Image Processing Reference
In-Depth Information
Table 2.4.
Examples of neighborhood.
Notation
Definition
N
[6]
{
(
i, j, k
)
}
{
(
i
+
p, j
+
q, k
+
r
);
|p|
+
|q|
+
|r|
=
1
}
6-neighborhood
N
[18]
{
18-neighborhood
(
i, j, k
)
}
{
(
i
+
p, j
+
q, k
+
r
);
1
≤|
p
|
+
|
q
|
+
|
r
|≤
2
}
N
[26]
{
26-neighborhood
(
i, j, k
)
}
{
(
i
+
p, j
+
q, k
+
r
);
p, q, r
=
,
±
1
}
0
3
×
3
×
3 neighborhood
N
333
{
(
i, j, k
)
}
{
(
i
+
p, j
+
q, k
+
r
); max(
|p|, |q|, |r|
)=
1
}
{
(
i
+
p, j
+
q, k
+
r
); max(
|
p
|
,
|
q
|
,
|
r
|
)
≤
1
}
[26]
=
N
{
(
i, j, k
)
}∪
(
i, j, k
)
K
×
L
×
M
neighborhood
N
KLM
{
(
i, j, k
)
} {
(
i
+
p, i
+
q, k
+
r
);
−
[(
K
−
1
)
/
]
≤
p
≤
[
K/
]
2
2
−
[(
L
−
1
)
/
2
]
≤
q
≤
[
L/
2
]
−
[(
M
−
1
)
/
2
]
≤
r
≤
[
M/
2
]
}
[
m
]
i
+
k
p, j
+
k
q, k
+
k
r
(
k
=
k
-th order
K
×
L
×
M
neighborhood
N
{
(
i, j, k
)
}
{
1
,
2
,
3
,...
)
}
N
ijk
((
i, j, k
))) is suciently small compared with the size of an input image.
If it is comparable to the image size, the process is called a
global operation
(Fig. 2.12). Obviously, only local information about an input image is used
to determine an output gray value of each voxel in a local operation. For
example, the discrete Fourier transform (DFT) is a typical global operation,
and most of procedures discussed in Chapters 3, 4, 5, and 6 of the text are
local ones.
Remark 2.7.
By the word
local
, we mean that only voxels close to the current
voxel (
i, j, k
) are referred to, as well as that the number of referred voxels are
small. Hence we do not regard the shift operator as a local one despite its use
of only one density value of an input image for calculating one output density
value.
Examples of the neighborhood frequently used are shown in Table 2.4. In
particular,
3
×
3
×
3
voxels centered at a voxel (
i, j, k
), that is,
N
ijk
((
i, j, k
)) =
{
(
i
+
p, j
+
q, k
+
r
);
p, q, r
=
0
,
1
,
−
1
}
(2.41)
are employed most frequently.
(b) Parallel type and sequential type
If the Eq. 2.39 has the form
g
ijk
=
φ
ijk
(
f
i
1
j
1
k
1
,f
i
2
j
2
k
2
,...,f
i
n
j
n
k
n
)
,
∀
(
i, j, k
)
∈
I
×
I
×
I
,
(
i
n
,j
n
,k
n
)
∈N
ijk
((
i, j, k
))
,
∀
n
(
N
≥
n
≥
1
)
(2.42)
where
φ
ijk
is an arbitrary
N
-variable function,
N
is an arbitrary integer, and
N
ijk
((
i, j, k
)) is the neighborhood of the voxel (
i, j, k
), an image operation is
called
parallel
(
type
)
operation
.
The operation is called
sequential
(
type
)
operation
if it is presented by the
equation
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