Biomedical Engineering Reference
In-Depth Information
criteria, which probably indicates a leakage. The force field
∇
R
has vectors
pointing toward the center of the region boundaries. The capture area of this
pure region force is quite small: only immediate areas close to region boundaries.
The vectors need to be diffused further away from the region boundaries to
create a larger capture field. To achieve this, we can diffuse
∇
R
resulting in
region forces with a larger capture area along the region boundaries. Hence, the
region force vector field [
R
(
z
)
=
(
u
(
z
)
,v
(
z
))
,
z
=
(
x
,
y
)] is obtained by solving
the following equations:
p
(
|∇
R
|
)
∇
2
u
−
q
(
|∇
R
|
)(
u
−∇
R
u
)
=
0
(10.10)
v
−
q
(
|∇
R
|
)(
v
−∇
R
v
)
=
0
,
p
(
|∇
R
|
)
∇
2
2
is the Laplacian operator with dimensions
u
and
v
,
p
(
·
) and
q
(
·
) are
weighting functions that control the amount of diffusion, and
∇
R
u
and
∇
R
v
are
the components of vector field
∇
R
along the
u
and
v
directions
5
. The weighting
functions are selected such that
p
(
·
) gets smaller as
q
(
·
) becomes larger with the
desirable result that in the
proximity
of large gradients, there will be very little
smoothing and the vector field will be nearly equal to the gradient of the region
map. We use the following functions for diffusing the region gradient vectors:
p
(
|∇
R
|
)
=
e
−
(
|∇
R
|
/
K
)
q
(
|∇
R
|
)
=
1
−
p
(
|∇
R
|
)
,
where
∇
(10.11)
where
K
is a constant and acts as a trade-off between field smoothness and gra-
dient conformity. The solution of (10.10) is the equilibrium state of the following
partial differential equations:
u
t
=
p
(
|∇
R
|
)
∇
2
u
−
q
(
|∇
R
|
)(
u
−∇
R
u
)
v
−
q
(
|∇
R
|
)(
v
−∇
R
v
)
,
(10.12)
2
v
t
=
p
(
|∇
R
|
)
∇
where
u
and
v
are treated as functions of time. These partial differential equa-
tions can be implemented using an explicit finite difference scheme. An iterative
process can be set up and guaranteed to converge with the following constraint
t
≤
x
y
(10.13)
4
p
max
,
, within the
image domain. However, practically we will only choose
x
and
y
directions corresponding to
image plane coordinates. Thus
5
Theoretically,
∇
R
can be diffused in any two orthogonal directions,
u
and
v
are equal to
δ
R
δ
x
and
δ
R
δ
y
R
u
and
R
v
respectively.
∇
∇
