Stereograms Portraying Ambiguously Perceivable SurfacesAuthor(s): B. Julesz and S. C. JohnsonSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 61, No. 2 (Oct. 15, 1968), pp. 437-441Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/59094 .

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STEREOGRAMS PORTRAYING AMBIGUOUSLY PERCEIVABLE, SURFACES

BY B. JULESZ AND S. C. JOHNSON

BELL TELEPHONE LABORATORIES, INCORPORATED, MURRAY HILL, NEW JERSEY

Communicated by J. R. Pierce, July 11, 1968

Ambiguously perceivable stimuli have an important role in the study of various perceptual phenomena. A well-known class of ambiguous stimuli can be ob- served in binocular depth perception. Gxrid-like structures containing vertical bars of constant periodicity (such as wallpaper, old-fashioned radiators) can be fused at multiple depth levels since the binocular disparity can be any integer multiple of the horizontal periodicity. vSuch periodic random-dot stereograms have been successfully used in recent studies,1 2 yet are limited in their scope, since only parallel planar surfaces can be portrayed.

This report discusses a general algorithm which generates a single stereogram portraying two (or more) specified surfacts. That such stereograms can exist is based on the fact that certain areas are seen by one eye only and thus can be freely selected for one surface. Also, segments in which the two (or more) sur- faces coincide add to the degrees of freedom, since these surfaces can be covered by any random texture at will. In general, there is no restriction on the surfaces to be portrayed simultaneously. If the number of surfaces is restricted to two and the resolution is fine (the number of samples being large), the freedom in choosing the texture elements is sufficient that the formation of monocularly perceivable short periodicities can be prevented.

The algorithm is an extension of the technique of random-dot stereograms by Julesz.3' 4 The following simple example will -give an insight into the workings of the general algorithm. The two surfaces) to be portrayed by the stereogram, A and B, are given in the x-z plane in Figure 1, and for simplicity are selected as cylindrical; i.e., z = fA(x), and z = fB(x), independent of y.

The right image of the stereogram is selected as the perpendicular projection in Figure 1. We may represent the texture of each position in the right image by a sequence of textural elements ti, t2, ... as indicated in Figure 1. We may think of the ti as the "colors" of the blocks; we have "colored" the figures so that the right stereogram is simply tl, t2, t3, t4 ... (see top row of Fig. 2).

The left image is supposed projected at 45?. Thus a point of height z has a horizontal displacement of z between the left and right images. Figure 2 gives the appearances of A and B in the left image. Notice that some areas are un- covered; they are denoted by the symbol *, whereas other areas are "in the shadow" and are not seen by the left eye (e.g., t3 of B).

In stereograms where only one surface at a time is represented, the ti may be chosen at random; here, we must satisfy a number of constraints in order to get a coherent left image; the constrainits may be read off immediately from Figure 2. We see that t3 = 13, t4 = t7, t3 = t1o, etc. Notice that the *'s add to the freedom in selecting the ti, since they impose no constraints. In addition, those places where the two figures coincide produce no effective constraints on the ti.

437

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438 PSYCHOLOGY: JULESZ AND JOHNSON PROC. N. A. S.

RIGHT IMAGE

LFT tlSI tl 16 IMAGE t,13 tK

_ t71i - SURF;ACE A

__t5 t6'

_, .t REFERENCE PLANE

t3i t4i t5

. 7- t-8SURFACE B

FiG. 1.-A simple example of two surfaces (cross sectioni in the x-z plane) to be portr ayed.

In the actual production of these pictures, the computer takes the mathemat- ical description of the two functions and, beginning with a random choice for the t4, modifies the t to satisfy each of the constrainits successively until the entire picture is determined. The left and right projections of both surfaces (A + B) are shown in the bottom row of Figure 2. All the constraints having been satisfied, it is merely a matter of bookkeeping to generate the images on microfilm. For cylin- drical surfaces, each row has the same constraints, yet within this limitation the brightness values of each line can be chosen differently. The number of inde- pendent choices possible on each line before all colors are determined is called the degrees of freedom of the figure.

In the forthcoming examples, we restrict ourselves to two cylindrical surfaces to be portrayed. In these ambiguous stereograms only one surface can be seen at a time, a fact that can be exploited for theoretical and practical applications. Obviously, the study of how such ambiguous stereograms are perceived and how the perceptual reversal may be influenced can be of considerable scientific interest. On the practical side, it is now possible to portray hidden surfaces of objects together with the visible ones. For instance, the reader could irnterpret the two surfaces of the stereograms shown in Figures 3 and 4 as being the front view of ani object and the rear view, respectively. As the perception of the two surfaces may be alternated at will, one can obtain an entire 360? impression of the object.

We have to restrict the demonstrationi to two characteristic examples. Sur- faces of greater complexity will be published elsewhere. The demonstrated stereograms have a 1000 X 1000-dot resolution. Each dot (picture element) can take three different brightness values. This has been achieved by using

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VoL. 61, 1968 PSYCHOLOGY: JULESZ AND JOHNSON 439

RIGHT IMAGE PROJECTIONS

tj t2 t53 t4 ts5 tk6 tt7 to t j tiol til ''CONSTRAIN TS

AFTER .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t t__ . tm*Sr:l. . _ _

t, t2 t3I | t3It t4 t8 t t3I CSTRAINTS

LEFT IMAGE PROJECTIONS

t / t2 / t3 t4 * t5 t6 : tI to '' . ''SURFACE A

r-i -- t4 t5j t7lO tIO Itll3 It lt-1I ItI,I ---SURFACE B

f t2 jt4 jt3 |t4 t8 I t3 t6 t4 tS t7I. . . SURFACE

FIG. 2.-The left and right projections of the surfaces given in Fig. I prior to and after the constraints.

three characters (blank, light-period sign, heavy-period sign) of the General Dynamics (Stromberg-Carlson) 4060 microfilm printer. The probability of us- ing the light- and heavy-period signs has been selected to be 0. 1; the probability for the blank has been selected to be 0.9. Thus, the average number of dots in the stereograms of Figures 3 and 4 is 101. This is within the resolution capabil- ities of the printing process.

Figure 3 shows a stereogram portraying a single pyramid- (wedge) behind the surround and two pyramids above the surround. The maximum depth is =t60 picture elements and to facilitate perceptual reversal, a 150-picture-element- wide margin in the upper and lower portions of the images contains the unambigu- ous surfaces A and B, respectively. A 50-picture-element-wide gap at zero depth level separates the unambiguous surfaces from the ambiguous organization. When the upper portion of the fused stereogram is looked at, the unambiguous ascending pyramid usually carries with it the percept of the ambiguous pyramid, whereas when the lower portion is looked at, the ambiguous organization reverses. Of course, these unambiguous margins serve onily as an aid, and the ambiguous organization can be reversed at will when the margins are covered.

In Figure 4, the upper margin contains the unambiguous surface A, which is a plane with 40-picture-element disparity, whereas the lower margin contains the unambiguous surface B, which portrays a descending pyramrid (wedge) having a maximum disparity of -60 picture elements. In this example, there is no gap between the ambiguous and unambiguous surfaces.- This examnple is particu- larly interesting, since organization A yields a strong percept. Indeed, our everyday experience would suggest that when each dot of a front plane is seen by both eyes without any hidden areas being present, then this plane should be the only possible percept. Yet, as Figure 4 demonstrates, it is relatively easy to ob- tain the other organization, too.5

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440 PSYCHOLOGY: JULESZ AND JOHNSON PROC. N. A. S.

FIG. 3. Ambiguous stereogram with unambiguous margins in the upper and lower areas. Surface A is a pyramid (wedge) behind the surround, while surface B is a double pyramid above the surround. Either one of the two surfaces can be perceived at will when stereoscopically viewed, yet reversal can be aided by viewing the upper or lower margins, respectively.

FIG. 4.-Ambiguous stereogram with unambiguous margins in the upper and lower areas. Surface A is a horizontal plane above the surround, while surface B is a pyramid (wedge) behind the surround. For viewing instruction, read the text of Fig. 1.

We note that in Figure 3, 359 degrees of freedom (out of 1000 samples) were obtained; in Figure 4, only 99 degrees of freedom. As shown, even this low degree of freedom yields adequate image quality without the excessive formation of perceivable periodic stripes.

When an attempt is made to portray more than two surfaces, the degrees of freedom rapidly diminish. On the other hand, some of the stereograms with two surfaces yield some additional percepts. For instance, in Figure 4, after the plane above the surround is perceived, sometimes the percept of an ascending pyramid above the plane can also be obtained.

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VOL. 61, 1968 PSYCHOLOGY: JULESZ AND JOHNSON 441

These ambiguous stereograms can be used to gain insight into how the visual system resolves ambiguities. For instance, it is possible to study how visual or verbal instructions influence the reversal, or whether the familiarity or simplicity of the surface shape affects perception. COne of the ambiguous organizations can be also biased by a certain per cent of unambiguous dots in order to counteract natural bias, a technique successfully used in reference 1.

The purpose of this short report, however, is not to dwell on particular psy- chological applications, but rather to draw attention to a new technique of image portrayal in which the quest of cubist art is achieved; instead of the many sides of objects being placed side by side in confusing ways, they are combined in a holistic organization in which the various percepts can be independently evoked. Thus, the multiple-surface stereogram shows some similarity to three-dimensional holograms. However, in the case of holograms the viewer has to move around the object in order to inspect it from various viewpoints, whereas for multiple- surface stereograms the viewer can sit still; it is his mind that panders around the object.

Summary.-An extension of the technique of random-dot stereograms that gen- erates the same stereogram for two (or more) selected surfaces has been devised. Prior to this development, the only known ambiguous stereograms have been the projections of periodic grid structures (i.e., wallpaper) that could be perceived at various parallel depth planes. The new algorithm, however, can portray two (or more) selected surfaces of general shapes with adequate degrees of freedom in coloring the random-dot texture. Besides its obvious use in perceptual studies, such ambiguously perceivable stereograins permit the portrayal of both the visible and the hidden surfaces of objects, since the perceptual organization can be altered at will.

I Julesz, B., "Binocular depth perception without familiarity cues," Science, 145, 356-362 (1964).

2 Julesz, B., "Texture and visual perception," Sci. American, 212, 38-48 (1965). 3Julesz, B., "Binocular depth perception of computer-generated patterns," Bell System

Tech. J., 39, 1125-1162 (1960). 4Julesz, B., and J. E. Miller, "Automatic st.ereoscopic presentation of functions of two

variables," Bell System Tech. J., 41, 663-676 (1962). 5 We wish to thank Miss Roseanne Hesse for 'writing a microfilm plotting routine for por-

traying our computer programs.

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