Biomedical Engineering Reference
In-Depth Information
of implementing an alternative formulation for our work, but also to compare Level
Sets with T-Surfaces within its context.
To answer this question we must review some details of the Level Sets for-
mulation [8]. The main idea of this method is to represent the deformable surface
(or curve) as a zero level set
x
∈
3
|
G
(
x
)=0
of an embedding function:
G
:
3
×
+
→
,
(9)
such that the deformable surface (also called the
front
in this formulation), at
t
=0,
is given by a surface
S
:
S
(
t
=0)=
x
G
(
x, t
=0)=0
.
∈
3
|
(10)
The next step is to find an Eulerian formulation for the front evolution. Fol-
lowing Sethian [8], 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
|
(11)
∈
+
of the propagating front
S
. From its implicit
definition given above, we have
Let us take a point
x
(
t
),
t
G
(
x
(
t
)
,t
)=0
.
(12)
We can now use the Chain Rule to compute the time derivative of this equation:
G
t
+
F
|∇
G
|
=0
,
(13)
F
where
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,
(14)
where
d
is the distance from
x
to the surface
S
(
x, t
=0)and the signal indicates
if the point is interior (-) or exterior (+) to the initial front.
Finite-difference schemes, based on a uniform grid, can be used to solve
Eq. (13). The same entropy condition of T-Surfaces (
once a grid node is burnt it
stays burnt
) is incorporated in order to drive the model to the desired solution (in
fact, T-Surfaces was inspired by the Level Sets model [9]).
