Left: When an observer looks at a point P, the two images of P fall on the foveas P. Q is a point that is judged by the observer to be the same distance away as P. The two images of Q (QL and Qr) are then said to fall on corresponding points. (The surface made up of all points Q, the same apparent distance away as P, is the horopter through P.) Right: If Q' appears closer to the observer than Q, then the images of Q' (QL'
and Qr') will be farther apart on the retina in a horizontal direction than they would be if they were corresponding points. If Q'
appears farther away, QL' and Qr' will be horizontally displaced toward each other.
The strategy of judging depth by comparing the images on our two retinas works so well that many of us who are not psychologists or visual physiologists are not aware of the ability. To satisfy yourself of its importance, try driving a car or bicycle, playing tennis, or skiing for even a few minutes with one eye closed. Stereoscopes are out of fashion, though you can still find them in antique shops, but most of us know about 3-D movies, where you have to wear special glasses. Both of these rely on stereopsis.
The image cast on our retinas is two-dimensional, but we look out on a three-dimensional world. To humans and animals it is obviously important to be able to tell how far away things are. Similarly, determining an object's three-dimensional shape means estimating relative depths. To take a simple example, circular objects unless viewed head-on produce elliptical images, but we can generally recognize them as circular with no trouble; and to do that requires a sense of depth.
We judge depth in many ways, some of which are so obvious that they hardly require mention (but I will anyhow). When the size of something is roughly known, as is so for a person, tree, or cat, we can judge its distance—at the risk of being fooled by dwarves, bonsai, or lions. If one object is partly in front of another and blocks its view, we judge the front object as closer. The images of parallel lines like railroad tracks as they go off into the distance draw closer together: this is an example of perspective, a powerful indicator of depth. A bump on a wall that juts out is brighter on top if the light source comes from above (as light sources generally do), and a pit in a surface lit from above is darker in its upper part: if the light is made to come from below, bumps look like pits and pits like bumps. A major clue to depth is parallax, the relative motions of near and far objects that is produced when we move our heads from side to side or up and down. Rotating a solid object even through a small angle can make its shape immediately apparent. If we use our lens to focus on a near object, a far one will be out of focus, and by varying the shape of the lens—by changing accommodation (described in Chapters 2 and 6)—
we should be able to determine how far an object is. Changing the relative directions of the eyes, adjusting the toeing in or toeing out, will bring the two images of an object together over a narrow range of convergence or divergence. Thus in principle the adjustment of either lens or eye position could tell us an object's distance, and many range finders are based on these principles.
Except for the convergence and divergence, all these depth cues need involve only one eye. Stereopsis, perhaps the most important mechanism for assessing depth, depends on the use of the two eyes together. In any scene with depth, our two eyes receive slightly different images. You can satisfy yourself of this simply by looking straight ahead and moving your head quickly about 4 'inches to the right or left or by quickly alternating eyes by opening one and closing the other. If you are facing a flat object, you won't see much difference, but if the scene contains objects at different distances, you will see marked changes. In stereopsis, the brain compares the images of a scene on the two retinas and estimates relative depths with great accuracy.
Suppose an observer fixes his gaze on a point P. This is equivalent to saying that he adjusts his eyes so that the images ofP fall on the foveas, F (see the left part of the diagram this page). Now suppose Q is another point in space, which appears to the observer to be the same distance away as P, and suppose QL and QR are the images of Q on the left and right retinas. Then we say that QL and QR are corresponding points on the two retinas. Obviously, the two foveas are corresponding points; equally obvious, from geometry, a point Q'
judged by the observer to be nearer to him than Q will produce two noncorresponding images QL' and QR' that are farther apart than they would be if they were corresponding (as shown in the right of the diagram). If you like, they are outwardly displaced relative to each other, compared to the positions corresponding points would occupy. Similarly, a point farther from the observer will give images closer to each other (inwardly displaced) compared to corresponding points. These statements about corresponding points are partly definitions and partly statements based on geometry, but they also involve biology, since they are statements about the judgements of the observer concerning what he considers to be closer or farther than P. All points that, like Q (and of course P), are seen as the same distance away as P are said to lie on the horopter, a surface that passes through P and Q and whose exact shape is neither a plane nor a sphere but depends on our estimations of distance, and consequently on our brains. The distance from the foveas F to the images ofQ (QL and Qp) are roughly, but not quite, equal. If they were always equal, then the horopter would cut the horizontal plane in a circle.
Now suppose we fix our gaze on a point in space and arrange two spotlights that shine a spot on each retina so that the two spots fall on points that are not corresponding but are farther apart than they would be if they were corresponding. We call any such lack of correspondence disparity. If the departure from correspondence, or disparity, is in a horizontal direction, is not greater than about 2 degrees (0.6 millimeters on the retina), and has no vertical component greater than a few minutes of arc, what we perceive is a single spot in space, and this spot appears closer than the point we are looking at. If the displacement is inward, the spot will appear farther away. Finally, if the displacement has a vertical component greater than a few minutes of arc or a horizontal component exceeding 2 degrees, the spot will appear double and may or may not appear closer or farther away. This experimental result is the principle ofstereopsis, first enunciated in 1838 by Sir Charles Wheatstone, the man who also invented the Wheatstone bridge in electricity.
It seems almost incredible that prior to this discovery, no one seems to have realized that the slight differences in the two pictures projected on our two retinas can lead to a vivid sense of depth. Anyone with a pencil and piece of paper and a few mirrors or prisms or with the ability to cross or uncross his eyes could have demonstrated this in a few minutes. How it escaped Euclid, Archemides, and Newton is hard to imagine. In his paper, Wheatstone describes how Leonardo da Vinci almost discovered it. Leonardo attributed the depth sensation that results from the use of the two eyes to the fact that we see slightly farther around an object on the left with the left eye and on the right with the right eye. As an example of a solid object he chose a sphere—
ironically the one object whose shape stays the same when viewed from different directions. Wheatstone remarks that if Leonardo had chosen a cube instead of a sphere he would surely have realized that the two retinal projections are different, and that the differences involve horizontal displacements.
The important biological facts about stereopsis are that the impression of an object being near or far, relative to what we are looking at, comes about if the two images on the retina are horizontally displaced outward or inward relative to each other, as long as the displacement is less than about 2 degrees and as long as vertical displacement is nearly zero. This of course fits the geometry:
an object's being near or far, relative to some reference distance, produces