Binocular Vis ion and the Perception of Depth

CHAPTER 8

8.1 up your thumb at arm's length. focus on a more distant object. You INTRODUCTION Now open that eye and close the will notice two somewhat transpar­ other. Notice that your thumb ap­ ent fingers in front of the distant - pears to move against the back­ object. Alterna tely, focus on th e fin­ Up to now we've been concerned ground as you alternate eyes-each ger, and you can see two images of with m onocular* optics; the cam­ eye sees a slightly different view. the dista nt object. The TRY IT offers eras we considered had one (pOSSi­ Nevertheless, normally you see another example.) bly compound) lens, the properties only one view--somehow your brain Why bother with two images of of vision we described required only combines the two images into one the same scene? These slightly dif­ on e eye, and the optical instru­ view of the world. In Figure 7 .3, we ferent images enable you to gauge m ents we discussed were mostly de­ showed the neural connections that the depth of a three-dimensionaL signed for the use of one eye. We allow this mixing of the signals scene. Among animals, predators now want to acknowledge that we, from your two eyes. Notice that a (such as cats) have their two eyes in like many other animals, have two given side of the brain gets signals front, with overlapping fields of eyes and ask what might be the ad­ from the corresponding pOints of view, to enable them to judge accu­ vantage of this binoculart arrange­ each eye, allowing it to combine rately the distance to their prey. On ment. them into one view of the world. the other hand, animals (such as One advantage of the second eye (There are occasions, however, when is that it provides us with an in­ your brain Isn't able to provide a creasedjield oj view. Close one eye single, smoothly combined, view, FIGURE 8,1 and you immediately notice that and you become aware of the t~o part of the scene previously visible images produced by your two eyes. Edward Hicks' "The Peaceable Kingdom" shows both predators and prey. They are is no longer in your field of view. Hold up a finger in your field of easily distinguis hable by the location of Many animals, such as fish or rab­ view, fairly close to your eyes, and their eyes. bits, have eyes set in oppos ite sides of their heads, each eye providing a separate view of the world. To­ gether, two such eyes, with practi­ ~~~o~rl~inthdr~~~ view, enable the animal to see a sweeping panorama. (The field of view of a rabbit is 360°, allowing it to see all around without turning its hea d , with only 24° overlap be­ tween the two eyes.) However, your two eyes, placed in th e front of your head, have fields of view that overlap considerably. (Your field of view is 208°, with 130° over lap.) Your eyes thus provide slightly d ifferent views of almost the same scene. Close one eye a nd hold

'Greek monos, single, plus Latin oculus, eye. tLatin bini, a pair. 207 CHAPTER B: BINOCULAR VISION AND THE PERCEPTION OF DEPTH

208 rabbits or deer) who are likely to be that provide conflicting visual that while you sleep an artist or someone else's dinner have non­ depth cues. By examining the re­ photographer has made an ex­ overlapping fields of view to give sulting illusions, we can learn tremely realistic picture of the view them the wide angle of view best for about the way the brain processes from your bedroom window and detecting predators (Fig. 8.1). Sim­ the variOUS cues in arriving at its pasted it to the outside of your win­ ilarly, animals that leap about the depth determinations. dow so when you awake you see the branch es of a tree (such as squir­ Let's examine a number of the picture (Fig. 8.2). How can you tell rels and our simian ancestors) techniques by which we visually whether you are looking at such a must be able to gauge the depth of fathom the depths of the world painting or at the actual scene out­ those branches, and correspond­ around us. We'll begin by separat­ side your Window? In the next few ingly h ave two eyes in the front. ing the depth cues that can be used sections, we'll discuss several tech­ Th e brain, as we1J see, uses many in a painting from those normally niques for distinguishing the two cues in its determination of depth. unavailable to the artist. Imagine alternatives. S ome require Signals from two eyes, but others do not. These latter In­ clude the cues that artists rely on FIGURE 8.2 when they convey a feeling of depth TRY IT Rene Magritte, "The Human Condition in two-dimensional pictures. It is I. " How can we visually distinguish FOR SECTIO N 8.1 also possible to playoff one cue between the artist's rendering of the against the other-to create scenes outside world and the world itself? Two eyes provide two views

Hold one end of a string against your upper lip and pull the other end straight in front of you. You will see not one but two strings stretching out in front of you and crossing. These correspond to the images from your two eyes, as you can readily confirm by alternately closing each eye. The point at which they cross is the point on the string at which you "aim" your eyes (the point toward which your eyes converge). Try looking at different points of the string, beginning close to you and moving away, and notice how the cross-over point moves away from you as you do this. (Having a friend slide a finger along the string may enhance this effect.)

8.2 -ACCOMMODATION Just as you can measure the dis­ tance to an Object by focusing you r camera on it and noting the lens' position, the amount of accommo­ dation necessary to focus your eye on an Object tells you the object's distance. [f you see an objec l clearly while your eyes are relaxed, you know that it is far from you. [f, however, you must tense your cili­ ary muscles to make the Object come into focus, then the object must be closer. Thus wh en looking out your window at the actual street scene, you would accommodate dif­ ferently for objects at different dis­ 8.4 PARALLAX

209 tances in the scene. The artist, at­ tions of gaze is bigger than if you 8.4 tempting to simulate that scene look at a distant object (Fig. 8.3). PARALLAX with a picture, must decide what is This angle is called the angle of in focus and what is not. Once the convergence of your eyes. (For an artist makes that decision, no object at the normal near pOint. 25 -To gauge depth, you rely much amount of accommodation on your cm, the angle of convergence takes more heavily on the fact that your part will s harpen the focus of an ob­ its maximum value of about 15°. It view is different from different po­ ject painted out of focus. is only 1° for an object about 4 m sitions-the phenomenon of paral­ If you were a cha meleon, you'd from your eyes.) If your brain keeps lax.* With one eye closed, hold you r rely very heavily on th e technique of track of the convergence of your thumb a few centimeters from this accommoda tion to gauge the dis­ eyes, it can determine the distance page, so it blocks your view of the tance to flying insects as you flicked to the object that your eyes are word "parallax." By moving your them w i th your tongue. If you cover vie\ving by uSing the rangefinder head, you can Change your view one eye of a chameleon, he main­ principle (Sec. 4.2D). As you scan a suffiCiently so that you can see the tains his high degree of accuracy in painted picture, your convergence, word. This is possible only because fly-flic king-binocula r vision is not like your accommodation, remains your thumb and the word are at d if­ important here. However, give a unchanged. because all the objects ferent distances from your eye. So, chameleon a pair of glasses that lie in the plane of the picture. How­ your view of different objects change the amount of accommoda­ ever, when you view objects at dif­ changes as you move, according to tion necessary and his tongue flaps ferent depths in an actual scene, their distance from you (Fig. 8.4 ). futilely at the fleeing fly. your convergence changes. No matter how carefully our artist Humans, on the other hand, Like accommodation, conver­ simulates other depth cues, as long make little use of this technique, gence is most effective for determin­ as she confines herself to a flat can­ possibly because our potential meals ing depth in nearby scenes, but you vas, she cannot overcome parallax. are gen rally more than a tongue's make relatively little use of it. That You need only move your head and dislance away. Accommodation as a you make some use of convergence, compare the relative positions of way of determining depth is at best however, can be seen from the TRY the distant scene and the window only useful for close objects. If our IT. glazing bars to determine if she has mischievous artist confines herself tried to trick you. Similarly, the to a distant scene and if you cannot painted finger of Lord Kitchener get too close to the window, then (Fig. 8 .5) always points at you, no both the p ainting and the actual matter where you move \vith respect scene would be in focus to your re­ T RY IT to the picture, but an actual finger laxed eye and you wouldn't be able pOints in one direction-when you to distinguish the one from the FOR SECTION 8.3 move out of that direction, it no other by accommodation. Convergence and depth longer pOints at you. Because your view of Lord Kitchener's finger Look at a distant street lamp. Cross your doesn't Change. your brain m ay in­ eyes (say by looking at your finger, terpret the image as if he were ro­ which you hold in front of and just tating as you walk by, so as always below the light). Notice that the light 8.3 to point at you. This provides the appears closer (and smaller) than CONVERGENCE before-the more your eyes converge, recruiting poster with a rather p er­ the closer. This is true even though you sonal touch. (The FOCUS ON X-Ray - see two images, but it may be easier if If you look at a near object, the an­ you position yourself so that only one *Greek parallaxis. change. gIe between your two eyes' direc­ eye can see the lamp.

FIGURE 8.3

The angle of convergence, , is (a) large Left eye for near objects, and (b) small for distant objects.

Right eye (al (b) CHAPTER 8, BINOCULAR VISION AND THE PERCEPTION OF DEPTH

2 10 This happens sometimes when o you're drunk or injured and your eyes don't properly converge to a '- ." 0 single Object. (As Shakespeare put Moon it, "Methinks I see these thin gs ¥ ! with parted eye, when everything View from position J A 8 seems double.") A Consider the two eyes viewing the 8 cube in Figure 8.6. The views s een Rays from the moon by the two eyes, shown in the fig­ Moon ' ------­------ure, are slightly differen t. A com­ Eye position 1 mon feature, say the front edge aA, B is imaged at corresponding pOints of the two retinas. Anoth er fe atu re, Rays from the moon Moon say the left edge dD, is imaged at ------~ ------locations on the two retinas that do B not correspond to each other. Som e Eye position 2 A cells in the visual cortex of the brain respond strongest wh en a o common feature occurs at corre­ sponding pOints of the retinas. Other cells have their strongest re­ sponses at particular differences between the locations of the two View from position 2

Left eye Right eye tJ Ii .... :1 )1" F1G URE 8.4 Tomography gives an application of :1 ::~ , parallax.) The locations of the images on the retina Q change as the eye moves from position 1 to position 2. The location of the retinal \\, \\ im age of the nearest object, A, changes 8.5 I" the most-from one extreme to the I \\ BINOCULAR DISPARITY 1 \ \ o ther. The image of a more distant , object, B, moves less, while the parallel 1 rays of the very distant moon are always I imaged on the same p lace on the retina. - Those of us blessed with two eyes This is why the moon seems to follow need not move in order to gain the you as you look at it, say, from the benefits of parallax for gauging window of a moving car. The closer the object is to you, the more it appears to depth. The two eyes, separated b y (a) move in the direction opposite to your about 61 cm and with significantly motion. overlapping fields ofview, see slightly c c different views ofany Object they look at. This difference between the views of the two eyes (binocular disparity) thus provides a way of TjJ~ JEr determining the distance of the ob­ D A A B ject in sight. Your brain attempts to View from left eye View from right eye reconcile the two views by ascribing the difference to depth. You "see" (b) one three-dimensional view of the world, rather than two two-dimen­ Sional views. If the images are too FIGURE 8.6 diverse, your brain cannot fuse them. and you get double vision. The two eyes looking at a cube see slightly different views of the cube. (a) View of eyes and cube. (b) Views seen by each eye. Note that the edge da FIGURE 8.5 subtends a smaller angle for the right eye than does ab, and consequently that eye Alfred Leete's 1914 recruiting poster of sees da as shorter than abo For the left Lord Kitchener, a British hero of the eye, the situation is reversed. Boer War. 8.5 BINOCULAR DISPARITY

2 11 retinal s timulations. Thus, one set blue and red patches separated and afterimages with similar results. The of s u ch cells in the visual cortex re­ surrounded by black or white. Blue and persistence of positive afterimages also sponds strongest to aA, while a dif­ red squares on one side of a Rubik's permits you to achieve an enhanced ferent set responds strongest to dD. Cube can be used, as can a picture of an depth. Use the technique described in Thi difference in response leads to American flag on a white background. the first TRY IT for Section 7.lA, but With pieces of cardboard or stiH paper, separately th e viewer 's percept ion that these here, since each eye is cover the outer half of each pupil while exposed to the window scene, care must features lie at different depths. (See you look at the colored patches. Notice be taken that the eyes are pointed in the the TRY IT's.) Under some circum­ which color appears closer, then same direction. After covering both eyes stances, two objects will be seen in suddenly remove the cardboard pieces with your hands for 30 seconds, expose dep th if the separations of their im­ and notice the change. Now cut a piece your right eye. To be sure that it is aimed ages on the two retinas differ by as of cardboard so it fits between your properly, momentarily squint and point little as 1 j..lm-Iess than the diam­ pupils, covering the inner half of each. your eye at a distant point, such as the eter of a cone! (If you cut one of the vertical sides of the top of a distant tree. Then open your Of course, binocular disparity is cardboard at an angle, you can get a right eye wide for three seconds, useful for d epth determination only good fit without preCision measurements keeping it pointed at the treetop. Next if your two eyes see different im­ by raising or lowering the cardboard until close and cover your right eye. it just doesn't block your view.) Look at Immediately move your head a few ages. A h orizontal clothes line looks the patches again, and notice that the centimeters to the left and similarly the same to both eyes, so when you depth has reversed from the previous expose your left eye to the scene, want to know where it is, it helps to case. pointing it at the same treetop. Expose tilt your h ead in order to introduce this eye for only two seconds. Close and some disparity. (Objects that repeat P ON DE R cover the left eye (the right should them selves, such as bars on a cage, already be closed and covered) and can offe r confusing binocular dis­ Why did you need to use the cards? watch the stereo afterimage develop. parity. If your eyes fuse the wrong Compare the depth in the afterimage to bars together, the cage appears at a the actual depth. different distance than it should.) Considering your half lenses as prisms, To appreciate fu lly this "superstereo," compare it to reduced stereo, achieved Dis tant scenes also present essen­ draw ray diagrams to convince yourself by moving your head to the right tially the same view to your two that the dispersion of the "prisms" is responsible for the eHects you saw. between the exposure of your eyes in the eyes, so, like the other techniques order described above. The reduced dlscussed so far, binocular dispar­ eHective separation between the eyes ity is of little use for such scenes. gives a flatter view. Moving your head But for relatively close objects it is about 13 cm to the right may result in a an extrem ely effective way of gaug­ Sec ond TR Y I T pseudoscopic view (Secs. a.SA and a.6HJ. ing depth. In fact, by presenting two different views to your two eyes, FOR SECTION 8,5 it is p ossib le to fool your brain into Increase your binocular disparity believing that there is depth even A. The stereoscope and wh en there actually isn't. A variety The amount of depth perceived depends related optical of optical instrumen ts, devices, and on the binocular disparity between your instruments a nd toys toys are b ased on th is idea. two views. The wider the separation of the points of view, the greater the How can we present two different apparen t depth. Thus, stereoscopic aerial p ictures to the two eyes, so that photographs are often taken from the each eye sees only the image in­ two wing tips of the airplane. If you find tended for it? A simple tech nique is first TRY IT the world too rhallow, you can use the periscope described in the TRY IT for that used in the nineteen lh-century fOR SECTION 8.5 Section 2.4C to increase your binocular stereoscope.· Here two photo­ Depth and chromatic aberration '. disparity. Hold the periscope horizontally graphs taken from slightly different and look through it with one eye while angles are placed side by side. You Th e lenses of your eyes have some also looking with the other, unaided eye view them through speCial lens­ chromatic aberration-they bend blue at the same object. A short periscope prism combinations, one for each light more than red. If you cover half works best; about 6 cm will double your eye (Fig. 8.7). The lenses p roduce your pupil, the resultant half lens acts eye separation. The diHerence in distant virtual images of th e photo­ like a prism, shifting the blue retinal apparent depth is most obvious if, after graphs, while the prisms cau s e image slightly away from the red image. looking through it, you quickly remove th ese virtual images to appear at By arranging that this shift is in opposite the periscope while continuing to look at directions for the two eyes, you can the scene. th e same place. Thus, the lenses as· produce binocular disparity-color Just as your brain interprets nerve sure the proper accommodation differences can therefore be translated signals from both of your eyes to into diHerences in apparent depth. produce the sensation of depth under In order to eliminate other depth cues normal viewing conditions, it can as much as pOSSible, you should look at interpret signals accompanying 'Greek stereos, solid. CHAPTER 8, BUVOCULAR VISION AND THE PERCEPTION OF DEPTH

212

Left eye Right eye of the population lack the s tereo­ scopic vision to see depth from stereo pairs, even if they have two good eyes.) Figure 8.8 shows a ste­ reoscopic pair. If you hold a piece of cardboard between your eyes so it blocks each eye's view of the other eye·s photograph (being careful to Lens-prism avoid shadows), you may be able to combination I fuse the two pictures even though \ I the convergence and accommoda­ \ I \ I tion is wrong. It may take a little \ I time and concentration befo re the \ I Photograph for left eye +_--1___ Photograph fo r right eye images fuse. If you are able t.o do this, you can then use this talent to \\ I" pick out small differences in other­ ---f-­ wise identical photographs, since Loca tion of virtual image formed only the different parts will ap pear by each lens-prism co mbination in depth. This depth effect h as been used to notice changes in the posI­ (a) tion of stars. Two photograp hs of the night sky, taken at different times, are viewed stereoscopically. Those stars that have moved appear in depth. (A better way is to View the two photographs in rap id suc­ ceSSion, repeatedly, and then look for the stars that appear to move­ Sec.7.7A.) Stereo pairs of pictur es m ay be obtained in a variety of ways. Cam­ eras have been made with side-by­ side lenses that produ ce two photo­ graphs simultaneou sly. You can make stereo pairs with an ordinary camera by photograph ing a still scene, then moving llie camera to one side, and taking another pic­ ture. Stereo pairs are produced on the scanning electron m Icroscope by taking a picture, tipping the m i­ croscopic sample sligh tly, and lak­ ing another picture. You can even make drawings, s uch as Figure 8.6b. (See the first TRY [T.) What happens if you in terchange the two pictures, say of Figure 8.6b, (b) so the right eye views the left eye's picture and vice vers a ? he left eye FI G URE 8 .7 ity resulting from differences in the will then see edge da as sh orter photographs often produces qUite a than ab, while the right eye sees (a) The principle of the stereoscope. The amount of depth perceived depends on realistic impression of depth. (The the reverse. This is just what lliese the disparity between the two modern "Viewmaster,'· used for sce­ eyes would see if they viewed the p ho tographs. (b ) A stereoscope from the nic views or as a toy, is just an in­ three-dimensional object shown in 1880s viewing a stereo pair of expensive version of Hermann von Figure 8.9. That is, instead of the photographs of Indians. Helmholtz's 1866 stereoscope.) original front edge aA appearing With a little effort, you can obtain closer to the viewer. it appears f ar­ and the prisms the proper conver­ depth from stereoscopic pairs of ther than the side edges dO and bB. gen ce for an object at one, average pictures without paraphernalia. Such a view, where the parts of th e distance. But the binocular dispar­ (That is, most of you can. About 2 % original Object that came forward 8 .5 BINOCULAR DISPARITY

213 Another way of providin g sep a­ rate images to the two eyes from two-dimensional pictures does not require that the two pictures be separated. The pictures are super­ imposed, and the viewer is given special eyeglasses that allow each eye to see only the image meant for it. This may be done in two ways: using color or using polarized lig ht (Sec. 1.3B and Chapter 13). Con­ Sider first the use of color. Here t he two eyes' views are printed in differ­ ent colors, say red and green. as in PLate 8.1. (Such a picture is called an anaglyph.*) The viewer is given a different colored filter for each eye. If the filter is red, t he green printing appears black, wh ile the FIGURE 8.8 present binocular disparity and hence a three-dimensional view to red printing is almost invisible. If Stereoscopic photograph: "1'\'lrs. Jones the observer. (The second TRY IT the filter is green. the reverse is Comes Back Unexpectedly" (1897). gives an unusual example.) The true. Thus, the eye with the red fil ­ most common such instrument is ter sees only the green p icture, Left eye Right eye binoculars-actually binocular tele­ while the eye with the green filter scopes, one for each eye. Binocular sees only the red picture. Desptte microscopes are more complicated. the difference in colors, the images Because the image produced by a can usually be fused into a three-di­ microscope is usually inverted (Fig. mensional view. (To produce three­ 6.9), simply providing each eye with dimensional shadows this way. s ee its own microscope directed at the the third TRY IT.) This technique same Object would produce a pseu­ was used in early "3-D" movies, but doscopiC view. because the color is used for !.be stereo effect the resultant three-di­ mensional view is not in full color. PONDER Full-colored 3-D movies can be achieved by using polarized light . Why does each eye viewing an inverted The idea here is to project the two image result in a pseudoscopic view? A a lens inverts the image (turns it upside images with light of different p olar­ down) by reversing both up·down and ization, say one with vertically po­ left-right. Draw these reversals separately larized light and one with horizon­ FIGURE 8.9 for the two views of Figure 8.6b, first a tally polarized light. The viewer left-right reversal and then an up-down looks through polarizing filters. The The two eyes looking at this object reversal of that. A mirror may help in filter over one eye allows only the would have views corresponding to each case. Which reversal causes the those shown in Figure 8.6b, but with the vertically polarized light to pass, stereo pair to result in a pseudoscopic views interchanged. The edge da now while the filter over the other eye, subtends a smaller angle for the left eye, image? oriented perpendicularly to the so that eye sees da as shorter than abo first, passes only the horizon tally polarized light-but ea ch "in liVin g For this reason, some binocular mi­ color." (Actually, the pola riza tion is croscopes do not offer a stereo view. usually at 45°, one running between appear to go back and vice versa, is They simply have a binocular eye­ upper right and lower left, and the caIled pseudoscopic.• piece so that you can look at the other betvleen upper left and lower Various other optical devices that same image with both eyes. Other right.) offer separate images to each eye binocular microscopes do offer a It is even possible to m al{e stereo proper stereo view by inserting an pairs of pictures where each mem­ erecting system in each of two mi­ ber of the pair. by itself, doesn't

214 ered with some pattern. To con­ first T R Y IT struct the right eye's view, we sim­ ply cut out a square (5) from the FOR SECTION 8.5A center of the left eye's view, slide it Constructing and viewing stereo pictures over, and fill the gap produced (Y) with more pattern. Any pattern will Stereo pictures may be made and viewed do, including a random array of in a variety of ways. For viewing with a dots, providing there is some pat­ stereoscope, you need two views from slightly separated positions. To do this tern that the brain can correlate be­ photographically, choose a scene that tween the two views. (A solid color has objects at a variety of distances from would not work because we are not the camera . Get the greatest possible going to draw any boundaries on depth of field by using the largest 5.) With random dots, each eye's f-number on your camera. It is b est to view shows no sign of the square 5 rest your camera on a flat surface, so yo U' when viewed monocularly. Never­ can carefully slide it to one side for the theless, by comparing the two ap­ second picture. The distance betwee n parently random patterns, the brain camera positions depends on the way the can notice the correlation (some of pictures are to be viewed and whether FIGURE 8.10 the dots are moved with respect to you want normal or exaggerated depth. If the object is distant (beyond 2 m) and The eye and brain can pick out the dog the others) and attribute this cor­ in this seemingly random array of dots. the pictures are to be viewed through relation to depth (Fig. 8.11b and magnifying glasses (as in most Plate 8.2). These random dot ste­ stereoscopes) so the image you see is look tik anything recognizable, but reograms, developed by Bela Julesz, distant, then you'll want to slide the when the pair is properly viewed, a have been extremely useful in sort­ camera sideways about 6-l cm, the three-dimensional figure becomes ing out which states of visual pro­ distance between your pupils. (The focal apparent. This can be done with cessing occur before stereoscopic length of the magnifying glasses should patterns of random dots. The eye fusion and which after. equal that of the camera lens so you see and brain are quite good at finding the same perspective the camera did.) Try taking several pictures at various order and sorting out familiar pat­ separations-the greater the separation terns when presented with a seem­ the greater will be the apparent depth, ingly random array of dots (Fig. but if the separation is too large, you 8 . 10). (See the fourth TRY IT for FIGURE 8.11 won't be able to fuse the two pictures. o t.her examples. ) If we present to Constructing a stereogram . (a) The views Very distant shots, from airplanes or tall the two eyes arrays of dots that, of the two eyes. The left eye sees X but buildings, are usually taken with a viewed monocularly, have no pat­ not Y. Everything else in the rectangle R separation between pictures of about % tern but do have a definite stereos­ is the same for the two views. Properly th e distance to the subject in order to viewed stereoscopically, the two eyes copic relation Wit h one another, the increase the apparent depth. For Close-up will fuse these views into a three­ photographs, you should rotate the arrays Will exhibit a pattern of dimensional image even if Rand S are camera as you slide it, so it is always depth when viewed properly binoc­ covered with random arrays of dots. directed at exactly the same point on the ularly. We can make such arrays by (b) The three-dimensional view is that of a square S floating above the rectangle R. object. For very close objects, you'll the method sh own in Figure 8.lla. For the left eye, S obscures the region Y want to slide the camera less than 6} cm. To construct the left eye's view, we of R. For the right eye. S obscures the (For magnifications greater than one, a need only make a rectangle (R) cov- region X of R. good rule of thumb is to divide 6) cm by

Left Right eye eye

9/ 1\ / 1\ I \ View fcom View from I \ left eye right eye

~l x y --' R DR

(a) (b) 8.5 BINOCULAR DISPARITY

215

the filters. You want colors that are easily seen through one filter but are invisible through the other. You can also use polarized light (Chapter 13) to proiect stereo pictures. You'll need four seperate polarizing Pi cture Picture filters and two projectors, as weI! as a " metallic" projection screen that does not destroy the polarization. You can use polarizing sunglasses, but you 'll either need four pairs, or you'll have to break them in half. (If yo u still have your glasses from a 3-D movie, you can get Mirrors away with two pairs them, only one of Mirror of which you need break.) Take two stereo (a) (b) pictures, as above, to make color slides. Tape a polarizing filter over the lens of each projector and project the two should draw a diagram like Figure 8.6a, slides, with their images the same size FIGU RE 8.12 but full scale, with the object at 25 cm and overlapping as much as possible. (a) Si ngle-mirror stereoscope. from the eye, and the eyes separated by View the screen with a polarizing filter (b) Wheatstone's stereoscope. 6I cm. To obtain the proper horizontal over each eye. These should be oriented distances for your version of Figure 8.6b, perpendicularly to each other (i.e., so draw a horizontal line through the object you can 't see through them when they the magnification.) Be sure to record in your version of Figure 8.6a and are held one behind the other). Rotate which picture was taken from the left measure the distances between the the filters over the projectors until each side and which from the right. locations where rays cross that line. For eye can see only the image from the Th e stereo pairs may be viewed example, the horizontal distance projector meant for that eye. without a stereoscope by the method between points c and d for the left eye described in Section 8. SA. Fusing the would be the distance, along that line, images may be easier if the views are between the rays that go from those crossed-that is, the photograph for the points to the left eye. Alternatively, if Second TRY IT left eye is placed on the right. (Looking you have a sufficiently simple real object at a pin held in front of you may assist and some artistic talent, you can directly you in crossing your eyes.) In another draw the pictures as seen by each of your FOR SECTION 8 .SA technique, you print one of the stereo eyes. The dark axle pair of photographs left-right reversed, Very simple figures can be drawn and instead of using a piece of cardboard without artistic talent. For example, two You 'll need a metal ring 6 cm across or b etween your eyes, you use a mirror circles, one inside the other, can be larger (like those used in macrame). In a (Fig. 8. 12a). Alternatively, you can use drawn with a compass. Make the outer dim room, illuminate a table top at a low Wh atstone's 1838 design to construct circle about 6 cm across and the inner angle, say with a flashlight. Spin the ring your own stereoscope. You will need circle 2 em across. For one eye's view, as you might spin a coin-it will look like only two small plane mirrors and some the center of the inner circle should be a a transparent sphere spinning around a way of supporting them (chewing gum, if few millimeters off center to the right, dark axle. The spherical appearance is all else fails). Position the two mirrors at and for the other eye's view, it should be due to perSistence of your vision in the an angle of 90°, with the reflecting off center the same distance to the left. dim light; the dark axle comes about surfaces facing outward (Fig. 8. 12b). The resultant stereo pair can be viewed because the ring looks extra dark when Looking into the mirrors from the vertex in all the usual ways. The Single-mirror its front hides its back. This occurs for of the angle, the left eye will see a view technique of Figure 8. 12a is particularly each eye when the ring is lined up with to the left and the right eye a view to the simple here, since for this case you draw that eye. Convince yourself that the right Position the photographs parallel to the inner circle off center in the same resulting binocular disparity locates the and facing each other and far enough direction for both eyes. (Why?) axle in the center of the sphere. from your eyes so that you can focus on Instead of a stereoscope, you can use them (25 cm, including the reflection) . If colored filters and make anaglyphs. The you have two identical converging colored filters can be the cheap plastic lenses, you can hold them in front of kind. You can make a photographic Third TRY IT your eyes and place the photographs at anaglyph by taking a double exposure on the focal plane of each lens. Since there color film, holding first one filter in front FOR SECTIO N 8.5A arc mirrors (which reverse the images), of the lens, and then the other, and Three-dimensional shadows you should print the pictures left-right moving the camera to the side between reversed or you'll get a pseudoscopic exposures. Alternatively, you can draw For this you'll need two lamps that throw view. an anaglyph by one of the above good sharp shadows, a red and a green You can also draw stereo pairs for your methods, using colored pens, pencils, light bulb, a white bed sheet, and a red stereoscope. This may be done in the or crayons, and drawing the two pictures and a green filter. (The filters should be same way as Figure 8.6 was drawn. If you one on top of the other. The colors with colors that make the biggest difference plan to view the pictures at 25 cm, you which you draw should be matched to when held in front of the two bulbs.) CHAPTER 8, BINOCULAR VISION AND THE PERCEPTION OF DEPTH

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_ ~m _ you move the loop across the screen­ the dots seem lassoed by the loop and move along with it. There seems to be a void immediately behind the loop as it slides through the sea of jittering dots. fuuQnfu=afu~w~ro~h~e~~ simply by moving your fingertip across the screen.

B. Lenticular screens

Another device that allows different images to be seen from a two­ dimensional picture is the lenticu­ Shadows lar screen. Consider a s heet of cy­ Sheet lindrical len ses (called lenticu1es) res ting on a flat picture at the focal p lane of the lenses Wig. 8.14). (S uch a sheet can easily and ch eaply be made of p lasUc. ) Light or iginating or scattered from any point in the picture plane is made parallel by the lenticules. The direc­ tion in which the light emerges Apparent location of three-dimensional structure from the len ticules depends on the location of the source in the picture plane. Light from a point slightly to the left of the axis of any of the lenses will emerge directed toward the r ight. Thus. ligh t from all the Viewer pOints labeled R in the figure will emerge headed to the right. as shown. Similarly, ligh t from the points labeled L will emerge di­ FIGURE 8.13 bright spots which, because the TV scans rected to the left. This means that through all 525 lines in .}~ sec (in North Set-up for three-dimensional shadows. America), change completely 30 times a an observer located at OF(' looking second. Therefore, the dots seem to slightly to the left. will see only the Arrange them as in Figure 8.73. With all jump about at random, as the eye pOints R, while an observer at OL' other lights off, an object behind the associates motion to the disappearance looking slightly to the right, will see scre en will cast two shadows on the of a dot at one pOint and the subse quent only the points L. sh eet, one due to each light. Viewed appearance of another dot at a nearby If the picture plane consists of al­ through the filters, the shadows will fuse location. Thus, the snow seems to into one three-dimensional shadow, ternating strips of two d ifferent pic­ consist of randomly placed dots tures, one at the pa ints R and the apparently in front of the screen. A good randomly jumping about-a truly chaotic other at the pOints L, then you see object to use is a person. As he walks scene. It is, however, quite easy to back from the screen toward the lamps, "train " these dots to behave in a more two different pictures, depending his shadow will appear to move forward orderly fashion, thanks to the ability of on the angle at which you view the from the screen toward the viewer. your brain to organize the visual lenticular screen. It may seem a bit information presented to it. All you need tedious to cut a picture into little do is give your brain the appropriate strips and paste them on the pLc­ hints. ture plane, b ut the s trips can be Fourth TRY IT Form a narrow channel between two made qUite simply photographically pieces of paper held against the TV by using the lenticular screen itself. FOR SECTION a.SA screen, separated by a few centimeters. Projecling a picture fro m O R toward Order oul of chaos Notice that the dots now seem to flow the pOints R will only expose the along the channel. Now make a loop out Ex cellent random arrays of dots can be of wire or string, about 5 to 6 cm across, pOints R. Another p icture can be easily found on your TV screen if you and hold it against the screen. Notice projected from OL toward the points simp ly tune to a nonbroadcasting how the dots jump around within and L, only exposing those pOints. In channel, so the screen is filled with without the loop, but do not seem to fact, it is possible to get several '·snow." This consists of randomly placed cross the loop. This remains true even if more pictures between those 8.5 BINOCULAR DISPARITY

217

FIGURE 8.1 4 The lenticular screen. A sheet of cylindrical lenses is backed by a picture, lying at the focal plane of the lenses . Light o riginating (or sca ttered) from the point labeled R is made parallel by the lenses and sent to the upper ri ght. light from the points labeled L is made parallel and sent to the upper left. If w e place two differen t p ictures , intermeshed , at the points R and L, we see the R picture only if we look from O R, the L picture only if we look from OL.

pOints. each corresponding to a dif­ feren l direction of exposu re. and la ter of viewing. Th e resultant de­ vice. whIch presen ts a d ifferent pic­ ture to you as you move past it, sLensheet { ---______~~UL ~~_1______ch angin your angle of view, is Picture plane sometimes seen in advertisements. picture pos tcards. and even politi­ c camp aign button s. It is reminis­ cent of the n Ineteenth-century puz­ zle pictures that were painted on a series of wedges (Fig. 8 .15). The FIGURE 8.15 a r tist Yaacov Agam has made mod­ em abstract (usually geometrical) (a) By painting one pi cture on the R faces of the w edge and a different one on the p ictures n such wedges. so that L faces , an artist can present two the viewer sees sev ral different pat­ pictures. Viewed from the right, the terns as sh e moves about. He has observer sees only the R picture; from also used len ticular screens simi­ the left, the observer sees the L picture. (b) Two views of such a Victorian puzzle larly to p resent a series of different p icture. (e) Another puzzle picture that images to the viewer. (The latter he shows three views. Two views are dra-,yn modesUy calls "Agamographs.") on opposite sides of the parallel slats . • (a)

(b) (el CHAPTER B, BINOCULAR VISION AND THE PERCEPTION OF DEPTH

218

If the R pictures and L pictures P _ Apparent motion correspond respectively to the right­ ----,--- ­ 1\ and left-eye views of a three-dimen­ 1 \

sional object . then a viewer with his P2 : \ P1 J( )( left eye at OL and his right eye at OR Actual moti on sees a three-dimensional view of the object. Each eye sees only the ap­ Apparent motion propriate view necessary for the correct binocular dispa rity. The an­ gle between the two views sh ould not be too great. corresponding to the natural convergence the viewer Dark filter Dark filter would have when looking at the ob­

ject. Such three-dim ensional piC­ Uncovered Covered Uncovered Covered tures can be foun d on picture post­ left eye right eye left eye right eye cards and som etimes on children's books. Stereo prints from the mul­ (a) (b) tiple pictures of the four-lens Nim­ slo camera are also made this way. It is important wh n viewing these pictures that the cylindrical lenti­ FIGURE 8.76 pendulum at two different positions cules should run vertically. If you (your covered eye "sees" the pendu­ rolate he pictu re so they run hori­ The Pulfrich pendulum . (a) The lum slightly in the past. compared pendulum moves to the left. The covered zontally. both you r eyes see the right eye sends a delayed signal to the to the uncovered eye). When the same image and the picture loses brain, so the viewer " sees" the pen dulum swings toward the side its depth. (However, if you then pendulum at P, with that eye at the same of your uncovered eye Wig. 8. 16a). '1 ... slowly Up the picture backward, the time that she "sees" it at P, with the the views of your two eyes, exten ded image you see will change.) uncovered left eye. Her convergence and backward . converge b eh ind the binocular disparity then tell her that the The picture may even appear in pendulum is at P, behind the actual plane of the pendulum. The conver­ front of the len ticular screen. Sup­ plane of motion of the pendulum. gence and binocular disparity infor­ pose that the picture plane is (b) The pendulum has now reversed its mation lead you to "see" the pen­ painted a s olid col r. say red. except direction . Again, the ri ght eye " sees" the dulum behind its actual posilion. pendulum at the delayed location, PJ , for two pOints (labled R J and LJ in which is now to the left of the location at When the pendulu m reverses its di­ Fig. 8 .14) that are painted green. If which the left eye " sees" the pendulu~ , rection Wig. 8.16b ). its apparent a viewer is positioned with his left P,. The point P, on which both eyes path is in front of its actual path. eye at OJ. and his right eye at OR' agree, is now in front of the actual Thus. the pendllJum a pp ars to both eyes s ee the same thing. red. pendulum plane. swing out of its actual plane. in an except a t the point where the rays ellipse.

from R J and L J intersect. This senting different views to the two This effect is named for Carl Pul­ pOint. shaded in the figure. appears eyes relies on the fact that the ia­ frich. who never saw it. He had green. Since this green point lies in tency time of the eye (Sec. 5.3C) been blind in one eye for sixteen front of the picture. the viewer sees varies with the intenSity of light. In yea rs before the p henomenon was a green spot fl oating in front of a dimmer light. the retinal cells re­ fi rst noticed as a nuisance In an op­ red baCkground. This effect has spond more slowly, as they must tical instrument that he des igned. been used by the artist Frank wait until there is suffiCient expo­ His colleague Ferdinand Fertsch ac­ Bunts, whose dizzying pictures may sure. Thus, an eye that is covered tually suggested the explanation. have spots floating as much as a with a dark filter will respond more The phenomen on was once used in foot in front of the picture plane. It slowly to the same subject than one diagnOSing syphilis. CurrenUy. it is i well suited to producing t.h ree-di­ that is not. used for the differential diagnosis of mensional geometrical patterns. Suppose you put a dark filter over optic neuritis occuring in multiple and Simplified versions are some­ one eye and look binocularly at a sclerosis. In this case, the dIffer­ times found In toys and novelties. pendulum swinging in a plane per­ en ce in response time of the two pendicular to your direction of vi­ eyes is actua l. so the patient s ees sion (Fig. 8.16 and the TRY IT). the effect without any filter. ·C. The Pulfrich Your brain gets the infon nation phenomenon about the location of the pen dulum from the covered eye a little later Another ph enom enon that pro­ than from the uncovered one. Th us. duces a stereoscopic effect by pre­ your two eyes locate the moving