Image Processing Reference
In-Depth Information
2.4.1 General form of image operations
In order to obtain an output image
of an image operation
O
,we
need to clearly determine the procedure to calculate the density value
g
ijk
G
=
{
g
ijk
}
of
the output image
for all (
i, j, k
)s. Thus a very general form of an
image operation is represented by
G
=
{
g
ijk
}
g
ijk
=
φ
ijk
(
F
)
,
∀
(
i, j, k
)
∈S
=I
×
I
×
I
,
(2.37)
where
φ
ijk
is a set of all integer
triads (
i, j, k
) corresponding to the numbers of rows, columns, and planes, and
F
is an appropriate multivariable function,
S
{
f
ijk
}
is an input image. For example, an image generation process in a
kind of imaging system is modeled by the equation
g
ijk
=
(
p,q,r
)
∈S
=
h
(
p, q, r
;
i, j, k
)
f
pqr
,
∀
(
i, j, k
)
∈S
.
(2.38)
Note here that the form of the function
φ
ijk
in Eq. 2.37 may depend on (
i, j, k
),
the position on the relating image space, and the calculation of one density
value
g
ijk
at a voxel (
i, j, k
) of an output image requires all of density values
in an input image
F
.
2.4.2 Important types of algorithms
An image operator may be implemented or executed by different types of algo-
rithms. From the viewpoint of algorithms, we describe here several important
aspects of an image operation characterizing how to implement an operation
defined as a mapping on an image set.
(a) Local and global operations
In some classes of image operations, density values of an input image
F
=
{
in a small finite area around a voxel (
i, j, k
) are used to calculate an
output density value
g
ijk
at the voxel (
i, j, k
). That is, the output density
g
ijk
is obtained by the equation,
f
ijk
}
g
ijk
=
φ
ijk
(
{
f
pqr
;(
p, q, r
)
∈N
ijk
((
i, j, k
))
}
)
(2.39)
where
N
ijk
((
i, j, k
))
≡{
(
i
+
p, j
+
q, k
+
r
); (
p, q, r
)
∈S
ijk
}
(2.40)
and
S
ijk
is an appropriate subset of the set of all integer triads I
×
I
×
I.
The subarea of the image space
N
ijk
((
i, j, k
)) is called the
neighborhood
of (
i, j, k
). Note that
N
ijk
((
i, j, k
)) may or may not include the voxel (
i, j, k
)
in it. The operation defined by Eq. 2.39 is called a
local operation
if the
size of the neighborhood
N
ijk
((
i, j, k
)) (= the number of voxels contained in
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