Biomedical Engineering Reference
In-Depth Information
models include Gradient, Balloon [44], the Distance Map [63], and Gradient Vector
Flow (GVF) [46]. The goal of modeling an external force is to find one kind of
force that has the capability of pushing the curve to significant concavities or
convexes, retaining a large capture range, stopping the evolving curve at the edge
gaps, and processing high-noise images. For our purposes, we will now give a
brief overview of the classic snake model and the GVF snake model.
2.1.1. Classic snake model
x
(
s
)=(
x
(
s
)
,y
(
s
)),
where
s
is the arc length, and
x
(
s
) and
y
(
s
) are
x
and
y
coordinates along the
contour, and the energy of the model is given by
Geometrically, a classic snake model is described by
E
snake
=
E
(
x
(
s
))
ds
=
E
int
(
x
(
s
)) +
E
ext
(
x
(
s
))
ds,
(1)
snake
snake
where
E
ext
is the external energy, and
E
int
is the internal energy, given by
α
(
t
)
2
,
2
∂
2
x
(
s
)
∂s
2
E
int
=
1
2
∂
x
(
s
)
∂s
+
β
(
t
)
(2)
where
α
and
β
are the coefficients that control the snake's tension and rigidity,
respectively. The goal is to find a snake,
x
∗
(
s
), that minimizes
E
snake
. The external
energy is in accord with the image features, and for a given image
f
(
x, y
),
γ
∇
G
σ
(
x,y
)
∗
f
(
x, y
)
,
E
ext
=
−
(3)
where
G
σ
(
x,y
)
is the two-dimensional Gaussian kernel with
σ
as the standard
deviation.
Solved by the variational method, the minimum of
E
snake
has to satisfy the
following Euler-Lagrange equation:
x
(
s
)+
β
x
(
s
)+
∇
−
α
E
ext
=0
.
(4)
Discretization of Eq.
(4) by the finite-difference method yields a linear system
[43]:
AP
=
L
,
(5)
where
denote the
discrete contour points vector and the forces at these points, respectively. From
the initial position of the contour, the following associated evolution equation can
be solved:
A
is a pentadiagonal matrix depending on
α
and
β
, and
P
and
L
=
P
L
P
t−
1
,
t
t−
1
+
τ
(
I
+
τ
A
)
P
(6)
x
∗
(
s
), can be achieved
where
t
is time, and
τ
is the time step. The final solution,
by solving Eq. (6) iteratively.
