STEREO.100

    Random-dot stereograms, invented by Bela Julesz (Bell Labs), consist of two
 squares filled with random pixels. The right square is identical to the left, 
except that some of the dots are shifted one or more pixels left or right. A 
viewer looks at the left square with one eye and the right with the other. When 
the two images 'fuse' the offset dots will appear to be higher or lower than 
the background. If the offset dots are arranged in a small square, as in 
STEREO.100, the viewer will see the square floating above the field, even 
though neither image alone contains any explicit image of a square.
    STEREO.100 contains three more 'grams, arranged in order of difficulty 
(of viewing). If you compile and run the program, it produces the floating 
square-- the easiest image for a beginner to 'fuse'. Be patient-- it 
takes about two minutes to compute the display arrays (which take up 16K of 
memory). Three other 'grams show a stepped pyramid, a descending ramp, and 
a hemispherical dome (as viewed cross-eyed: see below).
    Line 25 (GOSUB 1000) controls which 'gram is presented. To see the other 
three grams, change this line to read "GOSUB" 2000, 3000 or 4000. The 'gram is
 displayed until you press the BREAK key.

          VIEWING STEREOGRAMS

    The first (and simplest) way is with a stereogram viewer -- a small 
binocular magnifier. STEREO.100 produces the correct spacing between image pairs
 to let you use a viewer.
    You can also view the pairs either by crossing your eyes until the two 
images fuse or by 'wall-eyeing' them. To wall-eye a pair, center your head over 
the border between the two squares, and let your eyes drift out of focus to 
infinity. Images of each square will drift towards the center and eventually 
overlap. The trick is then to bring the two images back into sharp focus without
 losing the new convergence. Some people (like me) find it easier to cross eyes;
 others prefer to wall-eye. If you are nearsighted, you can refocus wall-eyed 
images much more easily by removing your glasses.
    Fusing images may take practice. It helps to first fuse a black edge, near a
 corner. Direct your attention inward, maintaining background fusion, until you
 start to see the 'subject' of the 'gram above or below the background

         GROWING YOUR OWN

    STEREO.100 uses two display arrays. "P" contains the values of each pixel's 
'height'. 0 is ground level, negative values will appear at lower levels (with
 cross-eyed viewing-- they'll look higher with the other two methods), and 
positive ones will be higher. The program zeroes P automatically-- the 
subroutines at lines 1000, 2000, 3000 and 4000 change some of the values in P.
    To create your own stereo image, decide which picture elements you want 
at what height, write a short subroutine to put the appropriate values into the 
appropriate locations of P, and change line 25 to GOSUB to your routine. 
Routine 1000 -- the floating square-- should give you the idea. Routine 3000 
introduces negative numbers (sunken images) as well as the use of random 
numbers to 'soften' the step between levels. Routine 4000 computes the points
 on an oblate spherical surface: Z = SQR(R^2-X^2-Y^2)/4. It looks more 
complicated because of various limit and symmetry tricks I used to reduce the 
computation time. It also includes a random smoothing factor.
    "Q" is computed automatically from "P" to tell it how to display 'edges'. A
 stereogram is inherently ambiguous at the point where the height changes, 
because the same information cannot exist in both images. Don't worry about 
Q, but you should understand that if you want to produce good 'grams:
    For example, if you plot a horizontal line 1 level up, there will 
be 1 pixel at each end whose height is 'undefined' so far as your eyes are 
concerned. For extended objects, your brain will ignore this and assign the 
pixel to either the upper or lower level, for continuity's sake. But, if 
you were to try and plot a vertical line one level up, it would disappear in the
 noise of undefined lines of pixels, to either side of it. If you plot a 
vertical bar four pixels wide, only the two center pixel lines will 
unambiguously appear at the desired height. A good rule of thumb is that 
your image should be 4N pixels wide for a change in height of N.
    Random number smoothing introduces more edges, so it also introduces more 
undefined pixels. Use it with caution-- if you "smooth" too much, you'll lose 
the image in noise and be unable to fuse the 'gram at all.
    Have fun!

               Alan Stein [71246,216]