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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?
From: Marcel Tschudin
Date: 2010 Jul 8, 15:36 +0300

```George,

It can be useful to understand something by relating it to a model. It
can however also lead to difficulties in understanding something if
the model is not appropriate or the drawn conclusion don't apply for
what one tries to understand.

> If we take axial symmetry for granted, then it seems simplest to define
> everything in terms of that central axis, and a radial line passing through
> the centre of the array. The incoming angle A, is measured from that axis,
> and the corresponding pixel count Px (in the x direction), is measured from
> the centre of the array, positive or negative along the x axis. If, in some
> implementation, pixels are instead counted from one edge of the array, a
> suitable offset is to be subtracted from that count. That relationship, Px
> = f(A) defines everything we need to know about the distortion of the
> system. That seems to be what Marcel refers to as the "conversion
> function", and I'll go along with that name.
>
> That is the function that has to be antisymmetric about the zero point, so
> that f(-A) = -f(A) : because of which, if it's a polynomial in A, it can
> not contain any constant term or any terms in even powers of A. So it can
> have only terms in A, A cubed, A to the 5th power, and so on. Another
> possibiliity is a tan function, which is also antisymmetric, passing
> through the origin at (0,0).

... doesn't apply - at least not in this sense - to what "calibration
function" and "conversion function" refer to.

May be the following explanations can help you to understand why or

"Calibration function":
In the context of lens distortion one refers - as you indicated
yourself some time ago - to barrel or pincushion distortion. Both of
them describe how a lens changes a straight line into a curved one. In
contrast to your model, as you described it above, these lines
correspond rather to symmetric functions, like e.g. a second order
polynomial. The "calibration function", which one derives for a lens,
corresponds to such a barrel or pincushion distortion line, directly
obtained from measurements.

"Conversion function":
available from the "calibration function", it only shows that
information in a different way. In order to better understand the
"conversion function" it could be helpful to look before, as an
intermediate step, at a "mean scale function" which represents for
different pixel ranges the mean scale in moa per pixel as derived from
the "calibration function", this the following way: In the figure of
the "calibration function" we select a pixel value which will be the
centre for all our pixel ranges. A data point for the "mean scale
function" is obtained by selecting a certain range and calculate for
it in the "conversion function" the mean value of the scale within
this selected range. The new "mean scale function" has as x-values the
pixel ranges and as y-values the mean scales (moa per pixel); it thus
shows now for the selected range-centre the mean scales for all pixel
ranges. From this "mean scale function" one arrives now at the
"conversion function" simply by multiplying the y-values with the
corresponding x-values, i.e. by multiplying the mean scale (moa per
pixel) with the corresponding pixel range, thus obtaining the angle
(moa) as a function of the pixel range.

===

The way how Greg proceeds for calibrating his lenses is slightly
different. By comparing in his photos of the sun above the horizon the
measured pixel heights with the calculated angular heights he obtains
directly the "conversion function". If he would draw his data directly
as HS as a function of pixel range, they would really be close to a
straight line. For some reason he however shows his data not as
directly obtained but as how they look as a sort of "calibration
function" which he wrongly tries to approximate linearly. There is a
difference between his "calibration function" and the one discussed
here. Here it's the scale (moa/Px) as a function of pixel position
whereas his one corresponds to mean scale as a function of pixel
range.

I hope all of this contributes to a better understanding.

Marcel

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