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09-07-2011, 02:01 AM   #1
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Math behind Pentax O-GPS1

Hi,

Can someone point out what is the math that applied behind this great tools? Especially in Astrophotography side.

Thanks.

09-07-2011, 02:31 AM   #2
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Hi kamayok3,
I'm not sure what you want to know. Do you want to know how GPS works?
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David
09-07-2011, 02:57 AM   #3
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How much do you want to know? I have a background in astronomy but haven't used an O-GPS1 so some of the details might be a bit off . I can give you a hand-wavy explanation ... the actual mathematics would take pages and pages!

GPS basically works like this -- a number of satellites orbit the earth. Each carries an atomic clock. Each clock sends out "ticks" as radio waves. Those take time to get to you. You will be closer to some satellites than other, so ticks from those ones arrive sooner. The GPS receiver measures the order all the ticks come and how long there is between the ticks from each satellite. Then it compares this with a prediction of what it expected based on where it thought you were. And if those two don't match up, it tries different positions in its model until the prediction lines up with the actual measurements.

The O-GPS1 also has an electronic compass and level sensor to know which way the lens is pointing.

Then it works out which way the stars should be moving across the image based on all that. Then it shifts the sensor to compensate so that each star lands on the same pixel all the time. Presumably it can only move the sensor so far so I guess there must be a maximum exposure time or maybe it pumps out a certain number of images per second (or minute), resetting the sensor between each - maybe even combining those in-camera to a single final image?
09-07-2011, 09:26 AM   #4
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It's rocket science.

09-07-2011, 10:42 AM   #5
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QuoteOriginally posted by russell2pi Quote
How much do you want to know? I have a background in astronomy but haven't used an O-GPS1 so some of the details might be a bit off . I can give you a hand-wavy explanation ... the actual mathematics would take pages and pages!

GPS basically works like this -- a number of satellites orbit the earth. Each carries an atomic clock. Each clock sends out "ticks" as radio waves. Those take time to get to you. You will be closer to some satellites than other, so ticks from those ones arrive sooner. The GPS receiver measures the order all the ticks come and how long there is between the ticks from each satellite. Then it compares this with a prediction of what it expected based on where it thought you were. And if those two don't match up, it tries different positions in its model until the prediction lines up with the actual measurements.

The O-GPS1 also has an electronic compass and level sensor to know which way the lens is pointing.

Then it works out which way the stars should be moving across the image based on all that. Then it shifts the sensor to compensate so that each star lands on the same pixel all the time. Presumably it can only move the sensor so far so I guess there must be a maximum exposure time or maybe it pumps out a certain number of images per second (or minute), resetting the sensor between each - maybe even combining those in-camera to a single final image?
I think I read somewhere something like 300 seconds maximum. I cannot tell you where that was (thread) but I'm pretty sure it was this forum. I would also assume that the time would depend on the lens used. The moon for instance (although it's a moving target itself) travels across the viewfinder a lot faster at 500mm FL than it does 50mm FL.

09-08-2011, 09:31 AM   #6
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QuoteOriginally posted by kamayok3 Quote
Hi,

Can someone point out what is the math that applied behind this great tools? Especially in Astrophotography side.

Thanks.
I think some abstract reasoning gives you the quickest start. I assume you have minimal knowledge in math as otherwise, you wouldn't have asked your question

Well, the sensor has to rotate during the exposure to do its trick. As it counters the Earth's rotation. What the camera needs to know is the virtual axis of rotation and rotational (angular) speed. If the virtual axis of rotation (e.g., near polar star) is very far off the sensor, the sensor movement will almost look like a translation.

It derives it from the Earth's axis rotation (known, 24h), camera optical axis and focal length.

The focal length f is transmitted via the lens mount to the camera and is a known.

The camera optical axis is described by 6 parameters where only 4 parameters do actually matter:
- the position in space (3 parameters, but altitude with respect to Earth's surface can be ignored, leaving 2 parameters): x, y
- the angle of the optical axis (3 parameters, rectascension, declination and roll where roll can be ignored): r, d

Now, the camera measures 4 knows to determine x,y,r,d:
- the GPS coordinate delivering x,y (using the O-GPS1 gps receiver).
- the camera pitch p (using roll only if the camera isn't level), using the built-in level meters.
- the camera yaw or heading h (using the O-GPS1 compass).

A formula returns x,y,r,d from x,y,p,h. From x,y,r,d,f follows the virtual axis of rotation for the sensor (and speed).

I would assume that the compass provides the most inaccurate measure. However, the sensor can only move that much (limiting max. exposure time depending on focal length and r,d) and therefore, may have been tuned to be exact enough.

As a first approximation, max. exposure time should decrease inversely proportional to the focal length ~1/f.


However, the real charme of the O-GPS1 is that you can take starry night images with a (slightly blurred) foreground. Therefore, the focal length wouldn't be too high. With a higher focal length, one may assume no foreground and it then is no problem to stack multiple images to achieve arbitrarily long exposures.

Last edited by falconeye; 09-08-2011 at 09:45 AM.
09-09-2011, 04:39 AM   #7
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Wow......Falconeye's answer is what I wish to see. Thanks a lot.
09-09-2011, 05:19 AM   #8
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Well, I hope to test that out this weekend (even though the moon is a bit too bright). I received my O-GPS1 this week, and the GPS function is very accurate - dropping the co-ordinates into Google maps pinpoints to within a few feet. But the Astrotracer function is why I got it.
Note to users - the calibration routine really twists up the neckstrap

09-09-2011, 05:54 AM   #9
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I hope I can receive mine in next week.
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