**FUNDAMENTALS of STEREOSCOPY**- The contents of the first part of this page until the "stereoscopic ratio" was presented at KAPiCA 2002, the international conference on kite aerial photography, on November 5th, 2002 and has been updated and completed since.
**FOREWORD**- The key of success in photographic stereoscopy is to have a clear understanding and knowledge of the stereoscopic process and of the parameters which control it. We also have observed that stereoscopy is more or less effective depending on situations, and we will see how to deal with it for successful pictures.

This page describes the fundamentals.

**CONDITION FOR STEREOSCOPY**- Stereoscopy is a physiological and geometric process.

To get a stereoscopic sensation, the image in our left eye and the image in our right eye must be similar, but with slight geometric differences. In this case, our brain will interpret the scene for stereoscopic effect. If the two images have no differences, the stereoscopic effect does't exist.

From this, it issues that stereoscopy is effective with the condition that:

- Dots P1 and P2 must be differentiated by our eyes
**PHYSIOLOGICAL LIMITS**__Keenness of eye__- The keenness is the ability of our eye to make out two close points. We cannot distinguish two points which are within an angle less than one minute. Mathematically, keenness is an angle expressed in radian unit: a = 0.0003 to 0.0004

__Color separation__- We can only distinguish colors if their area is large enough, and if the surroundings have a different tint. This may affect considerably the stereoscopic effect because the left and right image may not be different.

**PARAMETERS OF THE SCENERY**- There are one non-geometric and two geometric parameters to consider
__Background disposition__- If the background is of the same uniform color, there is no stereoscopic effect. It is the case of a seagull seen flying in an uniform deep blue sky. Our eye cannot see any difference between P1 and P2. If the background is variegate, with many different colored and contrasted areas, P1 and P2 are really different, and there will be stereoscopic effect. The background disposition is a non-geometric parameter.
- On a photograph, there is not only one foreground and only one background, but a succession of plans, one of them being the foreground of the next one, and the background of the one before.
- The formulas can be calculated for each couple of successive plans depending on their real arrangement.

__Distance between camera and subject__- This parameter will apply as angle between lines L1 and L2 must be greater than the keenness limit. As D increases, the angle decreases.

__Distance between subject and background__- The further is the background behind the subject, the more P1 and P2 will be distant. It means that just behind a far subject, there is an area where stereoscopy is non effective because the dots P1 and P2 will not be differentiated. It is named the
*neutral zone*.

**PARAMETERS OF THE OBSERVER**- The observer can use his eyes, but also binoculars, mirror devices, cameras, and so on. All these are optically different, and will affect stereoscopic effect. Applied to photography, the camera constituted of the lens and of the sensor, and the restitution device can change the stereoscopic rendering.

__The Base__- For us the base is the distance between our eyes, common value is 63mm. Using cameras, the base can be modified and it is the only parameter that we can change for stereoscopic effectiveness control. The larger is the base, the more stereoscopy will be effective

The orthostereoscopy is when the base is equal to the distance between the eyes.

**RULES AND GEOMETRIC FORMULAS**- Let's consider the distances B, D, L, and keenness a, when applying stereoscopic condition we have two basic relations:

**e/(D+L) > a**and**B/D = e/L**

- From these, formulas are easy to get:

Non effective zone**N = a . D² / (B - a . D)** -
**Base****B > a . D (D+L) / L**

**STEREOSCOPIC RATIO**- The base formula will give the minimum base to use. As we said, the stereoscopic sensation can be strong or weak. So, it is interesting for us to appreciate and scale this stereoscopic sensation, at least from the geometric parameters. For this, we will use the stereoscopic ratio SR

I must admit that I don't know any universal recognized rule or formula for this purpose. For years, I applied the formula that I published in Aerial Eye summer 1998, which I draw from a book. - But I observed that it was not working properly in some cases even it was satisfactory in most cases. After unsuccessful researches in a few books I went on my own, reconsidered the whole thing, expressed a new line of argument, and set a polyvalent formula.
__Note:__The formulas are set for a stereoscopic angle of view identical to the human vision which doesn't exceed 60°

- The minimum space of two dots that the eye can see on the background is s, and the gap of projected points of the subject on the background is e: let's compare it as angles: e/(D+L) and s/'D+L) or e/s.
- The Stereoscopic Ratio is
**SR = e / s**

If e=s Stereoscopic Ratio is 1 - With e = B L/D and s = a (D+L) we obtain:
**SR = B L / a D (D+L)**- Tables can be calculated. For the human eye:
- 0
< SR < 1 no stereoscopic
effect.

1 < SR < 10 weak stereoscopic effect

10< SR < 20 moderate stereoscopic effect

20 < SR < 35 heightened stereoscopic effect

35 < SR < 50 strong stereoscopic effect

50 < SR < 70 excessive stereoscopic effect - further the stereoscopy is disturbing or impossible.

**LIMITS of STEREOSCOPIC EFFECT**__Maximum Stereoscopic Ratio:__- Start from the formula: SR = B L / a D (D+L)
- and define t = L/D
- we get SR = B/ (a.D) . t / (1+t)
- It can be verified that 0 < t/(1+t) <1 which gives the formula of the maximum stereoscopic ratio.
- Mathematical considerations lean to:
- SR max = B / (a
**.**D) - which occurs when the background is at infinity and which varies decreasing with the distance to the field.

__Maximum stereoscopic distance of the eye:__

- The maximum stereoscopic effect is got when the background is at infinity. Let's find the maximum distance of a field to which the stereoscopic effect becomes imperceptible to the human eye. In this case, SR = 1

- The before formula is then written D = B / a
- The space of eyes is 63mm,
**Dmax**= 0.063 / 0.0004**= 157 m**- Behind this distance, everything is a neutral zone.

__Neutral zone:__- Already mentionned, the neutral zone is where there is no stereoscopic effect. It can be defined two ways:
- ¤ Zone for which the field behind the subject is too close.
- ¤ Zone further than D max
- Physically, in both cases, for the eye, it is the zone where the stereoscopic ratio is inferior to 1. Thus SR<1 is also e<s.
- .
__The extent of neutral zone behind the subject__is calculated by : **N = SR .****a .****D² / (B - a . D)**__The distance behind which all is neutral zone__is calculated by :**Dn = B /(a****.****SR)**- It is interesting also to be able to calculate the neutral zones for different stereoscopic ratios. The use of cameras with various lenses, of binoculars, of field-glass, makes the stereoscopic ratio in relation to their magnification factor.

__Nearest subject__- Any subject at a short distance D from the camera will not be visualize normally if SR is over its maximum value. It must be cared to evaluate the distance to the nearest plan and to determine the desired 3D level.
__distance D min as per L , B and SR__- For a known distance L and a planned base B we can determine the distance D-min where to set for the degree of relief chosen by the SR value..
- All the formulas can be calculated in the calculus sheets
__The background being very far__- L is great compared to D and D/(D+L) tends toward 1
- Thus D = B / (SR . a ) Now with the maximum value SR = 50 and a = 0,0004 there is D min = 50 . B which is the 1/50 rule.

**CONVERGENCE of OPTICAL AXIS**- In all before, it is supposed that the two cameras have their optic axis strictly
**parallel**. The stereoscopic pairs thus done are perfect for subjects moderately distant, or in the far. It is true for aerial photography, for landscapes. - It is quite different for very near subjects, and when the base is rather large.

If the last background plan is not too far from the subject, the convergence will be benefic. On the other hand, if the last background plan is very far, the two images of the background will be shifted, and it will be necessary to mask the lateral parts not superposed to avoid the visual trouble when looking at the views.

Beyond 50m our eyes don't converge any more. On the contrary, more near is the place we are looking at, the more our eyes are converging.

In general, each eye can converge up to 12° and will normally adapt to a converging angle of 5°. So, as the base is not greater than 1/5th of the distance to an object, B>D/5, we should theoretically be able to see the stereoscopic pairs.

However, it will depends on the means of visualization. With viewers, it will be possible. When projecting, it is more difficult.

**KEYSTONING EFFECT**- The horizontal and vertical lines are never displayed parallel, but they run to the infinity end of the axis lines, upward, or downward as well as toward the right and the left sides. It is well known that when turning the camera upward the vanishing of vertical lines is more pronounced.
- This keystoning effect bothers in stereophotography because the images cannot be exactly superimposed. When the two optical axis run to a near point, the keystoning is visible. In red the vanishing when the left camera is turned to the right, and in green when the right one is turned to the left.
- The bottom figure shows the difference when superimposed.

**CONCERGENCE RULE**- Let us recall that
- the convergence do not modify the stereoscopic ratio.
**The good rule is that the backgrounds are perfectly superimposed.**

- Thus,
- ¤ when the background is very far from the foreground, the optical axis will be parallel.
- ¤ when the background is close to the subject, it is possible to make converging the optical axis if assuring that the background is perfectly superimposed on both views.

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