Biomedical Engineering Reference
In-Depth Information
stays burnt
. A termination condition is set based on the number of deformation
steps in which a simplex has remained a transverse one.
7.2.4 Level Sets
It will be useful to review some details of
level sets
, which is the implicit for-
mulation presented in [46]. The main idea of this method is to represent the
deformable surface (or curve) as a level set
{
x
∈
3
|
G
(
x
)
=
0
}
of an embedding
function:
3
×
+
→
,
G
:
(7.15)
such that the deformable surface (also called
front
in this formulation), at
t
=
0,
is given by a surface
S
:
S
(
t
=
0)
=
x
∈
|
G
(
x
,
t
=
0)
=
0
.
3
(7.16)
The next step is to find an Eulerian formulation for the front evolution. Fol-
lowing Sethian [46], let us suppose that the front evolves in the normal direction
with velocity
F
, where
F
may be a function of the curvature, normal direction,
etc.
We need an equation for the evolution of
G
(
x
,
t
), considering that the surface
S
is the level set given by:
S
(
t
)
=
x
∈
|
G
(
x
,
t
)
=
0
.
3
(7.17)
Let us take a point
x
(
t
),
t
∈
+
, of the propagating front
S
. From its implicit
definition given above, we have:
G
(
x
(
t
)
,
t
)
=
0
.
(7.18)
Now, we can use the chain rule to compute the time derivative of this expression:
G
t
+
F
|∇
G
| =
0
,
(7.19)
where
F
=
F
is called the
speed function
. An initial condition
G
(
x
,
t
=
0) is
required. A straightforward (and expensive) technique to define this function is
to compute a signed-distance function as follows:
G
(
x
,
t
=
0)
=±
d
,
(7.20)
