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Fig. 33 Synthetic example for the gray level probabilistic model. a Empirical densities and the
estimated dominant modes. b Normalize absolute error. c LCG components of the gray level
probabilistic models. d Final estimated marginals densities
summarizes the estimation of the gray level probabilistic model (More details can
be found in [
38
]).
Figure
33
a shows the approximation of the given volume gray levels empirical
distribution H with a mixture of two Gaussians P
2
using conventional EM algo-
rithm. Figure
33
b illustrates the absolute of the deviations between H and P
2
, which
is approximated by a mixture of Gaussians P
n
using conventional EM algorithm.
Figure
33
c shows the approximation of the joint distribution P, which consists of P
2
and
þ
ve and
ve components of P
n
. The summation of a dominant mode and the
þ
ð
I
p
j
f
p
Þ
closest
ve and
ve components is the marginal distribution of a class P
as
shown in Fig.
33
d.
2.5.3 Spatial Interaction Model
Assuming the region map f
is a realization of random variables, for
which the joint distribution is presented as a Markov-Gibbs Random Field with
respect to a neighborhood system
¼
f
f
1
; ...;
f
jPj
g
. The Gibbs potential governing asymmetric
pairwise co-occurrences of the region labels can be described as follows:
N
V
ð
f
p
;
f
q
Þ
¼
cdð
f
p
6
¼
f
q
Þ;
ð
37
Þ
where
c
is estimated analytically using our approach [
37
], which is based on MLE of the
MGRF:
c
is the potential value specifying the Gibbs potential. This potential value
