Biomedical Engineering Reference
In-Depth Information
Figure 14.
The coupled boundary is defined by a set of coupled points
s
and
t
.
where
P
(
∇
I
s
,
∇
I
t
,t
∈
N
s
)=
P
(
∇
I
s
,
∇
I
t
,t
∈
N
s
|
s
∈
B, t
∈
B
)
·
P
(
s
∈
B, t
∈
B
)
+
P
(
∇
I
s
,
∇
I
t
,t
∈
N
s
|
s
∈
B, t /
∈
B
)
·
P
(
s
∈
B, t /
∈
B
)
+
P
(
∇
I
s
,
∇
I
t
,t
∈
N
s
|
s/
∈
B, t
∈
B
)
·
P
(
s/
∈
B, t
∈
B
)
+
P
(
∇
I
s
,
∇
I
t
,t
∈
N
s
|
s/
∈
B, t /
∈
B
)
·
P
(
s/
∈
B, t /
∈
B
)
.
The first term of (31) is the conditional boundary probability under the hypothesis
that
s
is on the boundary, the second term under the hypothesis that
t
is on the
boundary, and the last term is the a priori that constrains the distance between
s
and its associated
t
within a certain range.
The conditional boundary probability can be modeled as
P
(
∇
I
s
|
s
∈
B
)=
g
(
|∇
I
s
|
) and
P
(
∇
I
t
|
t
∈
B
)=
g
(
|∇
I
t
|
), where
are the magni-
tude of the gradient at
s
and
t
;
g
(
·
) is a monotonically increasing function, which
will be defined in the experimental section.
The last term restricts the coupled boundary within a reasonable range. It is an
|∇
I
s
|
and
|∇
I
t
|
important term that encodes a priori information. We model it as
P
(
s
∈
B, t
∈
B
)=
h
(dist (
s, t
)), where
h
(
·
) is defined as follows (Figure 15):
|
d
|−
d
min
d
1
−d
min
2
,
d
min
≤|
0
.
75
−
0
.
5
|
d
|−
d
min
d
1
−d
min
d
|≤
d
1
,
0
.
25
,d
1
<
|
d
|
<d
2
,
h
(
d
)=
|
d
|−
d
2
d
max
−d
2
2
|
d
|−
d
2
d
max
−d
2
3
,d
2
≤|
0
.
25
−
0
.
75
+0
.
5
d
|≤
d
max
.
Hence, coupled boundary definition greatly enhances the robustness and ac-
curacy of edge detection in three aspects:
1. It models the ribbon structure of the cortex, which moves the surface to
cross over noisy points or single lines.
