Information Technology Reference
In-Depth Information
Fig. 41 Sampling up to 3 standard deviations (from b
fi;
3
g
to b
fi;
3
g
) along the
first four principle
components (p.c.) (where
i
¼
f
1
;
2
;
3
;
4
g
for the
first four p.c.) from the mean for a set of a 80
training vertebral body shapes
U
n
¼
U þ
Y
n
B
t
;
n
¼
1
;
2
; ...;
N
:
ð
52
Þ
The shape variation of the
first and fourth p.c.s is shown in Fig.
41
. The shape
variation decreases from the
first to the fourth (or last) p.c. respectively. Hence,
selection of L value helps to capture the necessary shape variation with minimum
information.
Finally, the shape model is required to capture the variations in the training set.
This model is considered to be a weighted sum of the projected SDFs as follows:
X
N
w
n
U
n
:
U
p
¼
ð
53
Þ
n¼1
t
Let w
¼½
w
1
; ...;
w
N
to be the weighting coef
cient vector. By varying these
weights,
U
p
can cover all values of the training distance functions and, hence, the
shape model changes according to all of the given images [
44
].
2.6.2 Segmentation Approach
To estimate the initial labeling f*, we use the graph cuts which integrates the LCG
and MGRF methods as we discussed above. To segment vertebrae, we initially
labeled the volume based on its gray level probabilistic model. Then we create a
weighted undirected graph with vertices corresponding to the set of volume voxels
P
, and a set of edges connecting these vertices. Each edge is assigned a nonnegative
weight. The graph also contains two special terminal vertices s (source)
“
vertebrae
”
,
and t (sink)
“
background
”
. Consider a neighborhood system in
P
, which is rep-
resented by a set
N
of all unordered pairs {p, q} of neighboring voxels in
P
. Let
L
the set of labels {
}, correspond to the vertebrae and background regions
respectively. Labeling is a mapping from
“
0
”
,
“
1
”
P
to
L
, and we denote the set of labeling
by f
¼
f
f
1
; ...;
f
p
; ...;
f
jPj
g
. In other words, the label f
p
, which is assigned to the
