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 ( 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
 
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