Biomedical Engineering Reference
In-Depth Information
given by
d
:
∂
du
i
du
j
,
i
=
3
i
=
3
j
=
3
∂
∂
u
i
du
i
∂
u
i
,
∂
2
d
=
⇒
d
(7.34)
=
∂
u
j
i
=
1
i
=
1
j
=
1
2
is the square Euclidean norm of
d
. The matrix composed of the
coefficients
g
ij
=
∂
∂
u
i
where
d
,
∂
∂
u
j
is symmetric, and the extremes of the quadratic
form
d
2
are obtained in the directions of the eigenvectors (
θ
+
,θ
−
)ofthe
metric tensor [
g
ij
], and the values attained there are the corresponding maxi-
mum/minimum eigenvalues (
λ
+
,λ
−
). Hence, a potential function can be defined
as [57]:
f
(
λ
+
,λ
−
)
=
λ
+
−
λ
−
,
(7.35)
which
recovers
the
usual
edge
definition
for
gray-level
images:
(
λ
+
=
2
∇
I
,λ
−
=
0if
m
=
1).
Similarly to the gray-level case, noise should be removed before the edge map
computation. This can be done as follows [56, 57]. Given the directions
θ
±
,we
can derive the corresponding anisotropic diffusion by observing that diffusion
occurs normal to the direction of maximal change
θ
+
, which is given by
θ
−
. Thus,
we obtain:
2
∂
∂
t
=
∂
∂θ
−
,
(7.36)
which means:
2
2
∂
1
∂
t
=
∂
∂θ
−
,...,
∂
m
1
∂
t
=
∂
m
∂θ
−
.
(7.37)
In order to obtain control over local diffusion, a factor
g
color
is added:
2
∂
∂
t
=
g
color
(
λ
+
,λ
−
)
∂
∂θ
−
,
(7.38)
where
g
color
can be a decreasing function of the difference (
λ
+
−
λ
−
).
This work does not separate the vector into its direction (chromaticity) and
magnitude ( brightness).
In [67], Tang
et al.
pointed out that, although good results have been reported,
chromaticity is not always well preserved and color artifacts are frequently ob-
served when using such a method. They proposed another diffusion scheme to
address this problem. The method is based on separating the color image
into
chromaticity and brightness, and then processing each one of these components
