Stereoscopic Stretch and Squeeze
part 2

John Wattie

A) Part 1 of "stretch and squeeze"
Showed the phenomenon visually, with an anaglyph of Charles Darwin's table, but did not explain it formally.

B) Perceived Depth Maths.
Now we have derived the geometry of Stereoscopic Perceived Depth (Roundness) there is enough information to describe stretch and squeeze by equations.

Crossed disparity

T = M V / ( i / D + 1)

T: Thickness as perceived stereoscopically
M: Magnification of the stereoscopic image
V: Viewing distance from the screen
i: inter-ocular or distance between the observer's eyes
D: Disparity of the point in question, as measured on the screen.

Uncrossed Disparity

T = M V / ( i / D - 1 )


Stretch and Squeeze

Now we are in a position to explain experimental findings on stereoscopic roundness using algebra, instead of the perception methods I have described.

The formula shows that the distance you sit from the screen (V) has a major influence on the stereoscopic perceived depth, T.


The further you are away from the screen, (increase V) the perceived depth (T) also increases.
Objects become elongated in depth. A circular table looks like an oval table, with the longest distance stretching into the screen (z dimension).


The closer you sit to the screen (reduce V) the flatter the stereo image becomes. (Perceived depth, T, decreases in proportion.)
A circular table becomes an oval table, but now the largest dimension is in the horizontal plane (x dimension).

Size (age) of the viewer:

i, the distance between the observer's eyes, is bigger for large men and smaller for small children.
So children see stereo images made for adults as being too deep (stretch).
Stereo pairs made correctly for children look too shallow for adults (squeeze).
In practice, the distortion between adults and children is minimal and nobody worries about it. The children do not even know it is happening and the adults only know because of mathematics.

Lens stereoscope:

A lens stereoscope has the pair of images even nearer than a computer monitor, in fact too close for most people to focus them. But the magnifying lenses in the stereoscope take the image away. At the focus limit, the focal length and focus distance in the stereoscope are the same and the virtual image is taken to stereo infinity. Light rays reaching the eyes are made parallel. That is a bad arrangement for stereo vision and humans prefer to have the lenses focused closer, so the nearest object seems to be sitting about two meters away. Only better stereoscopes allow you to change the focus.

Size of the image and zooming:

Assume you are sitting at a constant distance from the screen.
Now change the image size on the screen by zooming in (how to zoom).
M, the magnification, increases.
The larger the image, the greater the perceived depth, T, because everything is magnified the same, including the disparity (D).

Despite the enlargement, objects retain their 3D shape or roundness. The perceived depth changes just as much as the change in size of the image, so that the ratio between width of an object and its thickness stays constant. A circular table stays circular, if that is how it looked at your current viewing distance.

That means you can enlarge a 3D image using your viewing software, or zoomed projector lenses and nothing becomes deformed, just bigger.

But there is a limit to this zooming. The stereo disparity (D) of the most distant objects increases and can become greater than your inter-ocular distance (i). Most people cannot handle that and the stereo image they see breaks down. (Surprisingly, experts can see in stereo with diverged eyes, especially if it is brought on gradually, as with a slowly increasing zoom. Most people cannot).

This constant ratio of depth to size as the image is zoomed becomes a problem if you project a stereo pair made for a computer onto a movie screen. Objects are not deformed, as long as your viewing distance is the same for movie and computer, but that is ridiculous. You cannot sit less than a meter from a movie screen. Of course very few people see a projected stereo image without distortion, because everybody is sitting at a different distance (V). Only one person is at the correct distance and lined up with the middle of the screen (in the ortho-stereo seat.)

Stereo images for computer viewing are frequently made as if they are to be projected onto a 2 meter distant screen, or seen in a lens stereoscope, by using the 1/30 rule for stereo base. So it is common to see flat computer stereo. Few people object because many of them cannot see a correct depth stereo image anyway, since it hurts their eyes.

Special case: Disparity equals the interocular distance: i = D:

T = M V / ( i / D - 1 )

If screen disparity and the interocular distance are equal, i/D is 1 and 1-1 = 0. You cannot divide by zero and get a sensible answer, but as D gets close to i, perceived depth gets closer to infinity. Maximum depth in the scene at stereoscopic infinity is common (clouds, or the moon).

Charles Darwin's house. His study with the round table is inside Stereo anaglyph by John WattieCharles Darwin's Down House.

In ordinary stereo photography viewed on a computer, infinity disparity (clouds) is rarely taken up to the interocular distance, since different people have different eye separation and it becomes impractical. Disparity of the most distant object on a computer screen is often lower than 1/10 of the interocular distance (65/10 = 6.5mm). The clouds behind Down House have a disparity of 9mm in the above anaglyph. That is totally unrealistic, but people accept it.

People handle the realistic 65mm disparity at the 3D movies, because they are a fair distance away from the screen and distant things, including disparity, look smaller (perspective.) Nearly every time you look at a stereo pair, your eyes are converging, like railway tracks seem to converge with distance. It is lucky trains also visibly shrink with distance, or they could never run on the tracks!