Image Processing Reference
In-Depth Information
2.3.2 Relations between image operators
We can define the relationships between two image operators based upon those
between images presented in 2.2.6.
Definition 2.6 (Relation between image operators).
For two operators
O
1
and
O
2
with the common domain
P
1
,
O
1
(
F
)=
O
2
(
F
)
,
∀
F
∈P
1
↔
O
1
=
O
2
,
(2.12)
O
1
(
F
)
>
O
2
(
F
)
,
∀
F
∈P
1
↔
O
1
>
O
2
.
(2.13)
Other relations such as
<
,
≤
≥
,and
between image operators are defined in
the same way.
2.3.3 Binary operators between images
An operator generating a new image from two input images also plays an
important role in the analysis of image processing algorithms.
Definition 2.7 (Binary operator).
A mapping:
P
1
×P
2
→P
3
,
(2.14)
where
P
3
are arbitrary subsets of the image space is called a
binary operator between images
(or simply
binary operator
). Here,
P
1
,
P
2
,and
P
1
×P
2
and
P
3
are
domain
and
range
of the mapping.
The notation,
H
=
φ
(
F
,
G
), or
H
=
F
∗
G
, is employed to state that an
image
H
=
{
h
ijk
}
is derived from images
F
=
{
f
ijk
}
and
G
=
{
g
ijk
}
by the
above mapping. If the following equation holds among
F
,
G
,and
H
,
h
ijk
=
φ
(
f
ijk
,g
ijk
)
,
∀
(
i, j, k
)
∈
I
×
I
×
I
,
(2.15)
where
φ
(
x, y
) is an arbitrary real function of two variables independent of
i
,
j
,and
k
, then the operation is called a
pointwise operation
on
F
G
.
The density value at a voxel (
i, j, k
) in an output of a pointwise operation
is calculated by using density values at the voxel of the same location (
i, j, k
)
in two input images and the function
φ
defining the operation. The form of
the function should be common to all voxels (Fig. 2.10).
Many pointwise operations can be defined by using different functions
φ
(
x, y
) in Eq. 2.15. In each of these specific cases we use an appropriate symbol
instead of
and
to represent the particular pointwise operation. Most commonly
used examples are given in Table 2.2.
A pointwise operation on two images is regarded as a set of arithmetic
operations performed on all voxels independently of each other. Thus they
satisfy several general rules similar to those of real numbers as follows:
∗
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