Image Processing Reference
In-Depth Information
level
t
is defined as follows.
S
t
(
f
)
≡{
(
x, y, z
)
∈
D
;
f
(
x, y, z
)
≥
t
}
,
(2.5)
where
D
is the domain of the function
f
= the region of all points (
x, y, z
)
such that their density values are larger than or equal to
t
.
Then, a 3D continuous function
f
(
x, y, z
) is represented in the form
f
(
x, y, z
)=sup
{
t
∈
R; (
x, y, z
)
∈
S
t
(
f
)
}
,
(2.6)
where R = set of all real numbers = supremum of the level
t
such that a
cross section includes a point (
x, y, z
). If all cross sections are given, then the
function
f
is fixed uniquely.
In a 3D image in which density values are quantized into
M
levels, there
exist
M
cross sections. The cross section at a level
k
is the set of all voxels such
that density values are larger than or equal to
k
. Inversely, the density value
of a voxel (
x, y, z
) is equal to the maximum level such that a cross section
includes the voxel (
x, y, z
).
Definition 2.2.
Consider a 4D space (
i, j, k, f
). Then, the set of points
(
i, j, k, f
) such that
f
(
i, j, k
)
t
is called
umbra
(or
projection
)oftheim-
age
f
(
i, j, k
) and denoted by
U
(
f
). That is,
≥
U
(
f
)
≡{
(
i, j, k, f
);
f
(
i, j, k
)
≥
t
}
(2.7)
Inversely, given the umbra
U
(
f
), then
f
(
i, j, k
) is determined by
f
(
i, j, k
)=sup
(2.8)
(= the maximum of
t
in the umbra, when (
x, y, z
)isfixed.)
{
t
;(
i, j, k, f
)
∈
U
(
f
)
}
Intuitively, the umbra is the subspace of the 4D space (
i, j, k, f
)belowthe
curved surface
t
=
f
(
i, j, k
) including the surface itself.
Thus both the cross section and the umbra (projection) are sets of points,
although the cross section is in the 3D space and the umbra in the 4D space,
respectively. We can consider a binary image that takes the value
1
on the set
(cross section or umbra) and takes
0
otherwise. According to the terminology
of the set theory, they are characteristic functions of sets called cross section
and umbra, respectively. All of cross sections or umbras are equivalent to the
original 3D image itself. In other words, a 3D gray-tone image can be described
equivalently by a 4D binary image or by a set of 3D binary images.
2.2.6 Relationships among images
For formal treatment of an image and an image operation we will give a formal
definition of a digitized image as follows.
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