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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Camera sextant? was: Re: On The Water Trial of Digital Camera CN
From: Marcel Tschudin
Date: 2010 Jul 5, 17:03 +0300

```George,

First I would like to thank you for having taken a look at it. Up to
now I didn't have had a possibility to share what I did with someone
who is able to notice possible flaws. I therefore appreciate your
effort to look more in detail at it. However, if I understand you
correct, your concerns don't seem to apply.

I didn't go into the theory to check which power expansion would
theoretically be the correct one. What I did was looking at the
measured data and used the polynomial which most reasonably described
those, considering also the errors in the initial measurements and
those of the approximations (fits). Doing this for different lenses
showed that what I did was consistent.

I'm not always sure to which function you referred to.

The most important function is the calibration function where the
measured scales in moa per pixels are approximated along the reference
line (see sheets Cal_Poly and Cal_Fig).

At the beginning I questioned myself whether some of these calibration
functions could actually require a 3rd order polynomial. But from all
lenses the figure where the measured scale was shown as a function of
the pixel positions along the reference line looked similar to this
one. In my opinion it is completely sufficient to approximate those
measured data with a second order polynomial. There could eventually
turn up some cases where the measured data can't be approximated
better than by a linear fit. This would require then some further

This calibration function is used in sheet "Observation" to calculate
the angular distance over any pixel range along the reference line.
The calculations in sheet "Observation" use only this calibration
function.

The calibration function is also used to derive a dataset for
calculating (fitting) a function to convert a centred pixel range
directly into the corresponding angle. For this conversion (centred
pixel range to angle) three different formulae are provided, a linear,
a quadratic and an arc-tan function, indicating also that the user
should select the one which suits best.

For the shown example the quadratic and arc-tangent conversion
function fit indeed much better to the dataset which is generated for
their calculation. This dataset is however generated using the
calibration function. The expected error of the conversion functions
can therefore not be better than the one of the calibration function.
Since the error of the fitted calibration function is considerably
larger than the ones from the fitted conversion functions, the total
error, resulting from the calibration function and the conversion
function, differ only  marginally between the three functions. For the
example in the Excel-file:
StdDev Linear conversion formula: +/-0.17 moa
StdDev Quadratic conversion formula: +/- 0.15 moa
StdDev arc-tan conversion formula: +/- 0.15 moa

This is true for the shown example where the quadratic calibration
function happens to be fairly symmetric around the middle of the
reference line (1940 Px). Using the same example: If you delete in
sheet Cal_Data the measurements #10, the maximum of the quadratic
function shifts to a value slightly less than 1500 Px, it becomes thus
fairly asymmetric. The fitted values will now certainly have changed
slightly but the difference in total error between the different
conversion formulae are still marginal, suggesting again that the
linear conversion formula is completely sufficient.

Do these explanations answer your concerns on the type of polynomial expansion?

Marcel

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