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

    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.
    Your following description ...
    > 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
    where your model fails:
    "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":
    This function contains no additional information than what is already
    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
    I hope all of this contributes to a better understanding.

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