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Stereo Base Formulae

By John Wattie

WARNING this section of the second edition is not finished and many links to individual equations do not work yet.
Links that do work are often to Edition 1.

Click on each equation number to find out more

* means multiply. It over comes the problem of confusing x with a variable called x.

00 Roundness

R = [Stereo Base Used] * V / n i

01 Brewster formula

B = n / 30

The widely quoted "one over 30 rule."


02 Pin-hole stereo camera formula with infinity in the picture

B = nP / F

Stereoscopic Parallax, Infinity in the picture

A simple formula, where m is infinity.


03 Pin-hole with infinity: for easy use in a pocket calculator

B = 1200n / F

(natural units: B in mm, n in meters, F in mm)

Actually the same as (2), but in a form easily solved with a pocket calculator.


04 Pinhole stereo camera formula, with infinity not in the picture

B = P / (F * ( 1/n - 1/m ) )

This formula works very well for lens cameras too. It is the Bercovitz formula, without correction for changing the focus, which is an impossibility with a pinhole.

Parallax formula, infinity in the picture


05 Bercovitz formula for lens cameras

B = P * ( mn / (m-n) ) * ( 1/F - ( m+n ) / 2mn )

P is usually 1.2 in 35mm equivalent photography and 1.2 is equivalent to a 1/30 rule or 2 degree rule, when using a standard lens.

Simplified Bercovitz formula:

when infinity is in the picture and the camera is focused at the hyperfocal distance:

B = n P / ( F - 1/2 )

 

Davis modification of the Bercovitz formula

If m < 2n

Then m = 2n

This was an arbitrary approximation by Davis and in my view is better replaced by the smaller of:
the base for perfect roundness, or the base for MAD.


Summary.

Concepts on this page are often found difficult to understand.

They are expanded and hopefully simplified when the number for each equation is clicked on the left.

The mathematics is simplified to the geometry of right angle similar triangles, avoiding matrices and analytical geometry, which real mathematicians use when discussing stereo. (The author makes no claims to mathematical ability beyond 5th form geometry from 60 years ago).

1) Viewing Distance plus Stereo Base determines ROUNDNESS

First decide how far away, in real or optical terms, your stereoscopic images are to be seen from.

Roundness or Stereoscopic Perceived Depth

is set by:

  1. Viewing Distance V
  2. Stereo Base B
  3. Distance to nearest object, n
  4. Interocular distance, i

Go here
to find the correct stereo base for undistorted roundness:

B = n * i / V

If you get roundness wrong, the 3D pictures can look too flat (cardboard) or too thick (hyper-plastic) instead of the desired orthoplastic (depth in true proportion to width). (Not everybody wants orthoplastic images: "photographic art" often changes the criteria).

Next worry about MAD

2) Stereoscopic Base during photography also determines MAD

Maximum Acceptable Disparity (MAD).

MAD varies with the person viewing the stereoscopic image. It is commonly set by the disparity (deviation) which is easily seen on a projected image. In professional practice, MAD is set by people who have poor stereoscopic perception for 3D images on a flat surface. If they are not catered for, they end up with eye strain and will never look at your 3D pictures again.

Disparity increases with magnification of the image.

Disparity is set by Parallax (P)

P = parallax, academically an angle, but generally expressed in mm, as the linear separation between homologous points on the two camera stereo receptors. (Commonly a maximum of 2 degrees, which becomes 1.2mm on a 35mm slide. It stays 1.2mm no matter what focal length lens is used, so the infinity to nearest object deviation stays constant in the viewing system or projected image. P only changes if the viewing distance is different from the standard 2 meters, (large screen movies), or experts are being catered for, who can handle bigger deviations).

B = stereo base (interaxial)

F = camera focal length (mm)

n = nearest distance imaged

m = maximum distance

stereo infinity: so far away that no parallax can be detected.

 


 

The best compromise stereo base is the smaller of the:

  • base for perfect roundness or
  • the base for MAD
    (Maximum Acceptable Deviation. )

 

06 Frank Di Marzio formulae

B = P ( mn / (m-n )) * ( 1/F - 1/H )

H = 2mn / (m+n)

(H is the distance focused on for maximum depth of field,
or the hyperfocal distance.)

Di Marzio simple calculator formula:

If P is 1.2, the Di Marzio Equation simplifies hugely to:

B = H / 60

Unfortunately, modern digital camera lenses do not have H marked on them

Di Marzio stereo close-up formula:

B = n/15

or: n = 15B

The Di Marzio macro formula presumes the furthest object is only twice as far away as the nearest: m < 2n. Stereo macro photography usually follows this requirement.

06cDi Marzio magnification formula:

B = F * (1+ 1/M) / 15

where M is magnification in the camera. A useful formula, worked out by Frank at my request, where macro lenses have magnification marked on them. It only works if m < 2n, which is usual in macroscopic photography.


 

H = Hyperfocal distance.

R = Magnification in the camera. (Reproduction ratio)

Mn: Magnification at nearest point
Mm: Magnification at maximum distance.

If E is the lens extension beyond the infinity position:
R = E/F

Frame magnification is:

(final image frame size)
(camera frame size

Magnification of the final image is:

R x (Frame magnification)

The Di Marzio equations depend on H, the hyperfocal distance.

Unfortunately, digital cameras use zoom lenses which do not have the hyperfocal ranges marked on them.

Digital cameras using full size sensors and fitted with "old" lenses can still use Di Marzio formulas.

07 PePax principle

How to handle a change in focal length. To maintain roundness, the base should increase proportional to the increased focal length. However that often causes a violation of Maximum Acceptable Deviation (MAD) unless the maximum distance (m) is reduced...

08 Jones et al

B = 2Z' tan a/2 dn N' / (W(Z' -N') + dn N' )

 

Follow the link to discover what the Jones equation is all about. So complex, it is best solved in a spreadsheet, which Jones et al provide.


 

 

A practical example:

Stream in a New Zealand Bush, running through Waitomo Limestone "pancakes".

Bush scenes look chaotic, with many superimposed pieces of small foliage.
You truly cannot see the forest for the trees, until the confusion is resolved by viewing it stereoscopically.

Anaglyph needs red-cyan filters to show up in 3DUse anaglyph glasses to see in 3D.

Move back from the computer screen to get more realistic depth on this 1024 x 807 pixel anaglyph.

(or use a digital projector or large TV on the large image)

Difficulty viewing index: D = 1, which is a conservative stereo base choice.

 

Waitomo Stream Anaglyph long base-line or hyperstereo

 

Anaglyph stereo by John Wattie
Link to full colour stereo pair.

Stereoscopic images often have 3D depth in the front, but perceived depth quickly fades (by an inverse square law) to flat in the distance. Many stereo photographs have a boring rock in the front, in excellent 3D, but the interesting distant objects are flat.
To increase the stereoscopic depth range, from the front of the anaglyph and further into the far distance:

        1. Move back, so foreground objects in the final image are actually a fair way off. (Increase n) This allows the stereo base to be increased, improving perceived depth (roundness), as the formulae show.
        2. Zoom towards telephoto, if the formulae allow that, because magnification increases stereoscopic disparity. (Increase F, cautiously.)
        3. Exclude infinity from the picture, with a hill or building, or in this case trees, because the mathematics only allows a small stereo base when far distances are included. (keep m small)
        4. Looking down from a high point, at around 45 degrees, is a good stratagem to keep infinity out of the view. (Not possible in this picture.) Aiming the camera at an angle to the earth surface from a high point also reduces the distance between near and far points (m-n is smaller), allowing a bigger stereo base.
        5. If the stereo base is fixed by the camera, the ability to control 3D space is sadly limited. Learn to do a long cha-cha! (Hyperstereoscopy). (Get B as big as MAD allows).
          The Fuji W3 allows cha-cha stereo using the A3D setting. If the left image is taken first, the result can be seen in 3D on the lenticular viewing screen. This ability to see the result of hyperstereoscopy in the camera is a huge advantage, allowing you to do it again if you have chosen a wrong stereo base.
        6. "Cheat" and use double rigging or depth maps.

    Just how long the base should be is computed from one of the above formulae, so algebra really is useful in 3D photography!

    More information on how to apply the mathematics comes later.


     

HDR

Forest photographs have so much confusing detail that stereoscopy is far better than flat pictures for carrying all the information.

The above image uses HDR (High Dynamic Range), which is a help in forest photography, where there is a big difference in brightness between the water highlights and sky, versus the shadows. Three pictures were taken in one second on "burst mode," with the exposure varying by +/- one stop from the mid value. Six pictures resulted (three for each channel) and they were fused in an HDR program (Photomatix) to two images (left and right). These became the stereo pair, later fused into a single anaglyph.

I have been asked "If this is a fusion of 6 photographs, how can I see detail in the rapids, when separate, fused images should have blurred it out?"

The white parts of the image, the rapids in the river, were derived from the under-exposed pair of images. The correct and over-exposed images had the light parts removed by the software, and so they made no contribution to the high-lights. The Sony V3 cameras fired synchronously, as arranged by the LANC Shepherd software trigger, so the bright rapids were in excellent sync. The bright parts show better in the red image (left eye), which has been set to show monochrome details to advantage, since red by itself makes no difference to colour discrimination in an anaglyph.

 

 

 

 

 


 

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