Autostereogram

An autostereogram is a single-image stereogram (SIS), designed to trick human eyes and brains into seeing a three-dimensional scene in a two-dimensional image. The simplest type of autostereogram consists of horizontally repeating patterns and is known as a wallpaper autostereogram. The Magic Eye series of books features another type of autostereogram called a random dot autostereogram. In order to 'see' 3D images in these autostereograms, the brain must decouple focusing operations of the eyes from convergence.

History

In 1838, Charles Wheatstone, a British inventor, discovered stereo vision (binocular vision) which led him to construct a stereoscope based on a combination of prisms and mirrors to allow a person to see 3D images from two 2D pictures (stereograms).

Around 1849-1850, David Brewster, a Scottish scientist, improved the Wheatstone stereoscope by using lenses instead of mirrors, thus reducing the size of the contraption. Brewster also noticed that staring at repeated patterns in wallpapers could trick the brain into matching pairs of them and thus causing the brain to perceive a virtual plane behind the walls. This is the basis of single-image wallpaper stereograms.

In 1959, Dr. Bela Julesz, a psychologist originally from Hungary, discovered the random-dot stereogram while working at Bell Laboratories on recognizing camouflaged objects from aerial pictures taken by spy planes. Julesz used a computer to create a pair of similarly looking random-dot images which when viewed under a stereoscope, caused the brain to see 3D shapes.

In 1979, Dr. Christopher Tyler, a student of Julesz and a visual psychophysicist, combined the theories behind single-image wallpaper stereograms and random-dot stereograms to create the first random-dot autostereogram (also known as single-image random-dot stereogram) which allowed the brain to see 3D shapes from a single 2D image without the aid of optical equipment.

How they work

Simple wallpaper autostereogram

The human brain accomplishes stereo vision by a complex set of mechanisms which attempt to relate the two slightly different two-dimensional images seen by the two eyes. The brain tries to assemble a three-dimensional impression by matching each point (or set of points) in one eye's view with the equivalent point (or set of points) in the other eye's view. It therefore assesses the points' positions in the otherwise inscrutable z-axis (depth).

When the brain is presented with a series of very similar points, such as in a repeating pattern like you might see on wallpaper, it has difficulty matching the two eye's views accurately. By looking at a horizontally repeating pattern, but converging the two eyes at a point behind the pattern, it is possible to trick the brain into matching one element of the pattern, as seen by the left eye, with another (similar looking) element, beside the first, as seen by the right eye. This gives the illusion of a plane bearing the same pattern but located behind the real wall. The distance at which this plane lies behind the wall depends only on the spacing between identical elements.

Autostereograms use this dependence of depth on spacing to create three-dimensional images. If, over some area of the picture, the pattern is repeated at smaller distances, that area will appear closer than the background plane. If the distance of repeats is longer over some area, then that area will appear more distant (like a hole in the plane).

People who have never been able to 'see' an autostereogram find it hard to understand remarks such as, "the 3D image will just pop out of the background, after you stare at the picture long enough", or "the 3D objects will just emerge from the background." It helps to illustrate how 3D images 'emerge' from the background from a second viewer's perspective. If the virtual 3D objects reconstructed by the autostereogram viewer's brain were real objects, a second viewer observing the scene from the side would see these objects floating in the air above the background image.

The 3D effects in the example autostereogram are created by repeating the tiger rider icons every 140 pixels on the background plane, the shark rider icons every 130 pixels on the second plane, and the tiger icons every 120 pixels on the highest plane. The closer a set of icons are packed horizontally, the higher they are lifted from the background plane. This repeat distance is referred to as the depth or z-axis value of a particular pattern in the autostereogram. The depth value is also known as Z-buffer value.

The brain is capable of instantly matching hundreds of patterns repeated at different intervals in order to recreate correct depth information for each pattern. An autostereogram may contain some 50 tigers of varying size, repeated at different intervals against a complex, repeated background. Yet, despite the apparent chaotic arrangement of patterns, the brain is able to place every tiger icon at its proper depth.

Depth maps

Autostereograms where patterns in a particular row are repeated horizontally with the same spacing can be read either cross-eyed or wall-eyed. In such autostereograms, both types of reading will produce similar depth interpretation, with the exception that the cross-eyed reading create larger depth differences between planes.

However, icons in a row do not need to be arranged at identical intervals. An autostereogram with varying intervals between icons across a row presents these icons at different depth planes to the viewer. The depth for each icon is computed from the distance between it and its neighbor at the left. These types of autostereograms are designed to be read in only one way, either cross-eyed or wall-eyed. All autostereograms in this article are encoded for wall-eyed viewing, unless specifically marked otherwise. An autostereogram encoded for wall-eyed viewing will produce incoherent 3D patterns when viewed cross-eyed. Most MagicEye pictures are also designed for wall-eyed viewing.

The following wall-eyed autostereogram encodes 3 planes across the X-axis. The background plane is on the left side of the picture. The highest plane is shown on the right side of the picture. There is a narrow middle plane in the middle of the X-axis. Starting with a background plane where icons are spaced at 140 pixels, one can raise a particular icon by shifting it n number of pixels to the left. For instance, the middle plane is created by shifting an icon 10 pixels to the left, effectively creating a spacing consisting of 130 pixels. The brain does not rely on intelligible icons which represent objects or concepts. In this autostereogram, patterns become smaller and smaller down the x-axis, until they look like random dots. The brain is still able to matches these random dot patterns.

The distance relationship between any pixel and its counterpart in the equivalent pattern to the left can be expressed in a depth map. A depth map is simply a grayscale image which represents the distance between a pixel and its left counterpart using a grayscale value between black and white. By convention, the closer the distance is, the brighter the color becomes. In an 8-bit grayscale image, there are 256 possible values. Normally 0 represents black, while 255 represents white.

Using this convention, a grayscale depth map for the above autostereogram can be created with black, gray and white representing shifts of 0-pixel, 10-pixels and 20-pixels, respectively. Depth map is the key to creation of random-dot autostereograms.

Random dot autostereogram

A software program can take a depth map and an accompanying pattern image to produce an autostereogram. The program tiles the pattern image horizontally to cover an area whose size is identical to the depth map. Conceptually, at every pixel in the output image, the program looks up the grayscale value of the equivalent pixel in the depth map image, and uses this value to determine the amount of horizontal shift required for the pixel.

One way to accomplish this is to make the program scan every line in the output image pixel-by-pixel from left to right. It seeds the first series of pixels in a row from the pattern image. Then it consults the depth map to retrieve appropriate shift values for subsequent pixels. For every pixel, it subtracts the shift from the width of the pattern image to arrive at a repeat interval. It uses this repeat interval to look up the value of the pixel-counterpart to the left and uses the value as the pixel's own value.

Unlike the simple depth planes created by simple wallpaper autostereograms, subtle changes in spacing specified by the depth map can create the illusion of smooth gradients in distance. This is possible because the grayscale depth map allows individual pixels to be placed on one of the 2^n depth planes where n is the number of bits used by each pixel in the depth map. In practice, the total number of depth planes is determined by the number of pixels used for the width of the pattern image. Each grayscale value must be translated into pixel space in order to shift pixels in the final autostereogram. As a result, the number of depth planes must be smaller than the pattern width.

The fine-tuned gradient requires a pattern image more complex than a standard repeating-pattern wallpaper, so typically a pattern consisting of repeated random dots is used. Autostereograms of this form are known as single image random dot stereograms (SIRDS).

Smooth gradients can also be achieved with an intelligible pattern, assuming that the pattern is complex enough and does not have big, horizontal, monotonic patches. A big area painted with monotonic color without change in hue and brightness does not lend itself to pixel shifting, as the result of the horizontal shift is identical to the original patch. The following depth map of a shark with smooth gradient produces a perfectly readable autostereogram, even though the image contains small monotonic areas. The brain is able to recognize these small gaps and fill in the blanks. This type of autostereogram is still known as a random-dot autostereogram, even though intelligible patterns are used.

Animated Autostereograms

When a series of autostereograms are shown one after another, in the same way moving pictures are shown, the brain perceives an animated autostereogram. If all autostereograms in the animation are produced using the same background pattern, it is often possible to see faint outlines of parts of the moving 3D object in the 2D autostereogram image without wall-eyed viewing; the contantly shifting pixels of the moving object can be clearly distinguished from the static background plane. To eliminate this side effect, animated autostereograms often use shifting background in order to disguise the moving parts.

How to see them

Much advice exists about seeing the intended three-dimensional image in an autostereogram. While some people can simply see the 3D image in an autostereogram, others need to learn to train their eyes to decouple eye convergence from lens focusing.

How the brain perceives objects in 3D

The depth perception of the brain is based on many operations. For objects relatively close to the eyes (within 18-20 feet), binocular vision plays an important role in depth perception. Binocular vision allows the brain to create a single Cyclopean image and to attach a depth level to each point in the Cyclopean image.

The brain uses coordinate shift (also known as Parallax) of matched objects to identify depth of these objects. The depth level of each point in the combined image can be represented by a grayscale pixel. The closer a point appears to the brain, the brighter it is painted. Thus, the way the brain perceives depth using on Binocular vision can be captured by a depth map painted based on coordinate shift.

The eye operates like a photographic camera. It has an adjustable iris which can open (or close) to allow more (or less) light to enter the eye. As with any camera except pinhole cameras, it needs to focus light rays entering through the iris (aperture in a camera) so that they focus on a single point on the retina in order to produce a sharp image. The eye achieves this goal by adjusting a lens behind the cornea to deflect light.

Stereo-vision based on parallax allows the brain to calculate depths of objects relative to the point of convergence. It is the convergence angle that gives the brain the absolute reference depth value for the point of convergence from which absolute depths of all other objects can be inferred.

How to trick the brain into seeing 3D images

Normally, focusing operations of the eyes are coupled to the convergence operations. That is, when looking at a faraway object, the brain automatically flattens the lenses and rotates the two eyeballs for wall-eyed viewing. It is possible to train the brain to decouple these two operations. The decoupling would have no useful purposes for the everyday life of a person, as it would prevent the brain from interpreting objects in the world in a coherent matter. To see a man-made picture such as an autostereogram where patterns are repeated horizontally, however, decoupling of focusing from convergence is crucial.

By focusing the lenses on a nearby autostereogram where patterns are repeated and by converging the eyeballs at a distant point behind the autostereogram image, one can trick the brain into seeing 3D images. If the patterns received by the two eyes are similar enough, the brain will consider these two patterns a match and treat them as coming from the same imaginary object. This type of visualization is known as wall-eyed viewing, because the the eyeballs adopt a wall-eyed convergence on a distant plane, even though the autostereogram image is actually closer to the eyes. Because the two eyeballs converge on a plane farther away, the perceived location of the imaginary object is behind the autostereogram. The imaginary object also appears bigger than the patterns on the autostereogram due to foreshortening.

The following autostereogram shows 3 rows of repeated patterns. Each pattern is repeated at a different interval to place it on a different depth plane. While there are 6 dolphins in the autostereogram, the brain should see 7 apparent dolphins behind the plane of the autostereogram. The two non-repeating lines can be used to verify correct wall-eyed viewing. When the autostereogram is correctly decrypted by the brain using wall-eyed viewing, the brain should see two sets of flickering lines as depicted in the second picture, when one stares at the dolphin in the middle of the visual field (the 4th apparent dolphin).

Due to foreshortening, the difference in convergence needed to see repeated patterns on different planes causes the brain to attribute different sizes to patterns with identical 2D sizes. In the autostereogram of 3 rows of cubes, while all cubes have the same physical 2D dimensions, the ones on the top row appear bigger, because their perceived locations are farther away, compared to those of cubes on the second and third row.

Techniques for improved viewing experience

As with a photographic camera, it is easier to make the eye focus on an object, when there is intense ambient light. With intense lighting, the eye can close down the iris, yet allow enough light to hit the retina. The more the eye resembles a pinhole camera, the less it depends on focusing through the lens. In other words, the degree of decoupling between focusing and convergence needed to visualize an autostereogram is reduced. This places less strain on the brain. Therefore, it may be easier for a first-time autostereogram viewers to 'see' their first 3D images, if they attempt this feat under a bright lighting condition.

The key to seeing 3D images is the eye convergence. Thus it may help to concentrate on converging/diverging the two eyes to shift images that reach the two eyes, instead of trying to see a clear, focused image. The brain by instinct tries to adjust the lenses to produce clear, focused images. One needs to fight this urge, and instead alternate between converging and diverging the two eyes, in the process seeing 'double images' typically seen when one is drunk or poisoned. Eventually the brain will successfully match a pair of patterns reported by the two eyes and lock onto this particular degree of convergence. The brain will also adjust eye lenses to get a clear image of the matched pair. Once this is done, the images around the matched patterns quickly become clear as the brain matches additional patterns using roughly the same degree of convergence.

When one moves one's attention from a depth plane to another (for instance, from the top row to the second row in the cube autostereogram), the two eyes need to adjust their convergence to match the new repeating interval of patterns. If the level of change in convergence is too high during this shift, sometimes the brain can lose the hard-earned decoupling between focusing and convergence. For a first-time viewer, therefore, it may be easier to see the autostereogram, if the two eyes rehearse the convergence exercise on a autostereogram where the depth of patterns across a particular row remains constant.

In a random dot autostereogram, the 3D image is usually shown in the middle of the autostereogram against a background depth plane (see the shark autostereogram). It may help to establish proper convergence first by staring at either the top or the bottom of the autostereogram, where patterns are usually repeated at constant interval. Once the brain locks onto the background depth plane, it has a reference convergence degree from which it can then match patterns at different depth levels in the middle of the image.

One way to help the brain concentrate on divergence instead of focusing is to hold the picture in front of the face, with the nose touching the picture. With the picture so close to their eyes, most people cannot focus on the picture. The brain may give up trying to move eye muscles in order to get a clear picture. If one slowly pulls back the picture away from the face, while refraining from focusing or rotating eyes, at some point the brain will lock onto a pair of patterns when the distance between them match the current convergence degree of the two eyeballs.

Another way is to stare at an object behind the picture in an attempt to establish proper convergence, while keeping part of the eyesight fixed on the picture to convince the brain to focus on the picture. A modified method has the viewer stare at her reflection on the shinny surface of the picture, which the brain perceives as being located twice as far away as the picture itself. This may help persuade the brain to adopt a wall-eyed convergence while focusing on a nearby picture.

This article analyzes autostereograms designed for wall-eyed viewing. For crossed-eyed autostereograms, a different approach needs to be taken. The viewer may hold one finger between his eyes and move it slowly towards the picture, maintaining his focus on the finger at all times, until he is correctly focused on the spot between him and the picture that will allow him to view the illusion.

It is estimated that some 2% of normally sighted people cannot see the illusion in autostereograms.

Terminology

• Stereogram was originally used to describe a pair of 2D images used in stereoscope to present a 3D image to viewers. The term is now often used interchangeably with random-dot autostereogram.

• Random-dot stereogram (RDS) originally described a pair of 2D images showing random-dots which when viewed with a stereoscope produced a 3D image. The term is now often used interchangeably with random-dot autostereogram.

• Single-image stereogram (SIS) is a single 2D image which when viewed with proper eye convergence causes the brain to fuse different patterns perceived by the two eyes into a virtual 3D image without the aid of any optical equipment.

• Autostereogram is a synonym of single-image stereogram.

• Wallpaper autostereogram is a 2D image where patterns are repeated at various intervals to raise or lower each pattern's perceived 3D location in relation to a virtual background plane.

• Random-dot autostereogram is also known as single-image random-dot stereogram (SIRDS). This term also refers to autostereograms where intelligible patterns instead of random-dots are used.

Reference

• Julesz, B. (1971). Foundations of Cyclopean Perception. Chicago: The University of Chicago Press. ISBN 0226415279

• N. E. Thing Enterprises (1993). Magic Eye: A New Way of Looking at the World. Kansas City: Andrews and McMeel. ISBN 0836270061

• Andrew A. Kinsman (1993). Random Dot Stereograms. Rochester: Kinsman Physics. ISBN 0963014218

• Pinker, S. (1997). The Mind’s Eye. In How the Mind Works (pp. 211–233) ISBN 0393318486

• Julesz, B. and J.E. Miller. (1962). "Automatic stereoscopic presentation of functions of two variables". Bell System Technical Journal, 41: 663-676; March.

• Marr, D. and Poggio, T. (1976). "Cooperative computation of stereo disparity". Science, 194:283-287; October 15.

• C.W. Tyler and M.B. Clarke. (1990) "The Autostereogram". SPIE Stereoscopic Displays and Applications, 1258: 182-196

• Julesz, B. (1963). "Stereopsis and binocular 3d Stereogram rivalry of contours". Journal of Optical Society of America 53, 994-999.

• Julesz, B. (1964). "Binocular depth perception without familiarity cues". Science 145, 356-363.

External links

• Stereogram FAQ (http://www-ai.ijs.si/sirds/SirdsFaq.html)
• Magic Eye Inc. (http://www.magiceye.com/)
• ASCII stereogram generator (http://aa-project.sourceforge.net/aa3d/)
• Stereogram Examples (http://www.netaxs.com/~mhmyers/rds-ex.html)
• Animated Autostereogram of two tori (http://www.mpi-sb.mpg.de/~petz/images/anim-2.gif)
• Freeware Stereogram Generator (http://www.traxxdale.com)