Measuring Stereoscopic Perceived Depth

John Wattie

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 x-y 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.

Measuring the Stereoscopic Perceived Depth or Roundness of a sphere.

80mm base 1 meter to grid

Measure the perception of depth

Poke 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.
  1. Set the pencil in a lump of plasticine or in a moveable clamp.
  2. Get your eyes at a measured distance from the screen and call it
    V for Viewing distance
  3. Slide the pencil back and forth until the point is precisely on the closest part of the sphere.
  4. Measure from the screen to the pencil point.
  5. Call this distance
    T, for Thickness

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 disparity

The 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.
Uncrossed disparity
(divergent) is how objects deep to the screen surface look.

Positive disparity is another term for uncrossed or far, according the original work of Wheatstone, but occasional authors currently define positive disparity as near, or in front of the screen. For that reason it is best to avoid "positive" and "negative" disparity and stick to crossed or uncrossed.

Convert stereoscopic deviation of the nearest swirl into a measure of the stereo depth.

  1. Consider the nearest and biggest swirl.
  2. Measure the distance between the red and cyan components of the swirl. (It is probably easier to measure the distance apart of the two "black eyes" in the middle of the red and cyan spirals ).
  3. Call that distance
    D, for Disparity (or Deviation)

Correct your measurement for magnification

  1. Measure 10 squares horizontally.
  2. Each square is 10mm, so divide the measurement by 100 to get the magnification.
  3. Call the Magnification, M

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 inter-ocular distance

This 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."

  1. Look at a far distant object. So far away that your eye lines are parallel.
  2. Place the two points of the caliper, one in front of each eye.
  3. Adjust the separation, until the two points are both lined up on the distant object and they fuse stereoscopically.
  4. The caliper measurement is your
    Inter-ocular distance, i



True diameter of the sphere

Measurement 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.

  1. Determine the magnification (M) of the 10mm grid as measured on your own screen.
  2. Set up your pencil 150xM mm from the screen.
  3. Move back and forth until pencil and image of the sphere's nearest part coincide.
  4. Measure the distance of your eye to the screen to find viewing distance (V) for seeing perfect roundness.

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 impression

We 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 disparity

Consider 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)
M: Magnification of the grid on the screen surface.
V: Viewing distance from the screen
i: inter-ocular, or distance between the observer's eyes
D: Disparity of the point in question, as measured on the screen. (Distance between the red and cyan components).


Measurement example on the above sphere

You 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
i = 68
D = 13

T = 1.08 x 890 / (68 / 13 + 1)

T = 154mm

The main source of error is measuring the viewing distance, V, to the outer canthus of my eye, all by myself.

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 disparity

This 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