Biomedical Engineering Reference
In-Depth Information
consider Dirichlet boundary conditions (i.e., the values
u
p
in boundary voxels are
not considered as unknown), then
i
=
i
l
,...,i
r
,
j
=
j
l
,...,j
r
,
k
=
k
l
,...,k
r
,
where
i
r
−
≤
N
1
−
2,
j
r
−
≤
N
2
−
2,
k
r
−
≤
N
3
−
2. We define the
i
l
j
l
k
l
N
1
space discretization step
h
=
, and for simplicity we assume that voxels have
cubic shape. For every co-volume
p
, the set
C
p
consists of
6 neighbours, west
u
i−
1
,j,k
, east
u
i
+1
,j,k
, south
u
i,j−
1
,k
, north
u
i,j
+1
,k
, bottom
u
i,j,k−
1
, and top
u
i,j,k
+1
, and the set
=
{
w, e, s, n, b, t
}
N
p
consists of 24 tetrahedra.
In every discrete time step
n
=1
,...,N
and for every
i, j, k
, we compute
the absolute value of gradient
u
n−
1
T
|∇
|
on these 24 tetrahedra. We denote by
G
z,l
C
p
the square of the gradient on the tetrahedra crossing
thewest, east, south, north, bottom, and top co-volume faces. If we define (omitting
upper index
n
i,j,k
,l
=1
,...,
4
,z
∈
−
1)
s
i,j,k
=
u
i,j,k
+
u
i−
1
,j,k
+
u
i,j−
1
,k
+
u
i−
1
,j−
1
,k
+
+
u
i,j,k−
1
+
u
i−
1
,j,k−
1
+
u
i,j−
1
,k−
1
+
u
i−
1
,j−
1
,k−
1
)
/
8
,
the value at the left-south-bottom NDF node of the co-volume, then for the west
face we get
u
i,j,k
−
2
s
i,j,k
+1
−
2
u
i−
1
,j,k
h
s
i,j,k
G
w,
1
i,j,k
=
+
+
h
u
i,j,k
+
u
i−
1
,j,k
−
2
,
s
i,j,k
+1
−
s
i,j,k
h
u
i,j,k
−
2
s
i,j
+1
,k
+1
−
2
u
i−
1
,j,k
h
s
i,j,k
+1
G
w,
2
i,j,k
=
+
+
h
s
i,j
+1
,k
+1
+
s
i,j,k
+1
−
u
i,j,k
−
u
i
−
1
,j,k
h
2
,
(24)
u
i,j,k
−
2
s
i,j
+1
,k
+1
−
2
u
i−
1
,j,k
h
s
i,j
+1
,k
G
w,
3
i,j,k
=
+
+
h
s
i,j
+1
,k
+1
+
s
i,j
+1
,k
−
2
,
u
i,j,k
−
u
i−
1
,j,k
h
u
i,j,k
−
2
s
i,j
+1
,k
−
2
u
i−
1
,j,k
h
s
i,j,k
G
w,
4
i,j,k
=
+
+
h
u
i,j,k
+
u
i−
1
,j,k
−
2
,
s
i,j
+1
,k
−
s
i,j,k
h
and correspondingly we get all
G
z,l
i,j,k
for the further co-volume faces.
In the same way, but only once at the beginning of the algorithm, we compute
values
G
σ,z,l
C
p
, changing
u
by
I
σ
in the previous expressions,
and we apply function
g
to all these values to get discrete values of
g
T
.
i,j,k
,
l
=1
,...,
4
,z
∈