Biomedical Engineering Reference
In-Depth Information
is given. A detailed description of our proposed approach to get an accurate
image model is then presented. Finally, we will apply the proposed model in the
segmentation of lung CT.
9.2.1 Statistical Framework
The observed image is assumed to be a composites of two random process: a
high-level process
X
, which represents the classes that form the observed image;
and a low-level process
Y
, which describes the statistical characteristics of each
class.
The high-level process
X
is a random field defined on a rectangular grid
S
of
N
2
points, and the value of
X
will be written as
X
s
. Points in
X
will take
values in the set (
1
,...,
m
), where
m
is the number of classes in the given
image.
Given
x
, the conditional density function of
y
is assumed to exist and to be
strictly positive and is denoted by
p
(
Y
=
y
|
X
=
x
)or
p
(
y
|
x
).
Finally, an image is a square grid of pixels, or sites,
{
(
i
,
j
):
i
=
1to
N
,
j
=
1to
N
}
. We adopt a simple numbering of sites by assigning sequence number
t
=
j
+
N
(
i
−
1) to site
s
. This scheme numbers the sites row by row from 1 to
N
2
, starting in the upper left.
9.2.2 Gibbs Random Fields
In 1987, Boltzmann investigated the distribution of energy states in molecules
of an ideal gas. According to the Boltzmann distribution, the probability of a
molecule to be in a state with energy
ε
is
1
Z
e
−
1
KT
ε
p
(
ε
)
=
(9.1)
,
where
Z
is a normalization constant, that makes the sum of probabilities equal to
1.
T
is the absolute temperature, and
K
is Boltzmann's constant. For simplicity
we assume that the temperature is measured in energy units, hence
KT
will be
replaced by
T
.
Gibbs used a similar distribution in 1901 to express the probability of a whole
system with many degrees of freedom to be in a state with a certain energy. A
discrete GRF provides a global model for an image by specifying a probability
