Biomedical Engineering Reference
In-Depth Information
we get from (12)-(13) our
3D fully-discrete semi-implicit co-volume scheme
:
Let u
p
,p
=1
,...,M be given discrete initial values of the segmentation function.
Then, for n
=1
,...,N we look for u
p
,p
=1
,...,M, satisfying
u
p
+
τ
q∈C
p
d
n−
1
p
a
n−
1
pq
(
u
p
−
u
q
)=
d
n−
1
u
n−
1
p
.
(18)
p
The system (18) can be rewritten into the form
+
τ
q∈C
p
τ
q∈C
p
d
n−
1
p
u
p
−
a
n−
1
pq
a
n−
1
pq
u
q
=
d
n−
1
p
u
n−
1
p
,
(19)
and, applying theDirichlet boundary conditions (which contribute to the right-hand
side), it gives a system of linear equations with a matrix
A
M×M
, the off-diagonal
elements of which are symmetric and nonpositive, namely
τa
n−
1
pq
=
−
,
q
∈
A
pq
A
pp
=
d
n−
1
C
p
,
A
pq
=0, otherwise. Diagonal elements are positive, namely,
+
τ
q∈C
p
p
a
n−
1
pq
, and dominate the sum of the absolute values of the nondiagonal
elements in every row. Thus, thematrix of the system is symmetric and a diagonally
dominant M-matrix, which implies that it always has a unique solution for any
τ>
0,
ε>
0, and for every
n
=1
,...,N
. The M-matrix property gives us the
minimum-maximum principle:
u
p
≤
min
p
u
p
u
p
u
p
,
min
p
≤
max
p
≤
max
p
1
≤
n
≤
N,
(20)
which can be seen by the following simple trick. We may temporary rewrite (18)
into the equivalent form:
τ
d
n−
1
u
p
+
a
n−
1
pq
(
u
p
−
u
q
)=
u
n−
1
,
(21)
p
p
q∈C
p
and let max(
u
n
1
,...,u
n
M
) be achieved in the node
p
. Then the whole second
term on the left-hand side is nonnegative, and thus max(
u
n
1
,...,u
n
M
)=
u
p
≤
,...,u
n−
M
). In the same way, we can prove the relation for
the minimum, and together we have
≤
max(
u
n−
1
1
u
n−
1
p
u
n−
1
p
u
p
u
p
u
n−
1
p
min
p
≤
min
p
≤
max
p
≤
max
p
,
1
≤
n
≤
N,
(22)
which by recursion implies the L
∞
stability estimate (20).
