Biomedical Engineering Reference
In-Depth Information
Figure 13.
Results from image gradient operators. Top: slices from the original MR data.
Bottom: results from a gradient operator.
being trapped in local minima in badly corrupted images. It is therefore the first
choice for cortical edge detection.
Let
s
be a point on the inner boundary,
t
the other point on the outer boundary
(see Figure 14), and
N
s
the set of points in the normal direction of
s
. Also, let
∇
I
t
be defined similarly.
Then the probability that a coupled edge exists in the neighborhood of
s
that lies
in the inner boundary can be represented by
I
s
be the gradient of
s
in the normal direction, and let
∇
P
(
s
∈
B,
∃
t
∈
B
|∇
I ,
∇
I ,t
∈
N
) = max
P
(
s
∈
B, t
∈
B
|∇
I ,
∇
I ,t
∈
N
)
,
s
t
s
s
t
s
(29)
t∈N
s
where
s
∈
B
means that
s
lies in the boundary and
t
∈
B
means that
t
lies in the
boundary.
Assuming that
∇
I
s
and
∇
I
t
are independent variables, we have
P
(
∇
I
s
,
∇
I
t
|
s
∈
B, t /
∈
B
)=
P
(
∇
I
s
|
s
∈
B
)
P
(
∇
I
t
|
t
∈
B
)
,
(30)
where
P
(
∇
I
s
|
s
∈
B
) is the probability of
∇
I
s
, given the condition that
s
lies
within the boundary.
Using MAP estimation, Eq. (30) can be rewritten as
P
(
s
∈
B,
∃
t
∈
B
|∇
I
s
,
∇
I
t
,t
∈
N
s
)
P
(
∇I
s
|s∈B
)
P
(
∇I
t
|t∈B
)
P
(
∇I
s
,∇I
t
,t∈N
s
)
∈
∈
= max
t∈N
s
P
(
s
B, t
B
)
(31)
∞
max
t∈N
s
P
(
∇
I
s
|
s
∈
B
)
P
(
∇
I
t
|
t
∈
B
)
P
(
s
∈
B, t
∈
B
)
,
