The purpose of any photograph is to restore on the adapted backing the image seen by eye. This restoration entails the saving, then the visualization of this record.
This page explores how the stereoscopic effect is kept, or modified along the various stages of the restoration process.
Any camera is constituted of a lens with focal length F and imager device, film or digital sensor.
The dots which make up the 3D image are distant of the stereoscopic gap e which has been defined on the page "fundamentals".
Recall here that the superimposition of two identical pictures increase the overall sharpness of the image, regardless of any stereoscopic effect.
Image of the gap:
Remind us the relation of the gap e:
e = s x SR = a x (D+L) x SR
visual keeness  a = 0,0004 = 0,4 c
  What becomes on the imager the stereoscopic gap?
The lens forms the image of this gap on the imager, and the gap e takes the dimension u.
 The optical relations are:
u / T = e / (D+L)  
1 / F =  [1 / T ] + [1 / (D+L)]
F focal length of the lens
T focusing distance
D distance from the camera to field
L space between field and next background
 By combination of here opposite relations:
u = SR . a . T
u = SR . a . F . (D+L) / (D+L-F)
Let's (D+L) / (D+L-F)  = W
u = SR . a . F. W
When the focus is on infinity,
T = F and W tends towards 1
therefore,  u = SR . a . F
or also  u = SR . 0,4 . c . F
This formula is sufficient because the error is below 5% for any subject 1m far away, and less than 0,5% where further than 10m.
The general formula is only useful for close-up photography.
Role of sharpness:
The sharpness comes from focus and diaphragm, which together work towards the depth of field. It also comes from the stillness of the camera, or the shutter speed in relation to shot conditions.
It first seems obvious that sharpness alteration will cause a deterioration of the stereoscopic effect.
At first, I have begun to try to establish mathematically this deterioration. Not getting tangible results, I came back to the fundamentals of stereoscopy and to the physiological processus of the stereoscopic effect.
The making of stereoscopic pairs intentionally out of focus has give me the opportunity to verify that the stereoscopic effect remains. moreover, the image appears sharper when viewed by stereoscopy.
With binoculars, it is easy to notice that a small variation of sharpness doesn't seem to change the stereoscopic perception.
A good example of blur from move is shown by Gerhard Kuhn with the pair of photographs Abb132 page 89 of his book Stereo Fotografie und Raumbild Projektion. The stereoscopic effect is excellent, and the picture effectively looks sharper when 3D viewed.
This is to compare to the still observation view per view of  the images of a movie with moving fields. Backgrounds, or people are blurred on still image function, but look clear in their move when viewing normally the movie.
As conclusion, the stereoscopic effect is not deteriorated by a visible but not excessive unsharp image.
Role of focal lenght:
The dimension u of the image of the stereoscopic gap e is straight proportional to the focal length F.
Let's compare the variations of focal length on the same imager. The normal focal length which corresponds to the human visual field 50° is equal to the diagonal Z of the imager.
 On the same imager, the stereoscopic effect is more important when the focal length is greater than the diagonal of the imager, and weaker when the focal length is shorter than the diagonal.
The variation of stereoscopic effect is faintly perceptible for a change of focal length lesser than 20% of normal one,and well perceptible above 50%.
Therefore, the decrease with wide angle is often not seen with usual lenses, when with long-focus lenses the more effective 3D effect is quickly felt.
Role of circle of confusion:
The stereoscopic ratio SR' of the picture on the imager is the ratio of the size u of the stereoscopic gap e with the circle of confusion c' proper to the imager.
SR' = u / c'
SR' = SR 0,4.c.F.[(D+L-F) / (D+L)] / c'
 SR' = SR . 0,4 c. F . W / c'
With focusing at infinity,
SR' = SR . 0,4 c. F / c'
The greater the circle of confusion, the lesser the stereoscopic effect.
The value of SR' at infinity focus is taken as reference value, being little different from usual values of photographs of fields more than 1m far.
As [a.F/c'] is always lesser than 1 so SR' is always lower than SR.
In some parts of the picture SR' being less than 1, the stereoscopic effect cannot be felt any more. The stereoscopic effect is kept in fields for which:
SR.a.F/c'  > 1
Thus   SR > c' / (a F)
which are the fields where 3D sensation is preserved.
The circle of confusion of the imager "kills" the lower values of the stereoscopic ratio and lessen the whole stereoscopic effect.



35mm film :
The value of the circle of confusion in 24x36mm size
is c' = 0.03mm.
Applied to a 35mm lens with angular field 54°, 
SR' = 0,47 SR
with  SR > 2,1
Digital sensor 1/1,8" :
The value of the circle of confusion of a 1/1,8" sensor 4 Mpx is c' = 0.006mm.
Applied to a 7,3mm lens with angular field 52° 
SR' = 0,48 SR
with  SR > 2,0
For other sensors, see the page circle of confusion.


To be viewed, the stereoscopic pairs are displayed on a restoration backing. These backings are various: photograph enlargement, digital print, computer screen, screen for show.


Only some cases are dealt with hereunder.
The significant criteria are the enlargement ratio P, and the circle of confusion.
Enlargement ratio:
It is the ratio between the size of sensor and the dimension of the picture on the restoration backing.
The dimensions are:
imager UxV with diagonal W
backing  XxY with diagonal Z
The enlargement ratio is:
P = X/U = Y/V = Z/W
Stereoscopic gap:
The stereoscopic gap u on the imager becomes t on the picture:
t =  u . P
t = SR . a . F . P
From this SR'' = t / c'' and the condition on SR:
  SR > c'' / (a.F.P)


Circle of confusion:
photographic prints
The circle of confusion is
c'' = c' . P
digital prints:
c'' =  2,5 x 25,4/n with n the number of dpi. Let's have a normal print at 96 dpi:
c'' = 0,26 x 2,5 = 0,65mm
computer screen
the dot-pitch of the screen is taken to calculate the circle of confusion. It is often 0,26mm, so:
c'' = 0,26 x 2,5 = 0,65mm
The dimension of the image on the screen is:
X = 0,26x number of pixels



The final viewing is done either by naked eye, either with a viewer.
The enlarged image of the object observed with a viewer is called the apparent size of the object.
Naked eye:
The visual circle of confusion is:
c''' = 2,5 . a . H
with eye keeness a = 0.0004
and H the distance of observation.
The conventional distance of observation conventionnelle is 250mm which gives the conventional circle of confusion:
cc = 2,5 x 0,0004 x 250 = 0,25 mm
An image of a circle of confusion c should be observed at the H = 0,4 c / a


The viewers are characterized by their ocular, with magnification ratio M, and the distance H of the ocular to the restoration backing. As for a lens, the magnification M is calculated with the focal lens R by:
M = 250 / R
The circle of confusion is:
c''' = 2,5 x a x H / M
c''' = a x H x R / 100
Stereoscopic gap:
The stereoscopic gap t on the print, or on the screen, is seen as a size which depends on the remoteness of the eye, or on the power of the viewer.s
 The apparent size h of the stereoscopic gap t is:
h = t . M = u . P . M
The last step of the restoration process, would it be through slide, on print, or on screen, depends on the restoration backing and possible viewing equipment. This combination modifies in its turn the perception of the image, and therefore the stereoscopic effect.
 The question is now:
 Which stereoscopic effect will be obtained at the end?
I named apparent stereoscopic ratio SR^ the ratio which is felt at the end of the restoration.
It is still to evaluate the stereoscopic effect, and to compare it to the initial stereoscopic effect SR.
The combination of the restoration backing and the visualization system compel to compare the circle of confusion of the one or the other, and to take for the calculation of the stereoscopic effect the most restricting of both. To that are added the restrictions caused by the successive losses of resolution.
 Examples will help to see the processus and to lay figures.
The reduced relation is t = SR.a.F.P
Les restriction conditions on SR are:
SR > c' / (a F)
SR > c'' / (a.F.P) or SR > c''' / (a.F.P)


The circle of confusion of backing is c''
The circle of confusion of viewing is c'''
the stereoscopic ratio is:
  SR^ =  t / c'''  if c'' < c'''
  SR^ =  t / c''   if c'' > c'''
Viewing slide:
In this case, without enlargement, P=1
t = u = SR . a . F
Viewer with ocular focal length 78mm to the distance H=65mm of the slide.
Apparent stereoscopic ratio:
SR^ =  SR . a . F / c'' = SR'' = SR'
Thus, SR^  = 0,47 SR
with  SR > 2,1


Circle of confusion of image on backing:
c'' = c' =  0.03mm
Circle of confusion of viewing:
c''' = a.RH/100 = 0.02mm
Thus c'' > c''' and SR^ = t / c''
The stereoscopic restoration on slides with viewer is good, and excellent with a quality viewer. Add it the full frame, the best contrast, the best intrinsic resolution.
Viewing on photographic print:
by naked eye:
Observe to naked eye corresponds to H=250mm (conventional distance) and G = 1
Example of a stereoscopic pair shot with a 35mm film camera and a lens focal length 35mm, viewed on two prints 10x15cm by eye-cross viewing.
Enlargement ratio is P=5
Apparent stereoscopic ratio:
SR^ =  SR . c . F . P / c'''
        = SRx 0,0004x35x5/0,25
Thus, SR^  = 0,28 SR
with  SR > 3.6


Circle of confusion of image on backing:
c'' = c' . P = 0,15mm
Circle of confusion of viewing:
c''' = 0.25mm
So c'' <  c'''  and SR^ = t / c'''
The enlargement ratio is P = 5 because the laboratories print by cutting large borders, fie our cared frames!
The stereoscopic restoration on print viewed by naked eye gives an acceptable result which is easy to achieve.
with a stereoscopic viewer:
Example of a stereoscopic pair shot with a 35mm film and a 35mm lens, viewed on two prints 6x9cm with a viewer focal length R=134mm to the distance H=110mm.
Apparent stereoscopic ratio:
SR^ =  SR . a . F . P / c''
        = SRx 0,0004x35x3/0,09
Thus, SR^  = 0,47 SR
with  SR > 2.1


The enlargement ratio is 90/36 = 2,5 but with a "laboratory" printing work, P=3 is more suitable.
Circle of confusion of image on backing:
c'' = c' . P = 0,09mm
Circle of confusion of viewing:
c''' = a.H.R/100 = 0.015mm
So c'' >  c'''  and SR^ = t / c''
The stereoscopic restoration on print and viewer gives a good theoretical result..
Viewing on computer screen:
Example of a stereoscopic pair which each view is 320x240 pixels for a crossed-eye viewing with naked eye. The image has been shot with a digital camera, sensor 1/1,8" length = 7,2mm of 4 Mpx (2272x1704) and 7,3mm lens.
Each view is cropped to 320x240 pixels to be viewed as stereoscopic pair.
Therefore, it is X=0,36x320=115mm
Apparent stereoscopic ratio:
SR^ =  SR . a . F . P / c''
        = SRx 0,0004x7,3x16/0,65
Thus, SR^  = 0,07 SR
with  SR > 14


 The enlargement ratio is:
 Circle of confusion of image on backing:
  c'' = 0,65mm
Circle of confusion of viewing:
  c''' = 0.25mm
So c'' >  c'''  and SR^ = t / c''
The stereoscopic restoration of a digital image on computer screen is far to have the restoration richness of the other processes. This is obviously caused by the low resolution of the screen. We have to remind that the low 3D ratio fields disappear and that a photograph all with 3D nuances  risks to wash out.

This methodology looks drudgery at first. It needs some elementary datas on the equipment used. However, it remains simple in its way. Moreover, it allows to quantify this 3D sensation, and consequently to improve our techniques for more fun and satisfaction!