Stereograms portraying ambiguously perceivable surfaces.

Ambiguously perceivable stimuli have an important role in the study of various perceptual phenomena. A well-known class of ambiguous stimuli can be observed in binocular depth perception. Grid-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. Such periodic random-dot stereograms have been successfully used in recent studies," 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 surfaces. 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) surfaces 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 t,, 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 ti, 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 uncovered; 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 constraints may be read off immediately from Figure 2. We see that t3 = t5, t4 = t7, ta = tio, 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.