Biomedical Engineering Reference
In-Depth Information
used
φ
as follows:
∇
I
1
+
[
∇
I
/
K
]
2
(7.32)
φ
=
,
as shall see next.
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 [77] and can be
defined through the following equation [78]:
∂
u
∂
t
=∇·
(
g
∇
u
)
+
h
(
u
−∇
f
)
,
(7.33)
u
(
x
,
0)
=∇
f
where
f
is a function of the image gradient (for example,
P
in Eq. (7.13)), and
g
(
x
)
,
h
(
x
) are non-negative 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 [77] and improves their convergence
to the object boundaries.
As the result of steps (1)-(6) in Section 7.5 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 [77] but
precision may be lost due to the nonselective blurring [52].
The anisotropic diffusion scheme presented above is an alternative smooth-
ing method that can be used. Such observation points forward the possibility of
integrating anisotropic diffusion and the GVF in 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 the diffusion of the initial field obtained by GVF. Besides, we ex-
pect to get a more stable numerical scheme for noisy images.
Diffusion methods can be extended for color images. In [56, 57] such
a theory is developed. In what follows we summarize some results in this
subject.
Firstly, the definition of edges for multivalued images is presented [57]. Let
(
u
1
,
u
2
,
u
3
):
D
⊂
m
be a multivalued image. The difference of image
values at two points
P
=
(
u
1
,
u
2
,
u
3
) and
Q
=
(
u
1
+
du
1
,
u
2
+
du
2
,
u
3
+
du
3
)is
3
→
