Information Technology Reference
In-Depth Information
c
¼
2
4
F
neq
f
ðÞ
;
ð
38
Þ
where
denotes the relative frequency of the non-equal labels in the voxel
pairs of that family.
F
neq
ð
f
Þ
X
;
1
jj
F
neq
ð
Þ
¼
d
f
p
6
¼
ð
Þ
f
f
q
39
f
p
;
q
g2T
where
T
is the family of the neighboring voxel pairs supporting the Gibbs
potentials.
Therefore, the region map unconditional probability distribution PðfÞ
ð
f
Þ
can be
speci
ed by the following Gibbs probability distribution:
X
1
Z
exp
P
ð
f
Þ
¼
ð
V
ð
f
p
;
f
q
ÞÞ:
ð
40
Þ
f
p
;
q
g2N
2.5.4 Final Energy Minimization Using Three Models:
After estimating the shape model and the appearance models our goal is to integrate
these models to
find the best labelling f i.e., the optimal segmentation. The MAP
estimate of f, using Eqs. (
33
), (
35
) and (
38
), is equivalent to minimizing the
following function:
X
X
X
E
ð
f
Þ
¼
log
ð
P
ð
d
p
f
p
ÞÞ
þ
log
ð
P
ð
I
p
f
p
ÞÞ
þ
V
ð
f
p
;
f
q
Þ:
ð
41
Þ
p
2P
p
2P
f
p
;
q
g2N
first term, in this function, measures disagreement with the shape infor-
mation for assigning a label f
p
to a voxel p. The second term measures the dis-
agreement with with the voxel intensity I
p
for assigning a label f
p
to a voxel p.
Finally, the penalty of the discontinuity between neighbors voxels p and q is
represented by V
The
.
In order to minimize the function Eq. (
41
), we construct a 3D graph and de
ð
f
p
;
f
q
Þ
ne
the weight of each edge as shown in Table
5
. The minimum cost cut on this graph
corresponds to the optimal segmentation boundary between the VB and its back-
ground. This minimum cost cut is computed exactly in polynomial time for two
terminal graph cuts with positive edges weights via s
=
t Min-Cut/Max-Flow algo-
rithm [
39
].
Although each given VB volume should be aligned with the ESP, to which our
training images are registered to create the shape model, it is a challenge to directly
align a VB from its CT slice (e.g., Figure
34
a). Therefore, a preprocessing step [
40
]
is used to detect the VB region (yellow box in Fig.
34
a) in the CT slice. After that
we can use the shape prior and apply our segmentation frame work on that region.
Details are given in the following algorithm.
