Graphics Programs Reference
In-Depth Information
The transformation for the right eye is similarly
⎛
⎝
⎞
⎠
1 000
0 100
000
r
−
=
T
right
.
e
001
It projects
P
to
P
right
.
The stereo pair is created by transforming each point
P
on the original image twice,
to the two points
P
left
=
PT
left
and
P
right
=
PT
right
. The value selected for
e
depends
on how the picture is to be viewed. For the dual-color method mentioned earlier, 2
e
should equal the distance between the eyes (about 60-70 mm). This is a small value, so
there is not much difference between
P
right
and
P
left
. The two images highly overlap.
For a general point
P
=(
x, y, z
), the projections for both eyes are
x
+
e
zr
+1
,
,
y
zr
+1
P
left
=(
x, y, z,
1)
T
left
=(
x
+
e, y,
0
,zr
+1)
→
x
.
e
zr
+1
,
−
y
zr
+1
P
right
=(
x, y, z,
1)
T
right
=(
x
−
e, y,
0
,zr
+1)
→
This means that the smaller
z
is (i.e., the closer the point is to the viewer), the greater
the difference between what the two eyes see. A good way to visualize this is to imagine
an object sliding past the viewer. The front of the object slides faster than the back,
an effect known as
parallax
.
As an example, consider the two points
P
=(5
,
0
,
1) and
Q
=(5
,
0
,
2). They differ
only in their
z
coordinate. Assuming that
e
=2and
r
= 3, their projections are
P
left
=
5+2
3+1
,
0
=
7
4
,
0
,
P
right
=
5
3+1
,
0
=
3
4
,
0
,
−
2
Q
left
=
5+2
2
·
3+1
,
0
=
7
7
,
0
,
Q
right
=
5
2
·
3+1
,
0
=
3
7
,
0
.
−
2
The difference between
P
left
and
P
right
is 7
/
4
−
3
/
4 = 1, whereas the difference between
Q
left
and
Q
right
is only 7
/
7
−
3
/
7=4
/
7.
Figure 3.42 is an example of a stereo pair of a polyline connecting the eight corners
of a cube. The
Mathematica
code that did the computations is also listed. Figure 3.43
shows the complete cubes.
A more sophisticated approach to generating a stereo image is shown in Fig-
ure 3.44a. The two eyes are located at (
e,
0
,
k
), and they view the
general point
P
=(
x, y, z
) from different directions. Point
P
is projected twice on the
projection plane, at points
P
L
and
P
R
, using the general rule for perspective projec-
tions. Assuming that the distance between the eyes is 2
e
, Figure 3.44c,d shows how to
calculate the
x
coordinates of points
P
L
and
P
R
, respectively. Using similar triangles,
−
k
)and(
−
e,
0
,
−
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