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The right number of images

May 2013: Let statistics work to the benefit of your pictures.
RELATED TOPICS: ASTROIMAGING
tony_hallas
Most astroimagers know that if you take lots of frames and combine them, you get better results. The classic formula states that the gain in signal-to-noise ratio is equal to the square root of the number of frames combined.

But I have found that this is too simplistic. A lot of the time, we didn’t have the best sky to work with or we misjudged exposure times. I used to think that a combination of the longest exposures and the most frames would give the best results. And in a perfect world, that would be true. In real life, however, this almost never happens. The reasons could be many. Some nights had bad seeing, others were cloudy, still others were too windy, etc. It’s a long list! Knowing how outlier rejection (getting rid of all the things that don’t belong in your picture, like cosmic-ray hits) and combining frames (to increase the signal-to-noise ratio) works is really important.

I use two kinds of rejection algorithms: Poisson Sigma Combine and STD Sigma Combine. You usually can find these in image-processing software like CCDStack. Both algorithms take a look at the statistics of your data and reject anything that falls outside parameters you choose. You can adjust the “attack” of this software to suit the degree of outliers in your image, but here is where you’ll notice a subtle difference between the two processes.
Satellite-trail
Compare the image on the left, on which the author did a STD Sigma Combine of three exposures, to the right image, a Poisson Sigma Combine of the same three exposures. Note how the satellite trail disappeared with the latter algorithm. // All images Tony Hallas
STD Sigma Combine works best with lots of frames, whereas Poisson Sigma Combine works with as few as three frames. With identical settings, STD Sigma failed to remove a satellite trail when given three frames, but Poisson Sigma did. (See images below.) So, if circumstances limit you to only a few exposures, use Poisson Sigma to clean up your outliers.

The mean (average) of a set of images works hand in hand with outlier rejection in two ways. First, the more frames you shoot, the more you dilute a particular outlier like a satellite trail. Average three frames together, and you will still see the trail; average 20 frames, however, and the satellite trail is now only one-twentieth that of the final image. It’s been “diluted out.” But there is a much more important reason to shoot a larger number of frames, and it has to do with statistics.
Noise
The left image shows a combination of three 15-minute exposures. Compare that to the right one, for which the author combined twenty 15-minute exposures. Notice how the 20-exposure stack reduced the overall noise.
STD Sigma Combine is a fantastic outlier identifier, but it needs a lot of frames to build up a solid statistical model. So, the answer to the question, “Is it better to shoot just a few long exposures or a lot of shorter ones?” is the latter. Not only does this dilute what outliers might show up in your frames, but it also gives STD Sigma enough rejection frames to work properly. The pair of images below shows how much better twenty 15-minute frames are compared to just three.

The fact is that you can combine up to 25 frames before the asymptotic noise boundary (the noise that no amount of combining will eliminate) begins to impinge on the “square root/signal-to-noise formula.” So, it is wiser to shoot a lot of shorter exposures and combine enough frames to allow statistics and the law of averages to perform their magic on your images.

It’s truly all in the numbers.
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