Biomedical Engineering Reference
In-Depth Information
Remark:
In the case of higher-dimensional HJ equations with symmetric Hamil-
tonians, numerical schemes can be built by replicating in each space variable
what we did for the one-dimensional case. Hence, the following equation with
convex (
H
is convex
H
smooth and
∂
2
H
(
p
)
⇔
∂p
i
∂p
j
≥
0(
p
=(
p
1
,p
2
,...,p
n
)) or
H
(
λp
+(1
−
λq
))
≤
λH
(
p
)+(1
−
λ
)
H
(
q
)
,
∀
0
≤
λ
≤
1
,
∀
p, q
∈
R
) Hamil-
tonian
U
t
+
H
(
U
x
,U
y
,U
z
)=0
can be approximated by
U
n
+1
i,j,k
U
i,j,k
−
∆
tg
(
D
−x
U
i,j,k
,
D
+
x
U
i,j,k
,D
−y
U
i,j,k
,D
+
y
U
i,j,k
,D
−z
U
i,j,k
,D
+
z
U
i,j,k
)
,
=
where
U
i−
1
,j,k
−
U
i,j,k
U
i−
1
,j,k
−
U
i,j,k
D
−x
U
i,j,k
=
,
and
D
+
x
U
i,j,k
=
.
∆
x
∆
x
The same thing applies for the other space variables, with ∆
y
and ∆
z
the spacings
in the
y
and
z
directions.
Example — first-order accuracy scheme
: The 2D version of level set Eq. (7)
can be approximated as
1
2
,
φ
n
+1
i,j
=
φ
i,j
−
∆
t
(max(
F
i,j
,
0)
∇
+
+ min(
F
i,j
,
0)
∇
−
)
where
i,j
,
0)
2
+ max(
D
−y
i,j
,
0)
2
+ min(
D
+
y
1
2
,
∇
+
= [max(
D
−x
i,j
,
0)
2
+ min(
D
+
x
i,j
,
0)
2
]
i,j
,
0)
2
+ max(
D
+
y
i,j
,
0)
2
+ min(
D
−y
1
2
.
∇
−
= [max(
D
+
x
i,j
,
0)
2
+ min(
D
−x
i,j
,
0)
2
]
where we have used shorthand notation in which
D
+
α
i,j
=
D
+
α
φ
i,j
with
α
=
x, y
. Note that this scheme chooses the appropriate
“upwinding”
[37] direction
depending on the sign of the speed fuction
F
.
Finally, all of the second-order curvature-dependent terms can be approxi-
mated, for instance, using central differences:
φ
i
+1
,j
−
2
φ
i,j
+
φ
i
−
1
,j
(∆
x
)
2
∂
2
φ
∂x
2
=
,
φ
i
+1
,j
+1
+
φ
i−
1
,j−
1
−
φ
i
+1
,j−
1
−
φ
i−
1
,j
+1
∂
2
φ
∂x∂y
=
.
4(∆
x
)
2
The same applies for the other derivatives and for higher dimensions. For more
details on the numerical techniques for level sets, see [40, 41, 42]. Estimating
the region parameters based on the level set function will be presented in the next
section so that they can be placed in the main evolution equation.