Center of Perspective

By John Bercovitz

Background

In stereo photography, it is important to view images from the correct 
perspective point in order to achieve a geometrically exact 
(orthostereoscopic) reconstruction.  A pinhole camera exposing an 8x10 
can serve as the model: The pinhole is the center of perspective for 
both the scene and the image.  In other words, the eye of the person 
viewing the image should be the same distance from the 8x10 as the 
pinhole was from the 8x10 in order that the angles between the image 
points at the eye be equal to the angles between object points at the 
pinhole in the original scene.

The problem

Most of the time perspective causes no problem for stereo photographers 
because stereo cameras usually use short lenses which are nearly 
symmetrical and in this case, the transparency is merely viewed with a 
lens which has the same focal length as the camera's lenses and the 
result is good enough.  

However, there are cases where more thought must be given to the locus 
of the center of perspective.  In particular, if macro shots are taken 
with a 35 mm SLR, and the lens on the SLR is of either retrofocus or 
telephoto design, the usual approach to selecting a viewing lens may 
result in a noticeable error.  The center of the entrance pupil of a 
lens is the center of perspective on the object side of the lens (see 
any number of books by Kingslake).  In a symmetrical lens, the primary 
principal point coincides with the entrance pupil (and the secondary 
principal point coincides with the exit pupil).  In this symmetrical 
lens case, the center of perspective for viewing the image is the 
secondary principal point, and for a short lens, the distance of the 
secondary principal point from the film is usually very nearly equal to 
the focal length of the lens.

The solution

Givens:
1) The center of the entrance pupil is the center of perspective on the 
object side of the lens.
2) If the film lies within its normal range of positions relative to 
the lens, then some distance in object space is in focus.
3) The usual formulae for magnification apply to the in-focus object 
and its image.
4) The in-focus image must subtend the same angle at the viewing eye as 
the in-focus object did at the entrance pupil.

Variables:
f = focal length of lens
L = distance from in-focus object to primary principal point
x = L-f (where x is Newtonian equation variable, as: xx" = f^2)
m = magnification = (in-focus image size)/(in-focus object size)

New variables:
PN = distance from primary principal point to entrance pupil,
     positive if primary principal point is left of entrance pupil
DCPI = distance of the (image's) center of perspective from the 
       image

Equations:
m = f/x      (a Newtonian form of the magnification equation)
m = f/(L-f)  (substituting equivalent of x)

DCPI = m(L+PN)  (L+PN is the distance of the object side center of 
                 perspective from the in-focus object.  m times 
                 this distance will be the DCPI.)

Combining the two equations:

        (L+PN)
DCPI = -------- * F
        (L-F)

To me, the interesting thing is that the center of perspective for 
the image side does not appear to land on any of the fixed Gaussian 
cardinal points or on a pupil.  It appears to be a new moving point.

You can see that as PN approaches 0, the center of perspective on 
the image side nears the secondary principal point.  If PN = 0, 
the two coincide.  So if PN is small, you can safely ignore this 
effect and still get orthostereoscopic reproduction with focal 
length of viewing lens equal to focal length of taking lens.  I 
believe some 35 mm SLR macro lenses are symmetrical as symmetry 
automatically cancels many aberrations especially when the 
magnification is near 1.

For checks of the validity of the equation, I set up the telephoto 
lens described in Dr. Kingslake's "Lens Design Fundamentals", Pg. 
265, in the ray tracing program, "Beam 3", from Stellar Software in 
Berkeley.  Also, I ran some older experimentally-determined data from 
a 15.5" Wollensak telephoto through the formula.  In all 
cases, agreement with the formula is excellent.

John Bercovitz