Latest update: March 19, 2012
Stereoscopic Roundness has been demonstrated as an illusion or perception, but this page describes how to measure it.
A sphere looks round if its perceived depth is equal to its diameter.
Diameter in the xy directions can be ~"measured"~ from the image, but how do you measure an illusion of depth in the z plane?
Actually, the sphere diameter can only be approximated by simple measurement from the grid. The diameter of the sphere is set above the grid, by half it's diameter. The sphere diameter is too big, by perspective, compared with the 1cm squares.
Measure the perception of depthPoke your finger at the sphere while viewing it as an anaglyph. The finger will go right inside the ball. Great fun! Now take a pencil and poke it at the sphere. The pencil point can be placed with great precision on the nearest swirl. It is tempting to make a mark on the swirl with the pencil, it is so realistic. Darn, I tried it and no mark! Keep the pencil fixed on the swirl. Then move your head back and forth. Moving back, the pencil will go deep to the swirl, while moving forward it will come out of the ball. You may be surprised just how sensitive this test is to tiny movements of your head. I was. Measure the stereoscopic perceived depth of the ball.
The Thickness measurement can be made with the sphere at different sizes. Just press ctrl + as many times as you desire and the internet browser program will magnify the sphere (ctrl will minify it again). So the measured depth you see is very sensitive to image size, but the perceived roundness is insensitive to size, as we found previously. Roundness of 1 means the diameter of the sphere and its stereoscopic depth look identical. But diameter is physical and can be measured with a ruler, while depth is a stereoscopic illusion. The diameter of the ball measures the same no matter how far your eyes are from the screen. The thickness illusion behaves very differently and is changed critically as the distance of your eyes from the screen changes. Notice the nearest swirl on the ball is bigger than the swirl sitting back on the grid. Not surprising of course, since near objects being bigger than far in a picture is called perspective. The most distant swirl is grey and it is not split into a red and cyan pair. Like the grid, it is on the screen surface, which has been set as the stereo window level. Separation into cyan and red components is the stereoscopic deviation or disparity, which is set at zero on the screen surface. The bigger the deviation, the further an object is from the screen surface. Crossed and uncrossed disparityThe anaglyph glasses have red over the left eye and cyan over the right eye. The big red swirl is on the right of the green swirl, the opposite way to how our eyes are arranged. This is called crossed disparity (convergent) and indicates that the object is in front of the screen.
Convert stereoscopic deviation of the nearest swirl into a measure of the stereo depth.
Correct your measurement for magnification
Realise this magnification only applies to the grid, on the screen surface. Magnification of the ball surface nearest to us is bigger. That does not matter as the subsequent formula uses disparity measurement on the screen.
Measure your interocular distanceThis is best done with a vernier caliper. A ruler gives a rough idea, using right and left fingers on the ruler as the points of a "caliper."

True diameter of the sphereMeasurement of the original sphere's diameter was 150mm. (Done by putting the sphere on a table, setting a box on each side then measuring the distance between the boxes. ) Measuring a roundness of 1 (perfect roundness)When I place the original sphere in contact with the screen, I can make the stereo image and the real sphere look the same, by moving backwards and forwards until they coincide. Being able to see through the sphere becomes a real bonus. You can do the same, despite not having the sphere, by knowing it is 150mm diameter.
In this way you measure the roundness, rather than estimating it stereoscopically. I found my estimation turned out pretty good. Movement parallax confuses stereoscopic depth impressionWe are very used to moving our heads sideways to check if something lies in front of another. That causes severe confusion in this experiment because the pencil point seems to stay still while the stereoscopic image of the sphere moves in the direction you swing your head. Movement of the front of the sphere is magnified and looks much bigger than the back of the sphere, on the screen surface. (Shear distortion). Movement parallax gives the strong illusion that the sphere is behind the pencil point even when stereoscopically the pencil is inside the sphere. The illusion is so intense and confusing, it can even be nauseating. Care must be taken not to move your head in any direction but forward and back when measuring stereoscopic depth. Formula for perceived depth by crossed disparityConsider the sphere shown as a stereoscopic anaglyph above. The sphere sits on the screen surface, where the grid lies and projects out towards the observer. The projecting part is seen by crossed disparity. By the geometry of similar triangles, the stereoscopic perceived depth, or Perceived Thickness of the sphere, is given by: T = M V / ( i / D + 1) T: Perceived Thickness (diameter in the z plane)
Measurement example on the above sphereYou will probably get different numbers on your computer screen, but on mine (Samsung 22 inch 1280x800pixels, 474mm wide) the result when rechecked January 2012 was: V = 890 T = 1.08 x 890 / (68 / 13 + 1) T = 154mm
I have been surprised by the sensitivity of stereo depth measurements, but should not be, since photogrammetry depends on it. Perceived depth formula for uncrossed disparityThis is the version of the formula given by Helmholtz in 1867, (according to Jones et al) but with my symbols of course: T = V / ( i / D  1 ) The Helmholtz/Jones formula is virtually the same as my derivation for crossed disparity, except +1 becomes 1. For the purposes of measuring the sphere, rather than just admiring its shape, I added Magnification to the Jones formula: T = M V / ( i / D  1 ) With uncrossed disparity, it is impossible to measure the perceived thickness directly, because you cannot get inside the screen! The only way to discover what you perceive quantitatively, which is an illusion deep in behind the screen surface, is from this equation. So I cannot check the Jones uncrossed disparity formula by measurement.

How these formulae explain stereoscopic stretch and squeeze