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As usual, Paul Hirose's contribution is perceptive and to-the-point.

I'll take his points in order-

=======================

| George Huxtable wrote:
| >
| > Well, all I was thinking of, really, was that if a zoom lens was being
| > used, that has a whole range of possible expansion factors, and
presumably,
| > a somewhat different lens-distortion for each one, so the "lot of work"
was
| > just to measure calibration curves at all these zoom settings. But I'm
| > aware that it's difficult, if not impossible, to return to exactly the
same
| > known zoom factor as before, in the absence of a precisely marked
| > scale-of-zoom. So I expect that the only way to use a zoom lens for
such a
| > purpose in practice is probably to use it at either maximum or minimum
| > zoom, not in-between, and hope it returns to exactly the same value
when
| > doing so.
|
| You can get pretty close by trial and error. Use the markings on the
| lens for the initial setting, then take a shot. My camera doesn't show
| the focal length in real time, but the value is recorded for each photo
| and can be viewed immediately after. The display sensitivity seems to be
| 2 mm, i.e., I can get it to read 48 mm and 50 mm, but not 49 mm. A
| definitely perceptible rotation of the zoom ring is possible without
| changing the readout. If the desired setting is one of the numbers
| marked on the zoom ring, I believe you could be repeatable within 1 mm
| simply by observing this ring.

Yes, but not good enough, as I understand the aim of Greg's project. He
wants to use the camera for making angle measurements in the sky. If it can
only distinguish between 48mm and 50 mm, that limits the accuracy with
which one could return to a pre-calibrated zoom position to +/- 2%. I doubt
if that would be anywhere near acceptable.

===========================
|
| > Presumably Greg's picture was taken from on land. I suggest that
observing
| > from the unsteady footing of a small craft would be a very different
| > matter, even at 1/500 sec.
|
| Many amateur grade cameras have a system of angular rate sensors and
| servos to remove most of the unsteadiness of the photographer's hands.
| The implementation varies among different manufacturers. For instance,
| Olympus puts the image sensor on a moveable x-y carriage, while Nikon
| uses a moving element inside the lens. At any rate, these systems
| typically let you use a shutter speed four to eight times slower than
| would be practical otherwise.
|
| The old 35 mm rule of thumb for the slowest safe handheld shutter speed
| was to take the reciprocal of the focal length. So with a 200 mm lens
| you would try to shoot at 1/250 or faster.

I still reckon that, except under millpond conditions, the motion in a
small vessel, counteracted as well as possible by hand-holding, will be
many factors of ten worse in angular rate than can be achieved on land.

==========================
| > "I could however imagine that there exist somewhere professional
| programs
| > which would help to do also this sort of calibration and analysis of
the
| > measurements."
| >
| > I can only agree. I'm pretty sure that he and I must be retracing
| > well-trodden ground, if only we knew where to look.
|
| Imatest sells the "studio" version of their program for $100.
| http://www.imatest.com/solutions
|
| According to their documentation, the basic distortion correction
| equation is an odd order 3rd degree polynomial. If Ru is the undistorted
| radius and Rd the distorted radius of a circle centered on the
| intersection of the optical axis with the image plane, Ru = Rd + k *
| Rd^3. The sign of k determines whether the distortion is barrel or
| pincushion.
|
| http://www.imatest.com/docs/iqfactors.html#distortion
|
| It's common nowadays for software (in the camera or on a computer) to
| apply distortion corrections based on the known characteristics of the
lens:
| http://www.dpreview.com/articles/distortion/
|
| In the case of my camera, manual correction "by eye" can be applied
| after the photo is taken, or you can select auto correction. For the
| latter option you must have a modern lens with the electrical contacts
| that allow the camera to identify the lens, and, I suspect, the zoom
| focal length.

Thanks for that link to those useful Imatest pages. I have found their page
at http://www.imatest.com/docs/distortion.html to offer good information.
The tan function correction, that we have taken to apply to apply to
imaging sky angles, is really a special case of the 3rd order correction
that Paul refers to, with k=1, if higher-order terms can be neglected, as
is the case at smallish angles.

=========================
|
| > Of course, it's a poor test of the correctness of the tangent model,
simply
| > because it's restricted to such a small  range of angles from the optic
| > axis, so the resulting distortion is so low. All we can really say is
that
| > it doesn't discredit that model by showing any disagreement with
| > observation.
|
| I see nothing wrong with George's desire to demonstrate it
| experimentally, but the mathematical basis for tangent formula can be
| found in books. For instance, Smart ("Textbook on Spherical Astronomy")
| derives a formula for the distance between the point where the optical
| axis intersects the photographic plate, and the image of a star on the
| plate. It equals the telescope focal length, times the tangent of the
| angle between the optical axis and the star.
|
| Green ("Spherical Astronomy") says, "It is seen that the process of
| imaging stars on to the photographic plate is similar to a central
| projection of the stars on to the tangent plane to the celestial sphere
| at A." (The point he calls A is the intersection of the optical axis
| with the celestial sphere.)

Indeed. By now, I'm very confident that the tan A correction model for
imaging sky angles is exact in itself, and Green's text confirms that. But
then, it has to be combined with any distortion that the lens system would
impose on a flat-plane image (by multiplying the transfer function,
presumably). Perhaps that lens-distortion function could be simply checked
on a bench by photographing a rectangular test-grid, as long as it's not
affected by altering the focus.

Although Green deals in great detail with other sources of distortion, he
does not seem to be bothered about lens distortions. However, we have to be
aware that the astrometric plates he is referring to subtend an angle of
the order of a degree; not the 40-degrees or more that measuring altitudes
in the sky might call for.

========================

Just to show the sort of distortion that a lens system can create, I attach
an uncropped photo (which I don't intend to enter for any photographic
prizes) of some brickwork on the side of my house. It was taken with a
cheapo Olympus C500 Zoom digital camera set to widest angle, in which case
the specification states that it's "equivalent to a 38mm focal length used
with a 35mm camera".

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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