Guess you see a 3D-scene through a glassplate. You can pick up a point P1 from the scene and mark it´s images on the glass as it is seen by the left and the right eye. Now you can pick another point P2 in a way that it´s right eye image is identical with the left eye image of P1. Because a single dot now serves as an image for two different points of the scene, we have to assume that they have the same color (otherwise I2 would have to be colored in two different ways at the same time). Of course I1 and I3 have to be colored the same way as I2.
Dependencies in the coloring of the scene
If we continued this procedure we would see that every dot on the glassplate serves
as an image for two points from the scene, depending on by which eye it is seen
(except for a few ones at the left and right edge of the plate). This creates chains
of dependencies for the color-ing of the dots. Fortunately, such are only possible among
objects within a horizontal plane.
One of the main tasks of our algorithm is to manage this constraints. The second one is
to calculate the positions of the two images of every visible point of the scene. There
also has to be a separate treatment for hidden surfaces, parts of the scene which are
visible only for one eye, while the other eye´s view is blocked by some other object.
Geometry for the algorithm
All the objects of the scene have to be placed between the near and the far plane,
which serves as a solid background. When viewing the stereogram you have to concentrate
on the far plane (a large equidistant target) to catch the sense of depth. Especially for
beginners this comes much easier when you converge at your reflection on the screen.
Assuming the distance from your eyes to the screen is D, the reflection lies D behind
the screen, and so the far plane should do.
The near plane defines the depth range of the scene. The greater the near-far distance
(mD), the greater the range of possible depth separations in the image. At the
first glance it should be possible to place the near plane as close to the eye as
the image plane is (m=1), but already a value of m=1/2 can lead to misinterpretations of
the image:
Misinterpretation of an image with m>0.5
In the situation shown in above the images of P1 and P2 could be interpreted as an image of a "ghost" point P3. Values close to 1/2 also cause difficulties for proper convergence of the eyes, so 1/3 is a
convenient value for m.
To portray a point of the scene on the image plane, we have to calculate it´s stereo
separation. Stereo separation is the distance between both images of the point. It is
a non-linear function of z, growing with descending values of z. Stereo separation relates
directly to stereo disparity, which is the difference in angle subtended at the eyes.
We will introduce a simplification into the calculation of stereo separation, assuming that we look straight at every point to depict. It simplifies the process of calculation and requires no preference for a special viewing point. The price of this simplification is that is introduces a parallax error into the image. Objects at the sides get pulled inwards slightly.
Error made by simplified algorithm
The formula for calculating stereo separation as a function of z[0,1] is based on the similar triangles PI1P´ and PE´E1(see Figure 6 for visualization):
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