Biomedical Engineering Reference
In-Depth Information
point in which the derivative
∂I
x
/∂t
has a sign opposite to
φ
(
I
x
).If
φ
(
I
x
)
>
0,
the slope of the edge point decreases with time. Otherwise, it increases, which
means that the border becomes sharper. Thus, the diffusion scheme given by
Eq. (16) allows to blur small discontinuities and to enhance stronger ones. In this
work, we have used
φ
as follows:
∇
I
,
φ
=
/K
]
2
(19)
1+[
∇
I
as can be observed from Eq. (8).
In the above scheme,
I
is a scalar field. For vector fields, a useful diffusion
scheme is the Gradient Vector Flow (GVF). It was introduced in [25] and can be
defined through the following equation [37]:
∂u
∂t
=
∇·
(
g
∇
u
)+
h
(
u
−∇
f
)
,
(20)
u
(
x,
0)
=
∇
f,
where
f
is a function of the image gradient (for example,
P
in Eq. (5)), and
g
(
x
)
,h
(
x
) are nonnegative functions defined on the image domain.
The field obtained by solving the above equation is a smooth version of the
original one, which tends to be extended very far away from the object boundaries.
When used as an external force for deformable models, it makes the methods less
sensitive to initialization [25].
As the result of steps (1)-(6) (Section 5) is in general close to the target, we
could apply this method to push the model toward the boundary when the grid is
turned off. However, for noisy images some kind of diffusion (smoothing) must
be used before applying GVF. Gaussian diffusion has been used [25], but precision
may be lost due to nonselective blurring [32].
The anisotropic diffusion scheme presented above is an alternative smoothing
method that can be used. Such an observation points toward the possibility of
integrating anisotropic diffusion and GVF within a unified framework. A straight-
forward way of doing this is allowing
g
and
h
to be dependent upon the vector
field
u
. The key idea would be to combine the selective smoothing of anisotropic
diffusion with diffusion of the initial field obtained by GVF. Besides, we expect
to get a more stable numerical scheme for noisy images.
