Biomedical Engineering Reference
In-Depth Information
It is this equation that we will consider for further analysis.
Now, let us represent
C
(
p, t
) as the zero level set of a smooth embedding
function
φ
:
R
2
×
[0
,
T
)
→
R
:
L
0
(
t
)=
{
X
∈
R
2
/φ
(
X, t
)=0
}
,
(3)
whereas the initial curve
C
0
is represented by an initial function
φ
0
.
Since
C
(
p, t
) is implicitly represented as the zero level set of
φ
, then
φ
(
C
,t
)=
0. By differentiation w.r.t.
t
, using the chain rule, we find,
∂
∂t
+
∂φ
∂t
(
C
∇
φ
(
C
,t
)
·
,t
)=
,
,t
)
·
(
βN
)+
∂φ
∂t
(
C
⇒∇
φ
(
C
,t
)=
.
where
denotes the gradient operator.
Using the fact that the unit normal vector is given, at each instant
t
,as
N
∇
=
∇φ
∇φ
−
, we get,
φ
∇
∇
∂φ
∂t
(
C
∇
φ
(
C
,t
)
·
β
(
−
)+
,t
)=0
,
φ
which results in the general motion form in the level set framework, namely:
∂φ
∂t
=
β
∇
φ
.
(4)
Depending on the problem to be solved, one can design the appropriate velocity
term
β
. Among several formulations proposed in the literature (see, e.g., [36, 35]),
we have chosen the following formulation:
β
=
ν
−
κ,
(5)
where
ν
=
±
1 controls the motion direction,
1 is a positive real number, and
κ
is the local curvature of the front defined in the 2D case as follows:
φ
xx
φ
yy
−
2
φ
x
φ
y
φ
xx
+
φ
yy
φ
x
∇
κ
=
.
(6)
φ
3
The latter term in (5) acts as a regularization term.
4.2.3. Our level set model
It is worth mentioning that several evolution models were proposed for seg-
mentation purposes, but most of them depend on a large number of parameters