Biomedical Engineering Reference
In-Depth Information
3.
PARAMETRIC DEFORMABLE MODELS
Recalling equation (1),
1
E
df
=
(
E
int
(
x
(
s
)) +
E
ext
(
x
(
s
)))
ds.
(16)
0
Depending on the specific definition of the external energy,
E
ext
, there are several
kinds of parametric deformable models. In this section we will introduce the
traditional deformable model, the balloon deformable model, the GVF deformable
model, and the robust color GVF deformable model.
3.1. Traditional Deformable Model
In the traditional deformable model [4], the external energy
E
ext
is simply
defined as the gradient of the graylevel image. As the gradient edge detector is
sensitive to noise, the actual external energy is defined as the gradient of smoothed
image using Gaussian filter, which is
E
ext
(
x
(
s
)) =
−
∇
G
σ
(
x,y
)
∗
I
(
x, y
)
,
(17)
where
G
σ
is the Gaussian filter with
σ
as the deviation, and
I
denotes the graylevel
image.
Using the calculus of variants theory introduced in the previous section, it can
be determined that in order to minimize the integral
E
df
in (17), it is necessary to
solve the following partial differential equation:
α
∗
x
ss
(
s
)
−
β
∗
x
ssss
(
s
)
−
γ
∗∇
E
ext
(
x
(
s
))=0
,
(18)
where
x
ssss
(
s
) are the second and fourth derivatives of the curve
with respect to the parameter
s
, and
x
ss
(
s
) and
E
ext
(
x
(
s
)) is the gradient. Notice that the
signs before each term are not important because the parameters
α
,
β
, and
γ
are
adjustable. In order to solve the PDE (partial differential equation), we can apply
the FDM introduced in the previous section and define
x
as the function of time
t
and
s
as follows:
∇
x
(
s, t
)=
α
∗
x
ss
(
s, t
)
−
β
∗
x
ssss
(
s, t
)
−
γ
∗∇
E
ext
(
x
(
s, t
))
.
(19)
After simple mathematical deduction, we are left with the following iterative equa-
tions:
∂E
ext
∂x
A
·
x
(
s, t
)+
γ
∗
=
−
(
x
(
s, t
)
−
x
(
s, t
−
1))
,
(20)
∂E
ext
∂y
A
·
y
(
s, t
)+
γ
∗
=
−
(
y
(
s, t
)
−
y
(
s, t
−
1))
,
(21)
