Image Processing Reference
In-Depth Information
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(
a
)
(
b
)
Fig. 7.6.
Geometry of mapping of a point onto a plane: (
a
) Perspective projection;
(
b
) parallel (orthogonal) projection.
selected arbitrarily. Here we assume that a viewpoint is at
Z
=
−
d
on the
Z
axis, that is, at (
0
,
0
,
−
d
), and that an image plane is on the
x
−
y
plane
(Fig. 7.6).
Denoting by P
=(
X
P
,Y
P
,Z
P
) a point on the image plane that corre-
sponds to the point P, the following two kinds of projections are most fre-
quently employed.
(1)
Orthogonal projection
(
parallel projection, normal projection
)
X
P
=
x, Y
P
=
y, Z
P
=0
.
(7.5)
In this case, a point in 3D space is projected in parallel to the
Z
-axis of
the world coordinate system. The position of P
does not depend on the
viewpoint. In other words, the viewpoint is regarded to be infinitely far
from an image plane.
(2)
Perspective projection
(
central projection
)
X
P
=
d
·
x/
(
z
+
d
)
,Y
P
=
d
·
y/
(
z
+
d
)
,Z
P
=0
.
(7.6)
In this case, the point P
is located at the intersection of the image plane
and a line connecting the point P and the viewpoint P
e
(= the view
direction).
By using these projections we can determine where on an image plane we
should draw each point in the 3D space. In this space each voxel is drawn on an
image plane and is determined in the same way, for example, by representing
each voxel by its center point.
Since we are discussing the digitized space in practice, both a 3D image
and a 2D image on a projected plane are digitized images. The coordinate
value (
x, y, z
) in the above explanation is a center point of a voxel or a vertex
of a voxel.
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