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George, you wrote:

> This question takes on a bit of importance because postings appear on this
> list, quite often, from proponents of the use of cameras for making
> celestial measurements. The geometrical distortions discussed here, that
> arise from portraying a spherical surface on to a plane array, add serious
> complications to interpreting measurements, and deriving scale factors;
> complications which are often neglected. It would indeed be possible to
> calibrate the transfer function of a lens system and remap images on a
> corrected grid, though this would call for a lot of work if a zoom lens is
> being used.

The "lot of work" may really depend on what you aim at. Calibrating a
camera-lens-system in order to make them useful for measuring the
altitude of the sun or moon above the apparent horizon to CN
accuracies requires a series of photos along the axis of interest. The
"lot of work" consists actually in measuring the dimensions in pixels,
which - for statistical reasons - are preferably measured several
times, e.g. about 6 times. By limiting ourselves in using for the
measurements always about the same (central) axis in the photo and aim
at having the middle of the distance sun-horizon in the centre of it,
we ended up for one lens with a simple linear relationship and for the
other with a second order polynomial for directly converting the
distance in pixels to the altitude in moa; the scatter of the data
points had in both cases a standard deviation of only about 0.2 moa.
Using for the one lens instead of the second order polynomial also a
linear relation ship increased the standard deviation to about 0.5
moa. These calibration photos were all taken at about the same time,
thus at about the same atmospheric conditions. Considering that under
different conditions this value may eventually be somewhat higher and
considering further also all the other observational errors one may
finally expect an accuracy of about 1 moa or even better, provided the
photos are made the same way as the calibration photos and that the
photos are measured using the same techniques as used for measuring
the calibration photos.

It becomes a bit more complicated if the axis sun-horizon doesn't have
to be in the middle (but is still parallel to the middle line of the
photo) and doesn't have to be centred. In this case the altitude is
calculated by integrating the differential scale over the pixel range.
The differential scales were in this case approximated with second
order polynomials using for one coordinate ax^2+bx+c. The dependency
from the other coordinate was approximated with a, b and c being also
second order polynomials, like uy^2+vy+w. The resulting standard
deviation (valid for the full x/y-plane) was about 0.5 moa. This
analysis can be done in a spreadsheet by using e.g. a template
containing all of the calibration parameters. The calibration itself
results in more work since it means measuring the pixel-dimensions for
several axis.

When you wrote a "lot of work" you may have thought to calibrate for
being able to measure any (oblique) distance within the x/y-plane.
Would this really be necessary for CN applications? 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.

Marcel

   

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