Image Processing Reference
In-Depth Information
Table 2.1.
Binary relations among images.
Relation
Definition
Notation
Equality
(
F
F
is equal to
G
)
f
ijk
=
g
ijk
,
∀
(
i, j, k
)
∈
I
×
I
×
I
F
F
=
G
Comparison (
is smaller than
G
)
f
ijk
<g
ijk
, ∀
(
i, j, k
)
∈
I
×
I
×
I
<
G
(
G
is larger than
F
)
(
F
is smaller than or equal to
G
)
f
ijk
≤
g
ijk
,
∀
(
i, j, k
)
∈
I
×
I
×
I
F
≤
G
(
G
is larger than or equal to
F
)
Definition 2.3 (Digitized image).
(A
3D digitized image
is defined as a
mapping
I
×
I
×
I
→
R
(2.9)
where I is the set of whole integers, and R is the set of all real numbers. An
element (
i, j, k
) of the direct product I
I is called
voxel
(or simply
point
),
and the image of the voxel (
i, j, k
) by this mapping is called
density
.
×
I
×
An image in which the density value at a voxel (
i, j, k
)isgivenby
f
ijk
is
F
{
f
ijk
}
denoted as
.
The set of all images is called an image space and is denoted by
=
P
.Ifa
density value
f
ijk
is equal to
0
or
1
for all voxels (
i, j, k
), then
F
{
f
ijk
}
=
is
called
binary image
. The set of all binary images is denoted by
P
B
.Theterm
gray-tone image
is used when we need to show explicitly that a density
f
ijk
takes an arbitrary value. A gray-tone image
F
=
{
f
ijk
}
is called a
semipositive
image
if
f
ijk
≥
0
for all
i
,
j
,and
k
and a
constant image
of the value
C
if
f
ijk
=
C
(a constant) for all
i
,
j
,and
k
. If an image
F
is a semi-positive
image, the set of all positive voxels and all 0-voxels in
F
is denoted by
R
(
F
)
and
R
(
), respectively. They also may be referred to as the
figure
and the
background
. A digitized image is simply called an
image
unless the possibility
of a misconception exists.
Relationships among these digitized images are defined below and can be
effectively utilized for analysis of image operators.
F
Definition 2.4 (Binary relation).
Binary relations between two images
F
are given as shown in Table 2.1, which are
based upon relations between density values
f
ijk
=
{
f
ijk
}
and
G
=
{
g
ijk
}
and
g
ijk
.
2.3 Model of image operations
We will now present a theoretical model of image operations and examine
basic properties of image operations. An image operation is defined as a map-
ping from a set of images to another set of images. By using this model and
relationships between images we introduce binary relationships between two
operators such as
equal to
and
greater
(
less
)
than
(in Section 2.3.2).
Search WWH ::
Custom Search