Image Processing Reference
In-Depth Information
They both can eliminate (or suppress) isolated random variations in an input
image; they change a curved surface
w
=
f
(
i, j, k
) transferring to an input
image into a smoother one.
Remark 3.4.
The following holds concerning the morphological operations.
Putting an input image by
F
=
{
f
ijk
}
and an output image by
G
=
{
g
ijk
}
,
(i) Erosion:
G
≤
F
. It eliminates an isolated protrusion smaller than a given
width.
(ii) Dilation:
. It eliminates an isolated cavity or a depression smaller
than the given size by filling them.
(iii) Opening:
G
≥
F
G
≤
F
. It makes an image smoother by whittling isolated
points, protrusions, and ridges smaller than the given size.
(iv) Closing:
G
≥
F
. It makes an image smoother by filling holes, depressions,
cavities, and valleys smaller than a given size.
By representing a 3D gray-tone image by a set of cross sections or a set of
umbras, we treat a 3D gray-tone image only by morphological operations on
a binary image or the operation on a set.
Remark 3.5.
We will introduce another expression of the dilation and the
erosion. Note that an input 3D image
F
=
{
f
ijk
}
can be regarded as a curved
surface
w
(
i, j, k
) in the 4D space. Let us consider the subspace
R
1
of this 4D space defined by
−
f
ijk
=
0
,
∀
R
1
=
{
(
i, j, k
);
z
−
f
ijk
≤
0
}
=umbraof
F
.
(3.16)
Next let us consider a weight function
{
w
ijk
∈N
((
0
,
0
,
0
))
}
defined on the
neighborhood
N
((
0
,
0
,
0
)) of the origin ((
0
,
0
,
0
)). We denote by
R
w
((
0
,
0
,
0
))
the region
z
). Then let us denote by
R
w
(
i, j, k
)
the result of the translation of
R
w
((
0
,
0
,
0
)) to (
i, j, k
). Then, the following
point set
−
w
ijk
≤
0
(= umbra of
{
w
ijk
}
S
S
=
{
(
i, j, k
);
R
w
(
i, j, k
)
⊂
R
1
}
(3.17)
is called
figure erosion
of the region
R
1
by the
weight function
(the
mask
function
,
structural element
)
R
w
. Furthermore, putting the region
{
(
i, j, k
);
z
−
by
R
2
,thepointset
¯
f
ijk
≥
}
S
S
), where
0
(= the complement of
S
=
{
(
i, j, k
);
R
w
(
i, j, k
)
⊂
R
2
}
(3.18)
is called
figure dilation
of the region
R
2
by the weight function (structure
element)
R
w
. The operations to derive the figure dilation and the figure erosion
are also examples of morphological operations (Fig. 3.1).
Remark 3.6.
Generally speaking, if an input image and a mask (a weight
function) are both binary, a morphological operation of the image is treated as
a morphological operation on the set, regardless of the dimension of an image
to be processed [Haralick87, Maragos87a, Maragos87b, Matheron75, Serra82].
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