Biomedical Engineering Reference
In-Depth Information
However, for noisy images the convergence of deformable models to the
boundaries is poor due to the nonconvexity of the image energy. This problem
can be addressed through diffusion techniques [18, 44, 52].
In image processing, the utilization of diffusion schemes is a common prac-
tice. Gaussian blurring is the most widely known. Other approaches are the
anisotropic diffusion [52] and the gradient vector flow [77].
From the viewpoint of deformable models, these methods can be used to im-
prove the convergence to the desired boundary. In the following, we summarize
these methods and conjecture their unification.
Anisotropic diffusion is defined by the following general equation:
∂
I
(
x
,
y
,
t
)
∂
t
=
div (
c
(
x
,
y
,
t
)
∇
I
)
,
(7.29)
where
I
is a gray-level image [52].
In this method, the blurring on parts with high gradient can be made much
smaller than in the rest of the image. To show this property, we follow Perona
et al.
[52]. Firstly, we suppose that the edge points are oriented in the
x
direction.
Thus, Eq. (7.29) becomes:
∂
I
(
x
,
y
,
t
)
∂
t
=
∂
∂
x
(
c
(
x
,
y
,
t
)
I
x
(
x
,
y
,
t
))
.
(7.30)
If
c
is a function of the image gradient:
c
(
x
,
y
,
t
)
=
g
(
I
x
(
x
,
y
,
t
)), we can define
φ
(
I
x
)
≡
g
(
I
x
)
·
I
x
and then rewrite Eq. (7.29) as:
I
t
=
∂
I
∂
t
=
∂
∂
x
(
φ
(
I
x
))
=
φ
(
I
x
)
·
I
xx
.
(7.31)
We are interested in the time variation of the slope:
∂
I
x
∂
t
.
If
c
(
x
,
y
,
t
)
>
0we
can change the order of differentiation and with a simple algebra demonstrate
that:
∂
I
x
∂
t
=
∂
I
t
∂
x
=
φ
·
I
xx
+
φ
·
I
xxx
.
At edge points we have
I
xx
=
0 and
I
xxx
0 as these points are local maxima
of the image gradient intensity. Thus, there is a neighborhood of the edge point
in which the derivative
∂
I
x
/∂
t
has sign opposite to
φ
(
I
x
). If
φ
(
I
x
)
>
0 the slope
of the edge point decrease in time. Otherwise it increases, that means, border
becomes sharper. So, the diffusion scheme given by Eq. (7.29) allows to blur
small discontinuities and to enhance the stronger ones. In this work, we have
