Biomedical Engineering Reference
In-Depth Information
10.5.1 Numerical Solutions for Region Force Diffusion
for RAGS
Initially, a mesh grid needs to be selected, with final accuracy directly dependent
on its resolution. However, due to the nature of a digital image, the grid resolution
is constrained to the pixel level. It was shown in Section 10.4.2 that the steady
solution of (10.10) can be achieved by computing the equilibrium state of (10.12).
The initial state of the region force vector field
R
is given by the gradient of
the region map
R
. Simple central differences can be used to approximate
∇
R
,
resulting in vectors that are then diffused. Let
x
and
y
be the grid spacing,
t
be the time step, and
i
,
j
, and
n
represent the spatial position and time. The
partial derivative of time can be approximated by forward difference as
u
n
+
1
i
,
j
−
u
i
,
j
t
u
t
=
.
(10.23)
The spatial partial derivatives can be solved using central differences ap-
proximation given by
u
i
+
1
,
j
+
u
i
,
j
+
1
+
u
i
−
1
,
j
+
u
i
,
j
−
1
−
4
u
i
,
j
x
y
2
u
=
∇
.
(10.24)
The solutions to partial derivatives of
v
(
x
,
y
,
t
) are similar to those of
u
(
x
,
y
,
t
).
The weighting functions given in (10.11) can be easily computed. Thus, substi-
tuting the partial derivatives into (10.12) gives the following iterative solution:
⎧
⎨
u
n
+
1
i
,
j
=
u
i
,
j
+
t
(10.25)
,
⎩
n
+
1
i
,
j
+
t
v
i
,
j
=
v
where,
p
(
·
)
i
,
j
x
y
(
u
i
+
1
,
j
+
u
i
,
j
+
1
+
u
i
,
j
−
1
+
u
i
−
1
,
j
−
4
u
i
,
j
)
−
q
(
·
)
i
,
j
(
u
i
,
j
−
R
x
,
ij
)
=
and
p
(
·
)
i
,
j
x
y
(
v
n
i
+
1
,
j
+
v
n
i
,
j
+
1
+
v
n
i
,
j
−
1
+
v
n
n
n
i
−
1
,
j
−
4
v
i
,
j
)
−
q
(
·
)
i
,
j
(
v
i
,
j
−
R
y
,
ij
)]
,
=
where
R
x
,
ij
and
R
y
,
ij
are partial derivatives of
R
. They can also be approximated
by central differences as
⎧
⎨
R
i
+
1
,
j
−
R
i
−
1
,
j
2
x
R
x
,
ij
=
.
(10.26)
⎩
R
i
,
j
+
1
−
R
i
,
j
−
1
2
y
R
y
,
ij
=
The convergence is guaranteed with the time step restriction of (10.13).
