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240 10. Stereopsis
riencing positive, negative or zero parallax. As the value of t approaches d,
the viewer can start to feel uncomfortable. As it exceeds d, divergent parallax
is being simulated, and the natural viewing perception is lost. For example,
sitting in front of a workstation where the average viewing distance is about
18 in., a good and comfortable stereoscopic effect can be obtained with a
parallax separation of about
1
2
in., equivalent to an angular parallax of 3
◦
.
When rendering images with the intention of providing a parallax sep-
aration of
1
2
in. on a typical monitor, it is necessary to determine the sepa-
ration between the cameras viewing the virtual scene. This is explained in
Section 10.1.1. When setting up the camera geometry, it is also advisable to
consider a few simple rules of thumb that can be used to enhance the per-
ceived stereoscopic effect:
• Try to keep the whole object within the viewing window, since partially
clipped objects look out of place and spoil the stereoscopic illusion.
• Render the objects against a dark background, especially where you
want negative parallax.
• Monocular cues enhance the stereoscopic effects, so try to maximize
these by using/simulating wide-angle lenses, which in essence open up
the scene to enable these comparisons to be made.
• The goal when creating a stereoscopic effect is to achieve the sensation
of depth perception with the lo west value of parallax. Often, the most
pleasing result is obtained by placing the center of the object at the
plane of the screen.
In practice, to get the best stereo results when viewing the rendered im-
ages, you should use as big a monitor as possible or stand as far away as possi-
ble by using a projection system coupled with a large-screen display. This has
the added benefit of allowing larger parallax values to be used when rendering
the images.
10.1.1 A Stereo Camera Model
Since parallax effects play such a crucial role in determining the experience
one gets from stereopsis, it seems intuitive that it should be adjustable in some
way at the geometric stages of the rendering algorithm (in CAD, animation
or VR software). That is, it has an effect in determining how to position the
two cameras used to render the scene. Since the required parallax concerns
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10.1. Parallax 241
viewpoints and projections, it will make its mark in the geometry stages of
rendering, as discussed in Section 6.6.6.
The most obvious geometrical model to construct would simulate what
happens in the real world by setting up two cameras to represent the left and
right eye viewpoints, both of which are focused on the same target. This tar-
get establishes the plane of zero parallax in the scene. Rendering is carried out
for the left and right views as they project onto planes in front of the left and
right cameras. If the target point is in front of the objects in the scene, they
will appear to be behind the screen, i.e., with positive parallax. If the target
is behind the objects, they will appear to stand out in front of the screen, i.e.,
negative parallax. Figure 10.5 illustrates the geometry and resulting parallax.
In this approach, we obtain the left view by adding an additional translation
by −d/2alongthex-axisasthelaststep(before multiplying all the matri-
ces together) in the algorithm of Section 6.6.6. For the right eye’s view, the
translation is by d/2.
However, when using this form of set-up, the projection planes are not
parallel to the plane of zero parallax. When displaying the images formed at
these projection planes on the monitor (which is where we perceive the plane
of zero parallax to be), a parallax error will occur near the edges of the display.
Rather than trying to correct for this distortion by trying to transform the
Figure 10.5. The two viewpoints are separated by a distance d,andtheviewsare
directed towards the same target. The origin of the coordinate system is midway
between the viewpoint with the x-axis towards the right, and the z-axis vertical—out
of the plane of the diagram.
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242 10. Stereopsis
projection planes onto the plane of zero parallax, we use an alterative method
of projection based on using parallel aligned cameras.
Using cameras with parallel axis alignment is known as off-axis rendering,
and it gives a better stereoscopic view than the one depicted in Figure 10.5.
The geometry depicted in Figure 10.6 sets the scene for this discussion. Two
cameras straddling the origin of the coordinate system are aligned so that they
both are parallel with the usual direction of view. Unfortunately, the projec-
tion transformation is different from the transformation for a single view-
point, because the direction of view vector no longer intersects the projection
plane at its center (see Figure 10.7). Implementing a projective transforma-
tion under these conditions will require the use of a nonsymmetric camera
frustum rather than a simple field of view and aspect ratio specification. The
geometry of this not considered here, but real-time 3D-rendering software
libraries such as OpenGL offer functions to set up the appropriate projective
transformation.
So far so good, but we have not said what the magnitude of d should be.
This really depends on the scale we are using to model our virtual environ-
ment. Going back to the point we originally made, we should always have a
parallax separation of about
1
2
in. on the computer monitor and the center of
the model at the point of zero parallax. In addition, the farthest point should
have a negative parallax of about −1.5
◦
and the nearest a parallax of about
+1.5
◦
.
Figure 10.6. Two cameras with their directions of view aligned in parallel, along the y-
axis. A point halfway between them is placed at the origin of the coordinate system.
We need to determine a suitable separation d to satisfy a given parallax condition
which is comfortable for the viewer and gives a good stereoscopic effect at all scales.
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10.1. Parallax 243
Figure 10.7. Using off-axis (parallel) alignment results in a distorted projection which
is different for each of the viewpoints. To achieve this effect, the camera frustum must
be made to match the distorted viewing projection. The OpenGL and Direct3D
software libraries provide ways to do this.
Consider the geometry in Figure 10.6 and the problem of determining d
in the dimensions of the scene given some conditions on the parallax we wish
to achieve. Two useful ways to specify parallax are:
1. Require the minimum value of parallax
min
= 0 and set a value
on
max
, say at about 1.5
◦
. This will give the appearance of looking
through a window because everything seems to lie beyond the screen.
2. Specify the point in the scene where we want the zero parallax plane
to lie. We shall define the location of this plane by its distance from
the viewpoint, D
zpx
. Therefore, everything that is closer to the cameras
than D
zpx
will appear in front of the screen.
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244 10. Stereopsis
To do this, we will have to ensure that the scene falls within a volume of
space bounded by known values: [x
min
, x
max
] and [y
min
, y
max
]. The vertical
extent of the scene (between [z
min
, z
max
]) does not enter the calculation. We
will also need to define the required field of view . This is typically = 40
◦
to 50
◦
. It is useful to define x = x
max
− x
min
and y = y
max
− y
min
.
From the angular definition of parallax (see Figure 10.4), we see that for
projection planes located at distances s
1
and s
2
from the viewpoint,
= 2arctan
t
1
2s
1
= 2arctan
t
2
2s
2
= 2arctan
P
2L
.
The measured parallax separation will be t
1
and t
2
respectively. The same
perceived angular parallax gives rise to a measured parallax separation of P
at some arbitrary distance L fr om the viewpoint. With the viewpoint located
at the world coordinate origin, if we were to move the projection plane out
into the scene then, at a distance L, the measured parallax separation P is
given by
P = 2L tan
2
. (10.1)
In the conventional perspective projection of the scene, using a field of
view
and the property that the scene’s horizontal expanse x just fills the
viewport, the average distance L
of objects in the scene from the camera is
L
=
x
2tan
2
.
For a twin-camera set-up, the parallax separation at this distance is ob-
tained by substituting L
for L in Equation (10.1) to give
P = 2L
tan
2
=
tan
2
tan
2
x.
Requiring that the scene must have a maximum angular parallax
max
and a
minimum angular parallax
min
, it is equivalent to say that
P
max
=
tan
max
2
tan
2
x;
P
min
=
tan
min
2
tan
2
x.
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