Biomedical Engineering Reference
In-Depth Information
α(
)
The energy of a snake
s
can be written as
E
snake
(α(
s
))
=
E
ext
(α(
s
))
+
E
int
(α(
s
)).
(4.1)
The external energy
E
ext
represents external constraints and image influence to
get the contour pulled towards desired image features.
E
ext
(α(
s
))
=
−
ω
1
|ⓦ
α(
s
)
|
ds
(4.2)
The internal energy expresses smoothness and tension constraints:
2
ω
2
ω
3
2
2
1
2
∂α(
s
)
∂
α(
s
)
E
int
(α(
s
))
=
+
(4.3)
s
2
∂
s
∂
where
ω
2
is the weight for the influence of stretching on the contour and
ω
3
weighs
the bending.
Xu et al. [
27
] have proposed the gradient vector flow (GVF, see Sect.
4.7.2.5
)as
a new external image force to improve segmentation results. This concept has been
extended by Cheng et al. [
28
] to a directional GVF. In 1993 Ivbins and Porril [
29
]
have shown a region growing segmentation that exploits a snake with pressure force
and statistical characteristics of the image. Mitrea et al. [
30
] have reviewed snakes
and proposed an iterative method for snake calculation. Wang et al. [
31
]showed
a method for muscle extraction from the leg using snakes. Kauffmann et al. [
32
]
proposed a snake-based method to quantify cartilage thickness and volume in MR
images.
Ip and Shen [
33
] proposed affine invariant active contour models (AI snakes) in
1998. AI snakes are an efficient method of establishing correspondence between
model and data. The basis for their energy function is formed by local and global
affine-invariant features.
In 1999, Vemuri and Guo [
34
] presented hybrid geometric active models. They
introduced a hybrid geometric snake to allow for topology changes. Their model
allows for the representation of global shapes with local details.
Another approach to topology adaptive snakes has been proposed by McInerney
and Terzepoulos in 2000 [
35
]. Their T-Snakes offer topological flexibility and sig-
nificantly extend conventional snakes. This approach has been extended further by
others [
36
,
37
].
4.3 Level Sets
Level sets have been introduced in 1988 by Osher and Sethian [
38
] to overcome the
difficulties that snakes have with changes in topologies. Their work has proven to be a
