.02	6.69	8.92	11.2	$\infty$
	$f$	0.03	0.06	0.14	0.24	0.39	0.62	0.76	0.87	0.94	0.97	1.00	1.00	1.00
$2^6$	e	0.00	0.98	3.76	7.89	14.7	27.8	38.9	48.4	56.7	60.2	63.6	64.0	64.0
	a	0.00	0.68	1.58	2.27	2.95	3.85	4.53	5.22	6.12	6.80	9.07	11.3	$\infty$
	$f$	0.02	0.03	0.07	0.14	0.24	0.44	0.61	0.76	0.89	0.94	0.99	1.00	1.00
$2^7$	e	0.00	0.99	3.88	8.41	16.5	35.4	55.8	77.9	102.	113.	126.	128.	128.
	a	0.00	0.69	1.60	2.28	2.97	3.88	4.57	5.26	6.17	6.85	9.14	11.4	$\infty$
	$f$	0.01	0.02	0.04	0.07	0.14	0.28	0.44	0.61	0.80	0.89	0.99	1.00	1.00
$2^8$	e	0.00	1.00	3.94	8.69	17.7	41.1	71.4	112.	169.	204.	250.	255.	256.
	a	0.00	0.69	1.60	2.29	2.98	3.90	4.59	5.28	6.19	6.88	9.17	11.5	$\infty$
	$f$	0.00	0.01	0.02	0.04	0.07	0.16	0.28	0.44	0.66	0.80	0.98	1.00	1.00