Above: Wheatstone's stereoscope, the original drawing of which is shown to the right.
outward or inward displacement of its images on the retinas, without any significant vertical component.
This is the principle of the stereoscope, invented by Wheatstone and for about half a century an object present in almost every household. It is the basis for stereo movies, which we view with special polarized glasses. In the original stereoscope a person looked at two pictures in a box through two mirrors so that each eye saw one picture. To make this easy we often use prisms and focusing lenses. The pictures are identical except for small, relative horizontal displacements, which lead to apparent differences in depth. Anyone can make photographs suitable for a stereoscope by taking a picture of a stationary object, then moving the camera about 2 inches to the left or right and taking another picture.
Not all people have the ability to perceive depth by stereoscopy. You can easily check your stereopsis by using the illustrations on this page: each of the diagrams shows two pictures that together would produce a stereogram for use in an ordinary stereoscope. You can place a copy of these in a stereoscope if you happen to have one, or you can try to look at one with each eye by putting a thin piece of cardboard between them, perpendicular to the plane of the page, and staring, as if you were looking at something far away; you can even learn to cross your eyes, by holding a finger between you and the pictures and adjusting its distance till the two fuse, and then (this is the hard part) examining the fused image without having it break apart. If this works for you, the depth will be reversed relative to the depth you get by the first method.
Even if you don't see the depth, because you have no stereoscope or you can't cross or uncross your eyes, you will still be able to follow the arguments—
you just miss the fun.
In the uppermost example, we have a square containing a small circle a little to the right of the center in one member and a little to the left in the other. If you view the figure with both eyes, using the stereoscope or partition method, you should see the circle no longer in the plane of the page but standing out in front of it about an inch or so. Similarly, you should see the second figure as a circle behind the plane of the page. You see the circle in front or behind the page because your retinas are getting exactly the same information they would get if the circle were in front or behind.
In 1960 Belajulesz, at Bell Telephone Laboratories, invented an ingenious, highly useful method for demonstrating stereopsis. The figure on the next page will at first glance seem like a uniformly random mass of tiny triangles—
and indeed it is except for the concealed larger triangle in the center part. If you look at it through pieces of colored cellophane, red over one eye and green over the other, you should see the center-triangle region standing out in front of the page, just as the circle did. (You may, the first time, have to keep looking for a minute or so.) Reversing the cellophane windows will reverse the depth. The usefulness of these Julesz patterns is that you cannot possibly see the triangle standing out in front or receding unless you have intact stereopsis.
To sum up, our ability to see depth depends on five principles:
1. We have many cues to depth, such as occlusion, parallax, rotation of ob jects, relative size, shadow casting, and perspective. Probably the most important cue is stereopsis.
2. If we fixate on, or look at, a point in space, the images of the point on our two retinas fall on the two foveas. Any point judged to be the same distance away as the point fixated casts its two images on corresponding retinal points.
3. Stereopsis depends on the simple geometric fact that as an object gets closer to us, the two images it casts on the two retinas become outwardly dis placed, compared with corresponding points 4. The central fact ofstereopsis—a biological fact learned from testing people—
is this: an object whose images fall on corresponding points in the two retinas is perceived as being the same distance away as the point fixated.
When the images are outwardly displaced relative to corresponding points, the object is seen as nearer than the fixated point, and when the displace ment is inward, the object is seen as farther away.
5. Horizontal displacements greater than about 2 degrees or vertical displace ments of over a few minutes of arc lead to double vision.
THE PHYSIOLOGY OF STEREOPSIS
If we want to know how brain cells subserve stereopsis, the simplest question we can ask is whether cells exist whose responses are exquisitely dependent on the relative horizontal positions of images falling on the retinas of the two eyes. We should begin by discussing how cells in the visual pathway respond when the two eyes are stimulated together. We are now talking about cells in area 17 or beyond, because retinal ganglion cells are obviously monocular, and geniculate cells, because of the left-eye, right-eye layering, are for all intents and purposes monocular: they respond to stimulation of either one eye or the other, but not both. In area 17 roughly half the cells are binocular, responding to stimuli in the left eye and to stimuli in the right eye. When tested carefully, most of these binocular cells seem not to be greatly concerned with the relative positions of the stimuli in the two eyes. Consider a typical complex cell, which fires continuously if a slit sweeps across its receptive field in either eye. When both eyes are stimulated together, the cell fires at a higher rate than it does to separate eyes, but it generally does not matter much if at any instant the slits in the two retinas fall on exactly the same parts of the two receptive fields. The best responses occur if the slit enters and leaves both eyes' receptive fields at about the same time, but if it enters one a little before or after the other, it doesn't matter very much. A typical curve of response (say, total number of spikes per pass) versus difference in slit position in the two eyes is shown at the top of the next page. The curve is rather flat, clearly indicating that the relative position of the slit in the two eyes is not very important. This kind of cell will fire well to an appropriately oriented slit whether it is at the distance someone is looking, or is nearer or farther away.
Compared with this cell, the cells whose responses are shown on page 38
are very fussy about the relative positions of the two stimuli and therefore about depth. The first cell (bottom of next page) fires best if the stimuli to the two eyes fall on exactly corresponding parts of the two retinas.
To prepare this figure, called an anaglyph, Bela Julesz begins by constructing two arrays of randomly placed tiny triangular dots, identical except that (1) one consists of red dots on a white background and the other of green dots on a white background, and (2) over a large triangular region, near the center of the array, all the dots in the grecn-and-whitc array are displaced slightly to the left, relative to the corresponding red and white dots. The two arrays are now superimposed with a slight offset, so that the dots themselves do not quite superimpose.
If the figure is viewed through a green cellophane filter, only the red dots are seen;
through a red cellophane filter, only the green dots are seen. If you view the figure with green cellophane over the left eye and red over the right, you will see the large triangle standing out about 1 centimeter in front of the page. Reversing the filters (green over the right eye and red over the left) causes the triangle to appear behind.
Right: Wheatstone's diagram of his stereoscope. The observer faced two-45 degree mirrors (A and A') and saw, superimposed, the two pictures, E through the right eye and E' through the left. In later, simpler versions, the observer faces the two pictures placed side by side on a screen at a distance apart roughly equal to the distance between the eyes. Two prisms deflect the directions of gaze so that with the eyes aligned as if the observer were looking at the screen, the left eye sees the left picture and the right eye the right picture. You can learn to dispense with the stereoscope by pretending you are looking at a distant object, thus making the two directions of gaze parallel so that the left eye sees the left picture and the right eye the right picture.