Image Processing Reference
In-Depth Information
(b) Shift operator
Definition 2.12 (Shift operator).
An operator
O
which derives an image
G
{
g
ijk
}
F
{
f
ijk
}
=
from an image
=
by the equation
g
ijk
=
f
i−p,j−q,k−r
,
(
p, q, r
)
∈
I
×
I
×
I
(2.30)
is called a
shift operator
and denoted by
T
[
p, q, r
]where
p
,
q
,and
r
are integer
parameters.
The shift operator
T
[
p, q, r
] translates an input image by
p
voxels in the
i
-direction,
q
voxels in the
j
-direction, and
r
voxels in the
k
-direction.
Property 2.2.
The following holds concerning a shift operator:
(1) Defining operators
D
≡
T
[
1
,
0
,
0
],
R
≡
T
[
0
,
1
,
0
], and
B
≡
T
[
0
,
0
,
1
],
the inverses
D
−
1
,
R
−
1
and
B
−
1
exist, and
T
[
p, q, r
]=
D
p
·
R
q
B
r
,
(
p, q, r
)
·
∈
I
×
I
×
I
(2.31)
(2) For two shift operators
T
[
p, q, r
]and
T
[
u, v, w
],
T
[
p, q, r
]
·
T
[
u, v, w
]=
T
[
u, v, w
]
·
T
[
p, q, r
]=
T
[
u
+
p, v
+
q, w
+
r
]
.
(2.32)
(3) The following left distributive laws hold:
T
(
F
∗
G
)=
T
(
F
)
∗
T
(
G
)
,
∀
T
∈T
,
∀
F
,
∀
G
∈P
,
(2.33)
T
·
(
O
1
∗
O
2
)=(
T
·
O
1
)
∗
(
T
·
O
2
)
,
∀
T
∈T
,
∀
O
1
,
∀
O
2
∈O
(2.34)
where
T
is the set of all shift operators,
P
is the set of all images,
O
is
the set of all image operators, and
is an arbitrary pointwise operation
of two images (or parallel composition of image operators).
∗
(c) Point operator
Definition 2.13 (Point operator).
Point operator
is defined as the operator
that calculates the output
G
=
{
g
ijk
}
by the equation
g
ijk
=
ψ
(
f
ijk
)
,
∀
(
i, j, k
)
∈
I
×
I
×
I
(2.35)
where
F
=
{
f
ijk
}
is an input image and
ψ
is an arbitrary real function.
Thus, in a point operator the output gray value at a voxel (
i, j, k
) is deter-
mined only by the input gray value at the voxel of the same location. Various
point operators are defined by selecting different functions for
ψ
. Some point
operators are denoted by symbols specific to them. Examples are shown in
Table 2.3.
Property 2.3.
The parallel and serial compositions of arbitrary two point
operators are point operators.
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