Biomedical Engineering Reference
In-Depth Information
graph of
ϕ
F
γ
level set
ϕ = 0
Figure 4.1:
Example of a level set representation of a circle.
be represented using a level set function. However, the level set method, with
some modifications, can even be applied to these cases as well. These variations
will be discussed in Section 4.3.
Once the level set function,
φ
, is constructed, the evolution equation for the
interface must be rewritten in terms of
φ
. Given the interface
, let
F
(
x
)bethe
speed of the interface in the direction of the normal (see Fig. 4.2). Let
x
(
t
)bea
point on the interface which evolves with the interface, then
φ
(
x
(
t
)
,
t
)
≡
0 for
all
t
. Differentiating with respect to
t
gives
d
x
dt
=
0
.
∂φ
∂
t
+∇
φ
·
(4.3)
Now, the evolution of
x
(
t
) can be described by
d
x
dt
=
F
(
x
(
t
)
,
t
)
n
,
(4.4)
where
n
is the unit normal to the interface. Use the fact that the unit normal can
also be computed to be
n
=∇
φ/
∇
φ
, and substituting this with Eq. 4.4 into
F
n
=
F
∇ϕ
||∇ϕ||
x
(t)
ϕ = 0
Figure 4.2:
Illustration of the relationship between
φ
(
x
,
t
),
x
, and
F
.
