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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
    adjustments in the Excel-evaluation.
    
    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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