Biomedical Engineering Reference
In-Depth Information
2.2. Geometric Deformable Models
In parametric deformable models an explicit parametric representation of the
curve is used, which can lead to fast real-time implementation. However, it is
difficult for parametric deformable models to adapt the model topology during
deformation. On the other hand, the implicit forms, i.e., geometric deformable
models, are designed to handle topological changes naturally.
Geometric deformable models were introduced independently by Malladi
et al. [52] and Caselles et al. [65]. These models are based on the theory of curve
evolution and the level set method, where the evolving curves or surfaces are
implicitly represented as a level set of a higher-dimensional scalar function, i.e.,
a level set function. Thus, geometric deformable models are also called level set
methods in much of the literature.
2.2.1. The level set method
The level set method views a moving curve as the zero level set of a higher-
dimensional function
φ
(
x
,t
) [48]. Generally, the level set function satisfies
φ
(
x
,t
)
<
0
in
Ω(
t
)
,
φ
(
x
,t
)=0
in
C
(
t
)
,
(10)
\
Ω(
t
)
,
R
n
φ
(
x
,t
)
>
0
in
where the artificial time
t
denotes the evolution process,
C
(
t
) is the moving curve,
and Ω(
t
) represents the region (possibly multi-connected) that
C
(
t
) encloses. An
evolution equation for the curve
C
moving with speed
F
in its normal direction is
given by
φ
t
=
F
(
x
)
|∇
φ
|
.
(11)
Here, the surface
φ
=0corresponding to the propagating hypersurface may change
topology, as well as form sharp corners.
A particular case is motion by mean curvature, when
F
= div(
∇
φ
(
x
)
/
|∇
φ
(
x
)
|
)
is the curvature of the level curve of
φ
passing through
x
. The above equation
becomes
div
∇
,
∂φ
∂t
=
|∇
φ
φ
|·
(12)
|∇
φ
|
with
φ
(0
,
x
)=
φ
0
(
x
) and
t
∈
(0
,
∞
).
2.2.2. Geometric deformable model
When utilized for image segmentation, the speed function is usually con-
structed by merging image features.
A geometric deformable model based on
