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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
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