One eye can only give a 2-dimensional image. Stereopsis, the vision of the third dimension (depth), is obtained through several automatic processes, one of these consists in comparing the differences of the images that each eye produces. Let's take this example:
If you look at this image through 3D-glasses (red filter for the left eye, blue filter for the right eye), then each eye sees a square, the square of the right eye being slightly shifted to the left. With both eyes together, you will have the impression to see a floating square (coming out of the screen)!
This effect was demonstrated for the first time in 1838 by Wheatstone.
There is a more sophisticated variant, where the red and blue filters of the glasses are replaced by horizontal and vertical polarizing filters, and the two polarized images are projected on a cinema screen.
Dr. Bela Julesz studied stereopsis with images like this:
Start with a surface filled with random dots, cut out a small square, move it a bit to the left, fill the resulting white space on the right with new dots.
The image pair is viewed with a stereoscope so that the left eye sees
the original image, and the right eye sees the transformed image: both
images fuse to one single image where a square seems to float above the
dotted surface.
(You may even arrive in seeing the floating square without a stereoscope,
if you look with each eye at one image, and squint so that both images
fuse to one; but this exercise is very straining for the eyes!)
The reason to use random dots was to prove that the brain can produce a 3-dimensional image, even when the original image doesn't contain any suggestive forms.
Christopher Tyler found a trick which made it possible to use only one image for both eyes, like in this example:
Instead of a dotted surface, we start with a background image consisting
of a vertical stripe large of about 4 cm.
This stripe is copied several times from right to left, but according to
the object to represent, the copied points are shifted more or less.
If you look relaxed at this image, and try to focus on a fictive point behind (or in front of) the image, then there will (hopefully) come a moment when your eyes get confused by the repeating stripes and will see only 7 of them instead of 8. This is somehow the contrary effect of squinting.
In practice, this object is a greyscale image, brighter points will result in a greater left-shift of the corresponding points of the background image. This explains why in books, the "solutions" of the "hidden images" are given as greyscale images! But this does not mean that every greyscale image can produce a good stereogram!
Sources:
Update: 1996-10-19, Patrick Hahn (phahn@vo.lu)