Image Processing Reference
In-Depth Information
Relationships between properties of a line figure in the continuous space
and its digitized version should be discussed under the assumption that dig-
itization satisfies at least several of the above conditions. For example, what
condition should be satisfied in order that a given 26-connected digitized
curve is a digitized version of a line segment in 3D continuous space? Al-
though a relatively clear solution has been obtained for a 2D line figure
[Rosenfeld74, Kim82a, Kim82b], it cannot be extended easily to a 3D fig-
ure [Kim83, Kim84]. Another example is the estimation of the length of an
original line figure from a chain code expression [Kiryati95, Chattopadhyay92,
Klette85, Kim83, Amarunnishad90]. These problems can be studied theoret-
ically by first assuming a method to map a curve in continuous space onto
digitized space.
2.2.5 Cross section and projection
A cross section of a 3D continuous image
f
(
x, y, z
) along an arbitrary plane
H
(= a distribution of density values on the plane
H
) is considered as a 2D
image. We call this
cross section
(
profile
) of an image
f
(by a plane
H
).
The integration of density values of a 3D continuous image
f
(
x, y, z
)along
the perpendicular line to the plane
H
is called
projection
of
f
to the plane
H
.
The projection to the plane
H
is also a 2D image on the plane
H
.
For example, the cross section by the horizontal plane
z
=
z
0
is given by
f
cross
(
x, y
;
z
0
)=
f
(
x, y, z
)
|
z
=
z
0
.
(2.1)
The projection of a 3D image
f
(
x, y, z
) to the plane
H
is represented as
∞
f
proj
(
x, y
)=
f
(
x, y, z
)
dz.
(2.2)
−∞
A cross section and a projection of a 3D digitized image are defined in the
same way (Fig. 2.8). For example, the cross section at
k
=
k
0
:
f
cross
(
k
0
)=
{
f
ijk
0
}
,
(2.3)
the projection to the
i
−
j
plane:
K
f
proj
(
k
)=
{
f
ijk
}
.
(2.4)
k
=1
Calculation of a cross section by a plane of an arbitrary direction and a
projection to an arbitrary plane are not always easy for 3D images. Various
algorithms to calculate them have been studied in computer graphics and
visualization. Both a cross section by the plane of an arbitrary direction and
a projection to an arbitrary plane as well as the cross section along a curved
surface are often used in medical applications.
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