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

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