Image Processing Reference
In-Depth Information
before executing the smoothing procedure. One idea is to divide the neigh-
borhood area into several smaller subareas and to consider the possibility that
each subarea contains a border. A suitable measuring of density variations in
each subarea must be applied to do this. A smoothing operator is executed
only in the subarea, regarded as such as “no border exists in it.”
The concept of edge-preserving smoothing was firstly developed for a 2D
image. The extension of the idea to be applied to a 3D image is not com-
plicated. However, the devision of the neighborhood and the estimation of
the existence of a border requires careful consideration, as the result may
be a serious increase in computation time. Different ideas were presented in
[Tomasi98, Wong04] with applications to 3D CT and MRI images of the brain.
3.2.4 Morphology filter
Let us consider an input image
F
=
{
f
ijk
}
and a gray-tone image
B
=
{
defined at the origin (
0
,
0
,
0
) and on its neigh-
borhood. Then we define the following two operations, dilation and erosion.
b
ijk
;(
i, j, k
)
∈N
((
0
,
0
,
0
))
}
Definition 3.2 (Dilation and erosion).
Dilation
of an input image
F
by
the structure element
B
is defined as
B
F
{
f
ijk
}→
G
{
g
ijk
}
Dilation DIL
[
]:
=
=
g
ijk
=max
(
p,q,r
)
{
f
pqr
+
bs
p−i,q−j,r−k
;(
p, q, r
)
∈N
((
i, j, k
))
}
.
(3.10)
F
B
Erosion
of an input image
by the
structure element
is defined as
Erosion ERO
[
B
]:
F
=
{
f
ijk
}→
G
=
{
g
ijk
}
g
ijk
=min
(
p,q,r
)
{
f
pqr
−
bs
p−i,q−j,r−k
;(
p, q, r
)
∈N
((
i, j, k
))
}
.
(3.11)
B
s
=
{
bs
ijk
}
B
where
is a gray-tone image symmetric to
with respect to the
origin.
The following notations are also used frequently.
Dilation of
F
by
B
=
F
⊕
B
.
(3.12)
Erosion of
F
by
B
=
F
B
.
(3.13)
Both are also called
morphological filters
with the structure element
B
(and
with the neighborhood
((
i, j, k
)). If
b
ijk
=
0
,(
i, j, k
), then the dilation and
erosion reduce to the maximum filter and the minimum filter, respectively.
Serial compositions shown below are called morphological operations, too.
They are sometimes called the
closing
and
opening
, respectively.
N
(
F
B
)
⊕
B
(operator expression
DIL
[
B
]
·
ERO
[
B
]) (
opening
)(3.14)
(
F
⊕
B
)
B
(operator expression
ERO
[
B
]
·
DIL
[
B
]) (
closing
) (3.15)
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