Biomedical Engineering Reference
In-Depth Information
deformable models. Instead of operating on the 2D contours directly, it embeds
the contour into a 3D function set, called the level set function. The level set
function evolved by solving a set of PDEs (partial differential equations), whereas
the contour was modeled by the zero level set function:
C
(
s, t
)=
{
s, t
|
Ψ(
s, t
)=0
}
,
(34)
where Ψ(
s, t
) is the 3D level set function and
C
(
s, t
) denotes the contour of the
object. The advantage of level set and geodesic deformablemodels are their power-
ful ability to handle arbitrary complex topological changes by merging or splitting
of contours automatically. They are also quite insensitive to initial (starting) po-
sitions, which we will be shown in our examples. The disadvantages are their
computational complexity and comparatively more difficult implementation. In
addition, because of the inaccuracy of computing the level set function numerically,
some algorithms may require reinitializations.
In this section, we will introduce the level set-based deformable models with
front propagation and the geodesic deformable models. They have already been
shown to be very similar [21].
4.1. Level Set-BasedDeformableModels Using Front Propagation
This shape model with front propagation was proposed by Malladi et al. [5]
and numerically implemented with the level set method. Let
γ
(
t
) be the contour
of the object modeled as the zero level set
{
Ψ=0
}
of a scalar Lipschitz function
Ψ such that
Ψ(
s, t
=0)
>
0 on Ω
1
,
Ψ(
s, t
=0)
<
0onΩ
2
and Ψ(
s, t
=0)=0on
C,
(35)
where Ω
1
is the region in which
s
locates inside the contour
C
, and Ω
2
is the
region denoting those
s
outside the contour
C
. The evolution of the contour
C
is
given by the zero-level set of the level set function
. Assume
the evolving speed function in the normal direction of contour
C
is defined as
F
;
we then have [30]
{
s, t
|
Ψ(
s, t
)=0
}
∂
∂t
=
|∇
Ψ
|
F,
Ψ(
s,
0)=Ψ
0
(
s
)
,
(36)
where Ψ
0
(
s
) is the initial position of the contour
C
. For certain types of functions,
F
, we obtain the standard Hamilton-Jacobi equations.
Some approaches have different definitions of the speed function
F
. In [5],
F
is separated as
F
A
and
F
G
, where
F
A
is named as the advection term that is
independent of the contour's geometry and
F
G
refers to the term that is related to
the geometry of the contour. If we set
F
G
=0to denote the situation of moving
the contour at a constant speed, then
F
=
F
A
. Let
F
A
=
γ
be an adjustable
parameter; we obtain
γ
M
1
−
F
=
γ
−
M
2
(
|∇
G
σ
∗
I
(
x, y
)
|−
M
2
)
,
(37)
