Biomedical Engineering Reference
In-Depth Information
co-volume scheme in a “finite-difference notation.” As is usual for 2D rectan-
gular grids, we associate co-volume
p
and its corresponding center (DF node)
with a couple (
i
,
j
),
i
will represent the vertical direction and
j
the horizontal
direction. If
is a rectangular subdomain of the image domain where
n
1
and
n
2
are number of pixels in the vertical and horizontal directions, respectively,
then
i
=
1
,...,
m
1
,
j
=
1
,...,
m
2
,
m
1
≤
n
1
−
2,
m
2
≤
n
2
−
2 and
M
=
m
1
m
2
.
Similarly, the unknown value
u
n
p
is associated with
u
n
i
,
j
. For every co-volume
p
, the set
N
p
consists of eight triangles (see Fig. 11.18). In every discrete time
step
n
=
1
,...,
N
, and for every
i
=
1
,...,
m
1
,
j
=
1
,...,
m
2
, we compute ab-
solute value of gradient on these eight triangles denoted by
G
i
,
j
,
k
=
1
,...,
8.
For that goal, using discrete values of
u
from the previous time step, we use the
following expressions (we omit upper index
n
−
1on
u
):
0
.
5(
u
i
,
j
+
1
+
u
i
+
1
,
j
+
1
−
u
i
,
j
−
u
i
+
1
,
j
)
h
2
u
i
+
1
,
j
−
u
i
,
j
h
2
G
i
,
j
=
+
,
0
.
5(
u
i
,
j
+
u
i
+
1
,
j
−
u
i
,
j
−
1
−
u
i
+
1
,
j
−
1
)
h
2
u
i
+
1
,
j
−
u
i
,
j
h
2
G
i
,
j
=
+
,
0
.
5(
u
i
+
1
,
j
−
1
+
u
i
+
1
,
j
−
u
i
,
j
−
1
−
u
i
,
j
)
h
2
u
i
,
j
−
u
i
,
j
−
1
h
2
G
i
,
j
=
+
,
0
.
5(
u
i
,
j
−
1
+
u
i
,
j
−
u
i
−
1
,
j
−
1
−
u
i
−
1
,
j
)
h
2
u
i
,
j
−
u
i
,
j
−
1
h
2
G
i
,
j
=
+
,
0
.
5(
u
i
,
j
+
u
i
−
1
,
j
−
u
i
,
j
−
1
−
u
i
−
1
,
j
−
1
)
h
2
u
i
,
j
−
u
i
−
1
,
j
h
2
G
i
,
j
=
+
,
0
.
5(
u
i
,
j
+
1
+
u
i
−
1
,
j
+
1
−
u
i
,
j
−
u
i
−
1
,
j
)
h
2
u
i
,
j
−
u
i
−
1
,
j
h
2
G
i
,
j
=
+
,
0
.
5(
u
i
,
j
+
u
i
,
j
+
1
−
u
i
−
1
,
j
−
u
i
−
1
,
j
+
1
)
h
2
u
i
,
j
+
1
−
u
i
,
j
h
2
G
i
,
j
=
+
,
0
.
5(
u
i
+
1
,
j
+
u
i
+
1
,
j
+
1
−
u
i
,
j
−
u
i
,
j
+
1
)
h
2
u
i
,
j
+
1
−
u
i
,
j
h
2
G
i
,
j
=
+
.
In the same way, but only in the beginning of the algorithm, we compute val-
ues
G
σ,
k
i
,
j
,
k
=
1
,...,
8, changing
u
by
I
0
in the previous expressions, where
σ
I
0
σ
is a smoothed image as explained in the paragraph above. Then for ev-
ery
i
=
1
,...,
m
1
,
j
=
1
,...,
m
2
we construct (north, west, south, and east)
