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    Re: Digital camera: stars in daylight
    From: Frank Reed
    Date: 2010 Sep 12, 14:27 -0700

    George H, you wrote:
    "Is Frank, here, discussing what's presently possible, in a navigational
    context, from the deck of a vessel, in ordinary sea-conditions?"

    Yes. That's what I'm interested in: a solution involving off-the-shelf hardware costing no more than a thousand dollars or so (apart from the cost of a laptop), but the software to make it happen would have to be developed --that's the hidden "cost".

    "Is he claiming that daylight shots are possible, in such conditions, that show such stars or planets when at a respectable altitude, with a clear horizon below in the same shot? With angular accuracy and resolution of, say, a (very) few arc-minutes, in the angle between them? In ordinary, clear-sky, weather, with no special requirement for extraordinary crystal-clarity of the sky?"

    Yes to all of that.

    And you wrote:
    "Those two plots differ by a factor of exactly tan (angle), which should be easy to allow for. Being axially antisymmetric in an (x,y) plot, it can involve only odd powers of pixel off-centre displacement."

    One should not be too religious about these things. Suppose the detector array is slightly tilted with respect to the optic axis of the lens. Then you no longer have that pure symmetry. It's best to work with a generalized transformation. I do agree that the usual transformation is from X,Y "out there" in the field of the image to x,y "in here" on the plane of the detector array while for our purposes we prefer theta, phi "out there" but that's just a variable transformation (your tan(angle) would certainly be part of it) after we're done with the calibration process.

    You wrote:
    "But, to use such a camera for our astro purposes calls for highly accurate
    calibration, perhaps more accurate than the camera maker ever envisaged.
    Although the maker's calibration curve might well be sufficient in terms of
    its shape, if the aim is to even approach the precision of a sextant, it's
    likely to be necessary to calibrate the overall scale-factor of each
    individual instrument."

    Yes, I'm not talking about something that the camera manufacturer provides. There is a well-established system used in photogrammetry and computer vision applications. Here's ONE example of this sort of thing:
    While the details differ, the basic procedure seems fairly similar. You photograph a standard target, frequently resembling a checkerboard, a few dozen times from various angles, and then the software generates a calibration at the sub-pixel level. Even ten years ago, these folks were generating 3d models with photogrammetric methods that had accuracies across the line of sight of one part in 30,000. That is, a point in the model would be correctly placed +/- 1cm at a distance of 300 meters. This ratio, you'll note, implies an angular accuracy of about 0.1 minutes of arc --and that includes the inaccuracy resulting from the 3d modelling algorithms.

    You wrote:
    "Even if the pixel array is exactly uniform, is the array pitch exactly the same in x and y directions?"

    No, not necessarily. It doesn't have to be. But I think you'll find that this possibility is included in the analysis methods.

    You asked:
    "Is the lens-to-array spacing exactly reproducible, between cameras?"

    No, it isn't. A camera has to be independently calibrated. A bigger problem may be that these arrays may be mounted on springs to prevent damage. If that's so, then you would have a slightly different calibration depending on the tilt of the camera relative to the horizontal. This would have to be tested, and it might exclude whole classes of cameras. We won't know until we try it out.

    And you wrote:
    "These are matters which over many years have been though about, and dealt with, in the case of the sextant, but may never have even been considered seriously for common usage of a camera."

    I think you may underestimate the huge amount of work that has gone into this, especially in the past twenty years. Computer vision is a big field with a lot of money and a lot of research interest. There's some hard-core mathematical analysis that's been thrown at this topic. Naturally, applied to celestial navigation there are unique issues, but there's a vast literature available to be tapped. As I said in an earlier post, the annoying part from our perspective is that mostly people have produced final products that "just work". This is not the sort of subject where end-users (e.g. generating accurate 3d models from photographs) care about the underlying principles. They just want results.

    You also wrote:
    "But such a calibration should not be difficult to make A single shot of a
    night-sky image should provide all that's needed."

    I agree. For the purposes we're discussing, this is the way to go. I would only add that, rather than a "single shot", dozens of shots of rich star fields would get us that sub-pixel accuracy that would eliminate one source of error from the game.


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