Autostereogram refers to a single-image form of stereograms that can be free-viewed to achieve a stereoscopic, three-dimensional (3D), depth effect without a stereoscope or any other artificial aids to binocular fusion (see Figure 1). The principle of the autostereogram is a horizontally repetitive image in which the repeated pattern is modulated in such a way that the image can be viewed with an abnormal convergence or divergence angle to generate the visual impression of a stereoscopic depth image within the space of the pattern. When viewing with normal convergence of the eyes on the physical plane, the image appears only as a flat repeating pattern. When the eyes either converge or diverge at the distance of the pattern repeat, small differences between adjacent pattern cycles provide binocular disparities that are interpreted by the viewer as differences in depth. The disparity structure may be designed to correspond to the depth map of any desired three-dimensional scene, which is perceived when the eyes are held at the appropriate convergence angle. In order to do so, however, the viewer must overcome the natural tendency of the eyes to focus at the convergence distance, and enable the eyes to refocus at the plane of the image.
The history of the autostereogram falls into five phases. The first phase was the early conceptual development, which began with Sir David Brewster in 1844. He observed, and worked out the principles of, the wallpaper illusion. To see this illusion, view a flat wall covered with a wallpaper showing a regular, horizontally repeating pattern then converge the eyes on a point in space closer than the wall. At a certain point, one part of the repeating pattern is imaged in central vision of the left eye and that part's neighbour, immediately to the left, is imaged in central vision of the right eye. Now the visual patterns viewed by the eyes are identical, except for the unpaired parts of the pattern in far peripheral vision, but the convergence angle is closer than the wall. One sees the wallpaper floating in space in front of its true position. Alternatively, one can relax the eyes to diverge them farther from the wall, so that the wallpaper appears to float at a distant location farther than its actual distance.
The second phase was the hand construction of coherent depth images based on systematic shifts in the horizontal position of adjacent elements of the repeated patterns of the wallpaper illusion. This approach seems to have been first used by Masuhiro Ito in 1970, was accurately developed in a modular 3D format by Edward Trent in 1972 (see Figure 2; Trent, 1972), and was patented in 1974 by Donald Peck (Peck, 1974).
The third phase was the development of the algorithm for mapping any arbitrary depth image into a random-dot autostereogram by Christopher Tyler in 1979 in conjunction with programmer Maureen Clarke. This technique, known as the single-image random-dot stereogram (SIRDS), was published by Tyler in a visual science textbook (Tyler, 1983), by David Stork in a computer science compendium (Stork, 1986), and by Dan Dyckman in a computer game magazine (Dyckman, 1990). These computer-generated 3D images used the computerized random-dot stereogram approach, developed by Bela Julesz in 1960, to camouflage the structure of the depth image being represented in the autostereogram when viewed directly. In 1968, Julesz extended this concept to camouflaging the dynamic structure of moving objects by generating stereoscopic movie sequences in which the random-dot pattern was randomly regenerated for every frame to avoid the spatial correlations over time that are formed by physical moving objects (Julesz & Payne, 1968).
The fourth phase of autostereogram development was the explicit, full-color images by Magic Eye and many other commercial companies. These usually required the eyes to overdiverge in order to get the 3D effect, which is rather harder for most people to achieve than overconvergence. In many cases, the designers of these autostereograms abandoned the idea of camouflaging the 2D information and instead organized the repeating patterns into multiple versions of recognisable images (Figure 3), often spatially related to the depth image. The requirements of the shifting horizontal positions in adjacent repeats to support the depth image stretch and degrade the recognisable images, especially where there are abrupt changes in disparity within the depth image. There are two basic techniques used with these full-color image base patterns. One is an elaboration of the Trent modular 3D concept, in which the depth structure repeats around the same repetition interval as the base pattern, with progressive variations to provide the requisite binocular disparities. This approach is limited to horizontally repeating 3D structures. The other approach employs the Tyler concept of a 3D structure that is independent of the pattern repeats, and just uses them as a base pattern on which to project the 3D depth map of arbitrary structure. In both cases, 3D images of full complexity and intersecting transparent overlay may be depicted.
The fifth phase of the technique is the generation of dynamic autostereograms, initiated in 1988 by Dan Dyckman with his movie 'Echodots' (Figure 4). The basic concept is to generate a series of autostereograms depicting successive views of a scene as it evolves in time and string them together to be viewed as a 3D movie without any specialized 3D equipment. However, just as the Julesz random dot stereogram concept masks the perception of the 3D object structure until the stereo image is binocularly fused, the true dynamic autostereogram should be masked to the movement of the underlying repeating pattern as the successive frames appear. To do so, the repeating random-dot pattern is replaced by a new, uncorrelated pattern in each frame. This technique creates the impression of a field of local swirling dots without coherent pattern motion, allowing the motion of the 3D autosterogram structures to be clearly perceived. Dyckman's movie included both in-plane and depth motion of the depicted 3D structures. This technique is known as the dynamic uncorrelated single-image random-dot stereogram (DUSIRDS)
Subsequent efforts to generate dynamic autostereograms are available in various formats, such as YouTube and animated GIFs, but very few of them employ the full DUSIRDS approach; they either use the same base repetition pattern allowing strong correlated motion cues, or provide additional uniform motion in an attempt to mask the correlated motion cues. This correlated motion can make it difficult to appreciate the 3D structure from the disparity cues. As with static autostereograms, dynamic autostereograms come in two flavors, either the Trent cellular repeating technique or the Tyler independent depth map technique may be employed. In general, it should be noted that human stereopsis is a fine-grain spatial processing system and requires very high-resolution images for optimal effect.
In addition to the random-dot camouflaged autostereogram and the non-camouflaged structured-image autostereogram, Tyler and Clarke (1979)  described four other types of free-fusion figure deriving from the same concept, although none have gained comparable recognition. Each requires computational tricks to achieve the novel conceptual structure. One is the continuous-depth autostereogram, in which the depth structure cycles around on itself to give an array of overlapping depth structures as the eyes converge on different parts of the image (Figure 5). Another is the wallpaper autostereogram, with a cyclic pattern that matches at the two ends to allow the infinite tiling of a plane, or wall (rather than the monocular confusion that occurs at the edges of standard autostereograms). A third is the two-way autostereogram, which contains a pair of independent cyclopean depth structures when viewed at rotations 90º apart (Figure 6). The final one is the autolustergram, which contains a cyclopean image depicted in dichoptic luster rather than defined binocular disparities. These latter give a ghostly, evanescent appearance rather than the smooth surfaces of the autostereograms.
Explanation of the Autostereogram
This entire section is excerpted with permission from Steven Pinker's How the Mind Works, with minor modifications. Steven Pinker retains copyright. The content in this section is not licensed under any Creative Commons license.
Stereo vision was not discovered until 1838, by Charles Wheatstone, a physicist and inventor after whom the “Wheatstone bridge” electrical circuit is named 1. ... Wheatstone proved that the mind turns trigonometry into consciousness when he designed the first fully three-dimensional picture, the stereogram. The idea is simple. Capture a scene using two of Leonardo's windows, or, more practically, two cameras, each positioned where one eye would be. Place the right picture in front of a person's right eye and the left picture in front of his left eye. If the brain assumes that the two eyes look at one three-dimensional world, with differences in the views coming from binocular parallax, it should be fooled by the pictures and combine them into a cyclopean image in which objects appear at different depths ...
The stereoscope became the television of the nineteenth century. Victorian-era families and friends spent cozy hours taking turns to view stereo photographs of Parisian boulevards, Egyptian pyramids, or Niagara Falls….
These technologies all force the viewer to don or peer through some kind of apparatus. The illusionist’s dream is a stereogram that can be seen with the naked eye—an autostereogram. The principle was discovered by David Brewster, the Scottish physicist who also studied polarized light and invented the kaleidoscope and the Victorian-era stereoscope. Brewster noticed that the repeating patterns on wallpaper can leap out in depth. Adjacent copies of the pattern, say a flower, can each lure one eye into fixating on it. That can happen because identical flowers are positioned at the same places on the two retinas, so the double image looks like a single image. In fact, like a misbuttoned shirt, a whole parade of double images can falsely mesh into a single image, except for the unpaired members at each end. The brain, seeing no double image, is prematurely satisfied that it has converged the eyes properly, and locks them into the false alignment. This leaves the eyes aimed at an imaginary point behind the wall, and the flowers seem to float in space at that distance. They also seem inflated, because the brain does its trigonometry and calculates how big the flower would have to be at that depth to project its current retinal image. ...
An easy way to experience the wallpaper effect is to stare at a tile wall a few inches away, too close to focus and converge on comfortably. (Many men rediscover the effect as they stand at a urinal.) The tiles in front of each eye easily fuse, creating the surreal impression of a very large tile wall a great distance away. The wall bows outward, and as the head moves from side to side the wall rocks in the opposite direction. Both would have to happen in the world if the wall were really at that distance while projecting the current retinal image. The brain creates those illusions in its headlong attempt to keep the geometry of the whole hallucination consistent.
Brewster also noticed that any irregularity in the spacing of a pair of copies makes them protrude or recess from the rest. Imagine that the flowers pierced by the lines of sight in the diagram are printed a bit closer to each other. The lines of sight are brought together and cross each other closer to the eyes. The images on the retina will splay out to the temples, and the brain sees the imaginary flower as being nearer. Similarly, if the flowers had been printed a bit farther apart, the lines of sight will cross farther away, and their retinal projections will crowd toward the nose. The brain hallucinates the ghost object at a slightly greater distance.
If you don’t already know how to fuse stereograms, try holding the book right up to your eyes. It is too close to focus; just let your eyes point straight ahead, seeing double. Slowly move the book away while keeping your eyes relaxed and “looking through” the book to an imaginary point beyond it. (Some people place a pane of glass or a transparency on top of the stereogram, so they can focus on the reflections of distant objects.) You should still be seeing double. The trick is to let one of the double images drift on top of the other, and then to keep them there as if they were magnets. Try to keep the images aligned. The superimposed shapes should gradually come into focus and pop in or out to different depths. As Tyler has noted, stereo vision is like love: if you’re not sure, you’re not experiencing it.
Some people have better luck holding a finger a few centimetres in front of the stereogram, focusing on the finger, and then removing it while keeping the eyes converged to that depth. With this technique, the false fusion comes from the eyes crossing so that the left eye sights a boat on the right while the right eye sights a boat on the left. Don’t worry about what your mother said; your eyes will not freeze into that position forever. Whether you can fuse stereograms with your eyes crossed too much or not enough probably depends on whether you are slightly cross-eyed or wall-eyed to begin with. …
The trick behind the wallpaper stereogram—identical drawings luring the eyes into mismatching their views—uncovers a fundamental problem the brain has to solve to see in stereo. Before it can measure the positions of a spot on the two retinas, the brain has to be sure that the spot on one retina came from the same mark in the world as the spot on the other retina. If the world had only one mark in it, it would be easy…. Add more marks, and the matching problems multiply. With three marks, there are six ghost matches; with ten marks, ninety; with a hundred marks, almost ten thousand.
This “correspondence problem” was noticed in the sixteenth century by the astronomer Johannes Kepler [see Figure 7], who thought about how stargazing eyes match up their thousands of white dots and how an object's position in space could be determined from its multiple projections. The wallpaper stereogram works by coaxing the brain to accept a plausible but false solution to the correspondence problem. Until recently, everyone thought that the brain solved the correspondence problem in everyday scenes by first recognizing the objects in each eye and then matching up images of the same object. Lemon in left eye goes with lemon in right eye, cherries in left eye go with cherries in right eye. Stereo vision, guided by the intelligence of the whole person, could head off the mismatches by only joining up points that came from the same kind of object. A typical scene may contain millions of dots, but it will contain far fewer lemons, maybe only one. So if the brain matched whole objects, there would be fewer ways for it to go wrong. But nature did not opt for that solution. The first hint came from another of Adelbert Ames’ wacky rooms.
This time the indefatigable Ames built an ordinary rectangular room but glued leaves on every inch of its floor, walls, and ceiling - the Leaf Room. When the room was viewed with one eye through a peephole, it looked like an amorphous sea of green. But when it was viewed with both eyes, it sprang into its correct three-dimensional shape. Ames had built a world that could be seen only by the mythical cyclopean eye, not by the left eye or the right eye alone. But how could the brain have matched up the two eyes’ views if it had to depend on recognizing and linking the objects in each one? The left eyes view was “leaf leaf leaf leaf leaf leaf leaf leaf.” The right eyes view was “leaf leaf leaf leaf leaf leaf leaf leaf.” The brain was faced with the hardest correspondence problem imaginable. Nonetheless it effortlessly coupled the views and conjured up a cyclopean vision.
The demonstration is not airtight. What if the edges and corners of the room are not perfectly masked by the leaves? Perhaps each eye had a rough inkling of the room's shape, and when the brain fused the two images it became more confident that the inklings were accurate. The airtight proof that the brain can solve the correspondence problem without recognizing objects came from an ingenious early use of computer graphics by the psychologist Bela Julesz. Before he fled Hungary for the United States in 1956, Julesz was a radar engineer with an interest in aerial reconnaissance. Spying from the air uses a clever trick: stereo views penetrate camouflage…. Since a camouflaged object, by definition, is near-invisible in a single view, we have another example of the cyclopean eye seeing what neither real eye can see.
The proof had to come from perfect camouflage, and here Julesz went to the computer. For the left eyes view, he had the computer make a square covered with random dots, like television snow. Julesz then had the computer make a copy for the right eye, but with one twist: he shifted a patch of dots a bit over to the left, and inserted a new stripe of random dots into the gap at the right so the shifted patch would be perfectly camouflaged. Each picture on its own looked like pepper. But when put in the stereoscope, the patch levitated into the air. Many authorities on stereo vision at the time refused to believe it because the correspondence problem the brain had to solve was just too hard. They suspected that Julesz had somehow left little cut marks behind in one of the pictures. But of course the computer did no such thing. Anyone who sees a random-dot stereogram is immediately convinced.
All it took for Julesz' collaborator, Christopher Tyler, to invent the magic-eye autostereogram was to combine the wallpaper autostereogram with the random-dot stereogram. ... The first commercial autostereograms used coloured squiggles and the Japanese ones use flowers, ocean waves, and, taking a leaf out of Ames’ book, leaves. Thanks to the computer, the 3D shapes that are depicted don’t have to be flat cutouts like in a diorama. By reading in the three-dimensional coordinates of the points on a surface, the computer can shift every dot by a slightly different amount to sculpt the solid shape in cyclopean space, rather than shifting the entire patch rigidly. As long as the computer has sufficient resolution, smooth, bulbous shapes materialize, looking as if they are shrink-wrapped in leaves or flowers.
Why did natural selection equip us with true cyclopean vision—an ability to see shapes in stereo that neither eye can see in mono—rather than with a simpler stereo system that would match up the lemons and cherries that are seeable by each eye? Tyler points out that our ancestors really did live in Ames’ leaf room. Primates evolved in trees and had to negotiate a network of branches masked by a veil of foliage. The price of failure was a long drop to the forest floor below. Building a stereo computer into these two-eyed creatures must have been irresistible to natural selection, but it could have worked only if the disparities were calculated over thousands of bits of visual texture. Single objects that allow unambiguous matches were just too few and far between.
Geometry of the Autostereogram
The autostereogram principle is illustrated in Figure 7 for the simple depth map of a step in depth at the visual midline. The image in the autostereogram plane is indicated by the horizontal series of black dots, representing the repeat interval of the repetitive pattern. In this example, the viewer’s right eye is fixated on the central dot, but the left eye is fixated on the adjacent repeat to the right. This gives a convergence posture of the eyes that yields the perception of false images in front of the autostereogram plane (all the blue dots), in this case a flat plane to the left (left, solid blue line) and a nearer, flat plane to the right (right solid blue line). The dotted lines show all the lines of sight to the black dots, but they also provide a grid of intersections representing all possible locations in space where a black dot could be perceived, due to spurious correspondences between the images of all pairs of black dots. This are all the false matches Kepler realised must occur; the technique of finding them Kepler invented is now referred to as the Keplarian projection field.
For the full autostereogram pattern of dense texture between the repeats, there would be thousands of such intersections in the Keplarian projection field. If our eyes consisted of only the foveas (F), they could fixate at any one of these locations (red lines) and receive the optical information of the presence of a dot in space at that location. The grid of intersections in Figure 7 shows that the autostereogram generates the optical information for a dense array of possible depth image interpretations both in front of and behind the physical autostereogram plane (such as the blue and yellow dotted planes). The eyes are depicted as focusing at one of these locations away from the physical plane to emphasize that the eyes are free to move anywhere in space in the vicinity of the autostereogram plane, in order to view different aspects of this disparity structure.
Given this superfluity of valid depth locations, how is it that the brain yields perception of the depth structure of just one 3D image? Although there are aspects of this question that still remain to be resolved, the basic explanation may be understood in terms of the spatial projection of the binocular neurons in early visual cortex (Fischer & Poggio, 1977), which have 3D receptive fields or sensitive zones covering small regions of space (diagrammed as the green circles in Figure 8). These 3D sensitive zones lie close to the Vieth-Müller circle of zero disparities keyed to the fixation location (dashed black circle passing through the eyes and the fixation point), indicated by the intersection of the red lines of sight from the two foveas at the back of the eyes (F). Only targets falling within the scope of the 3D receptive field array (green circles) will be perceived as points in depth. This array thus selects out a narrow zone of 3D space in which depth structure may be perceived, and some form of neural interaction mechanism effectively suppresses all other possible targets (Tyler & Kontsevich, 2005). The neural processing system shown in Figure 8 should be envisaged as superimposed on the optical array shown the previous figure, selecting one dot at the foveal intersection, and excluding the original autostereogram plane (dashed line). The neural processing also incorporates a mechanism for interpolating a continuous surface through the empty spaces between the dots (Li et al., 2013).
At a far distance, the region of space encompassed by the zones around the Vieth-Müller circle is approximately planar, but at close viewing distances it has noticeable curvature, so it is advantageous to view the autostereograms on a curved surface. This surface is not, however, spherical around the fixation point but cylindrical, with a forward skew passing roughly through the feet (see Tyler, 1991), so it is easy to curve a flat printed autostereogram into the correct cylindrical format for optimal viewing of the depth image throughout the visual field.
The depth-to-change-in-repetition-interval conversion may be achieved for the coloration of any point by setting it to that of the point one repetition interval to the left (or right, depending on how the autostereogram is constructed). The position of that point is determined by the differential shift prescribed by the depth map at that point. (Strictly, it should be the shift prescribed by the depth map at a location in space half of the base repetition interval back from that point.) In this way the full extent of the autostereogram can be built up from the initial vertical strip of seed dots. The only other requirement is to avoid the pattern replication that arises from going to dense repetitions and then switching to sparse ones at an edge. Thus, when the current look-back exceeds the previous one, the overlap should be replaced with new random dots rather than those provided by the standard look-back. The resulting region of unpaired dots has an uncorrelated dot-cloud appearance that is an inevitable result of the binocular viewing of vertical edges, which will always have a region visible to one eye that is hidden from the other due to the binocular viewing geometry.
1792. Charles Wells describes the fusion of the images from the two eyes into a single perceptual image, perceived as being located in the “cyclopean eye” centered halfway between the two physical eyes.
1844. David Brewster rediscovers the wallpaper autostereogram and remarks that defects in the repeats form a disparity-based depth image.
1858. Frenchman Joseph D’Almeida develops the color anaglyph technique for stereo projection from a single image.
1893. William Friese-Greene demonstrates the first 3D movie with an anaglyphic 3D movie camera of his own invention.
1901. Spanish neuroanatomist Santiago Ramon y Cajal invents an early form of random-dot stereogram, as a tool to study the stereoscopic depth processing of binocular vision.
1939. Adelbert Ames III constructs the Leaf Room to camouflage the monocular depth cues to, in order to study the perception of the 3D shape of the room primarily from disparity cues.
1939. In Russia, Boris Kompaneysky publishes the first camouflaged random-blob dual-image stereogram (for the Russian Academy of Fine Arts).
1960. Bela Julesz develops the computerized random-dot dual-image stereogram for encoding any specified disparity profile, perfectly-camouflaged.
1962. Bela Julesz and Joan Miller develop the iterative algorithm for simultaneous encoding of two independent disparity surfaces in a dual-image stereogram. When one of the two surfaces is flat, the depth of the second surface may be seen by free-fusion on one eye’s image alone, forming a random-dot autostereogram. There is no record that Julesz was aware of this single-image capability for viewing the two-surface stereograms.
1968. Bela Julesz and Richard Payne introduce the dynamic stereo movie in random-dot dual-image format and show that stereoscopic features have distinct temporal processing characteristics.
1970. Masayuki Ito uses hand-construction methods to design a fully camouflaged random-dot autostereogram of protruding flat-plane squares.
1972. Edward Trent publishes complex line autostereogram images with continuous depth variation in the Bulletin of the Stereoscopic Society.
1972-3. Roger Ferrogallo develops stereoscopic painting based on a repeating tile structure that generates alternating depth images at the repeat interval.
1974. Roger Ferrogallo publishes an article on his stereoscopic painting technique in /*Leonardo*/.
1974. Alphonse Schilling invents the stereogram triplet for either crossed or uncrossed free-fusion and hand-paints flat-plane autostereograms.
1979. Donald Peck obtains US Patent 41235502 for the block-image autostereogram technique.
1979. Christopher Tyler and Maureen Clarke develop the direct, ‘look-back’ algorithm to convert any specified disparity profile into a random-dot autostereogram.
1983. First publication of Tyler and Clarke’s autostereograms, generated on an Apple II computer and a dot-matrix printer.
1986. David Stork uses a random-dot autostereogram that he devised with Chris Rocca as the frontispiece of ‘Seeing the Light’ by David Falk, Dieter Brill and David Stork, introducing the technique to the physics and computer science communities.
1988. Dan Dyckman generates the first dynamic autostereoogram movie 'Echodots'.
1991. Christopher Tyler and Clarke publish the random-dot autostereogram algorithm, together with two-way autostereograms, depth-cycling autostereograms, and autolustergrams. This paper describes the necessity for inserting uncorrelated dots when increasing the repetition period to correct for the accumulated 'pattern ringing' errors that are found in many later descriptions of the uncorrected algorithm.
1992. Geoffrey Slinker and Robert Burton publish an article on the autostereogram generation that includes the first description of the animation of random dot autostereograms.
1993. Publication of 'Magic Eye: A New Way of Looking at the World' by Tom Baccei, Cheri Smith and Bob Salitsky, the first of an extended series of Magic Eye books.
1994. Dan Dyckman publishes the first autostereogram game book: 'Hidden Dimensions: Use Your Deep Vision to Solve Mazes, Riddles, and Other Perplexing Puzzles'.
1994. Stuart Inglis, Harold Thimbleby and Ian Witten rediscover the Julesz and Miller (1962) algorithm for generating autostereograms without the accumulated 'pattern ringing' errors of the uncorrected look-back algorithm.
1994. Publication of ‘CG Stereogram’, edited by Seiji Horibuchi, with a variety of high-resolution, full color and image-based autostereograms and extended analysis of their historical development by Christopher Tyler and by Itsuo Sakane.
2001. Harry Potter Magic Eye autostereogram books available in Germany and many other countries.
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