So what does yet another article on the relationship of stereo camera base and near point, far point, lens focal length and on-film deviation offer?
It presents (and derives for those interested) the most general solution to the geometry of a stereo camera. That means that the least number of assumptions are made to derive the equation. You can set the camera focus anywhere, allow for best depth of field, include far points not at infinity, use lenses of various focal length etc.
The article goes on to compare the results from this general equation with special case approximations that have been previously published, and also compares it with the "1-in-30" rule.
Figure 1 shows the general geometry of a stereo camera with two objects in the field of view - a near object and a far object. Here's a summary of the variables (or cast of characters):
Advanced mathematicians will note that as the
value of (a) approaches the value of (f), the distance from the lens to
the film must become infinite to maintain sharp focus.
The aim is to make use of the available 1.2mm on-film deviation effectively so as to maximise the stereo effect within the view or at least not decrease the stereo base when it is unnecessary.
To obtain an ortho stereo result, there is no requirement to calculate the stereo base or lens focal length - it is necessary to always use a 65mm stereo base, and always use camera lenses with focal length equal to the viewer lens. The only variables under some degree of control are the distance to the near point, far point, and camera focus. When shooting for an ortho stereo result it is not always possible to obtain an on-film deviation of 1.2mm, as there are more constraints.
Other situations may demand control over some of the other variables discussed in the previous Section. If it is possible to vary the camera lens base, or the focal length (as well as near or far points) then a better understanding of the relationships between the variables is required. If desired, it is possible with the right equipment to make any stereo photo have an on-film deviation of 1.2mm between near and far point. In general though, this is not usually the aim, as other factors such as "hyper/hypo-stereo scaling" may make such an end result undesirable.
It is up to the interpretation of the photographer as to how to go about using the variables that can be controlled. An understanding of the formulas presented here will enable the photographer to make full use of these variables if required.
The derivation of the general solution is presented in Appendix 1 of this article. But for most readers it is probably more than enough to quote the end result, discuss some of the aspects, and compare this general solution to some popular special cases.
The camera lens base is related to the other variables in Figure 1 by the following equation:
Using this equation, the camera lens base can be calculated in terms of all the variables that are likely to be known to the photographer when shooting in-the-field. This is the most general form of the solution to the stereo camera geometry shown in Figure 1.
As an explanatory example of the use of this equation, suppose the near point (an) is 2m, and the far point (af) is 4m, the lens focal length (f) is 0.05m, the camera is focussed on 2.67m to maximise the use of sharp focus, and the desired deviation (d) is 0.0012m, then the camera base can be set to:
If the camera base is fixed by the design of the available equipment at 65mm, the equation could instead be used to calculate any of the other variables under the photographer's control. For example, it could be used to calculate the closest allowable near point distance, given values for all the other variables. To do this requires algebraic manipulation of the formula, which if done for all 6 variables in the formula would make tiresome reading.
There are special case solutions for Eqn 1, some of which have been published previously by other authors (Ref 1, 2). For these special cases, assumptions are made about some of the variables.
As is well recognised in stereo photography, it is important to maximise the use of the depth of field provided by the lens settings. This is done in two ways: by working with the maximum practical f-number, and by setting the camera focus between the near and far point distances so that the near object and far object are equally un-sharp. The necessary focus setting to achieve the latter is given by:
Substituting Eqn 2 into the general solution given in Eqn 1:
Equation 3 is probably the most practical and useful form of the solution. It is accurate for conditions when the camera lens is focussed to make the near and far objects equally un-sharp (ie: according to Eqn 2). Of course if you have to solve Eqn 2 anyway, you may as well then go on and solve Eqn 1. However, if your camera lens has a DOF scale on the lens barrel, you won't need to solve Eqn 2 as that scale will do the job for you.
If desired, Eqn 3 may be simplified by assuming the far point to be at infinity. Setting af = infinity, we get:
This equation is accurate when: the far point is at infinity and the focus is set to twice the distance to the near point to make the near point and infinity equally un-sharp (ie: a = 2.(an) in accordance with Eqn 2).
If we assume the far point is always at infinity, then Equation 1 reduces to:
And if we further assume that the camera is focussed on the near point object by setting a = an (although this would not make use of the available depth of field efficiently) then Eqn 5 further reduces to:
This form of the equation agrees with that given previously by other authors (1, 2). It is accurate when the far point is at infinity, and the camera is focussed on the near point.
This is not really a special case of the general solution, but it is included in a separate Section here because it is a well known method for approximating the required camera base, and because in the next Section it is compared with the other methods already discussed. The rule states that the camera base is equal to one-thirtieth part of the near point distance (3).
Depending on the assumptions made, or the accuracy required, the camera base could be calculated with any of the methods given by Special Cases 1, 2, 3 or 4. As Special Case No. 1 is the most useful solution, it is interesting to compare it against the other methods for some example situations.
Figure 2: The ratio (an/b0) plotted versus (an) for various far points and focal lengths. The curves are all calculated for an on-film deviation of 1.2mm. The curve numbers correspond directly with the Special Case Number in the text.
To compare the four methods, the camera base has been calculated and then divided into the near point distance. This yields the ratio an/b0 (which of course for Special Case No. 4 will always equal 30, as the base is calculated using the 1 in 30 rule). For the other three cases, the ratio an/b0 varies considerably depending on what the near point distance is. The assumed on-film deviation is 1.2mm for all these graphs, which is an acceptable maximum value for most 35mm stereo formats. (Note: The graphs could also have been plotted as camera base (b0) versus near point distance (an), but the large range of values obtained requires a logarithmic scale for the b0 axis, and a log scale at first glance appears to reduce the difference between the various methods.)
To test the truth of the General Solution using a practical example, a stereo pair was taken with a near point distance of 1.8 m, a far point distance of 2.2 m, a separation of 0.2 m, lenses of .05 m, and focussed on 2m. Although this is a contrived situation, it was chosen because under these condition the General Solution is predicting a camera lens base about three times more than the other methods (eg: the 1-in-30 rule would allow 60mm base, not 200mm). The result is perfectly viewable (Fig 3) though no equation other than the General Solution would have predicted this.
In discussing the results, the curve of Special Case No. 1 (labelled as #1 in each graph) is taken as the reference against which the others are evaluated. This is because it is the most accurate for practical use - the least number of assumptions are made, and the depth of field is maximised.
For the 1-in-30 rule (#4 in each graph), this appears to only be applicable to the focal length of 35mm. For the 70mm focal length lens, it is predicting too much camera base, and it would be more appropriate to use a 1-in-60 rule. When using the 35mm format, it has been suggested (Ref 4) that perhaps a better way to state the rule is "1-in-(the lens focal length in mm)". Thus for the 70mm lens, the rule for 35mm format images would become 1-in-70, which gives an approximate but conservative calculation of the required base.
The graphs also show that the most well known method presented in Special Case No. 3 (labelled as #3 in the graphs) can provide an under-estimation of the allowable stereo base under certain conditions. As it is conservative, there is no harm in using it, unless it is desired to maximise the stereo effect, or unless it is desired to maintain an ortho stereo base under macro conditions. The underestimation of the base is most apparent for close up subjects, or for distant subjects when the near and far point are close together.
The method described by Special Case No. 2 shows similar characteristics to that shown by No.3.
The curves for Special Case No1 (curve #1 in all graphs) shows that the assumptions made to derive the simpler equations for Special Cases 2 and 3 may under certain conditions cause the camera base to be underestimated. Given that it is desired to maximise the depth of field (using Eqn 2) by setting the focus for equal un-sharpness of the near and far points, then this equation is the most accurate method to calculate the base. If the focus is not set in accordance with Eqn 2, then the general solution given by Eqn 1 would give the most accurate calculation of the stereo base. Unless you're taking a closeup, however, (a) will be much greater than (f) and so Eqn 1 won't be much affected by changes in (a).
The methods of base calculation given by the General Solution, or by Special Case No. 1 are the most accurate methods. They should be particularly useful for close-up subjects (eg: macro), or for more distant subjects where the near and far points are quite close together.
This paper was developed as a result of discussions that took place via the "photo-3D" Internet email-list provided by the East Texas State University (Computer Science Department). The support of the email-list by the East Texas State University is hereby acknowledged and appreciated.
For those curious enough to see how the various equations used in this article are derived, this Section should contain enough High School algebra to satisfy them. Figure 1 shows the general layout of the geometry for a stereo camera. Contained within the field of view of the camera is the closest (near) object and the most distant (far) object By using the similar triangles shown in Figure 1 as shaded areas it is possible to write down expressions for the near point on-film separation, and the far point on-film separation. For similar triangles T2 and T1:
Equation 7 gives the distance between the left and right near-point images on the film. For similar triangles T4 and T3:
Equation 8 gives the distance between the left and right far-point images on the film. Now the on-film deviation (d) is by definition the difference between the near point image spacing (bn), and the far point image spacing (bf):
Substituting Eqn 7 and Eqn 8 into Eqn 9:
Equation 10 enables you to calculate the camera base provided you know the necessary variables - and these are: (af) the far point distance , (an) the near point distance, (d) the on-film deviation limit and (a') the distance from the lens center to the film plane. Of these variables, the first two are determined by the picture composition, the deviation limit is usually taken to be 1.2mm maximum for the 35mm film format, and the last one is not known unless it is calculated from the camera focus setting (a). The equation that can be used to calculate (a) is the "lens equation":
Substituting the right side of Eqn 11 for the term 1/a' in Eqn 10 we get:
This is the most general form of the solution to the stereo camera geometry shown in Figure 1 .