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dmilligan
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Reged: 11/21/13

Re: How much difference do more subs make?
[Re: tazer]
#6272897  12/25/13 11:00 PM

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Stacking does not reduce noise, it increases noise  for a given total integration time. Stacking simply allows a longer effective total integration time when single exposure time is limited by some other factor (tracking, LP, etc). It's the longer effective integration time that reduces noise, not the stacking. If you take a single exposure of some given time and compare it to a stack of some N > 1 subexposures with the equivalent total integration time, the single frame will win in terms of SNR.
If you understand what stacking actually is mathematically you will understand why. Imaging is basically just counting photons. As time passes, photons (actually photoelectrons) accumulate on the sensor, at some point we end the exposure, and convert the photoelectrons into a digital number that is related to the number the photons that landed on the pixel during that time (plus some amount of error due to the electronics). Mathematically this accumulation is simply addition. We add up all the photons that land and end up with a total. Stacking is the same thing, simply addition. We add up all the counts in each subframe to come up with a total (then we normalize it to some range, usually 0 to 1, but that's beside the point). The reason that stacking works is that it is the exact same process that is happening in the physical realm (photons accumulating, i.e. addition), in the digital realm (addition).
The problem is that when we stack we add in the digital realm, so there is repeated quantization and read errors (i.e. sensor read noise) as well as rounding errors in the mathematical computation (though this is usually negligible). When we simply take a longer single exposure, the photons 'add' in the physical realm and we only get read and quantization error 1 time. Using bigo notation we would say that read noise grows at O(n) when we stack. For a single exposure the read noise is simply O(1). If the total exposure time is the same, the inherent photon noise (or shot noise) will also be the same. Photon noise diminishes with time at a rate O(sqrt(t)).
With proper calibration we can eliminate a lot of the read noise, but it is impossible to eliminate all of it. So it may be a small contribution, but it is still linear ( O(n) ). The improvement from more time is sublinear O(sqrt(t)), and while it will grow a lot faster than the read noise contribution at first, there will be a point at which the increase in read noise catches up and eventually dominates, this is the point of 'diminishing returns'.
So lets say that a single frame exposure time is limited to some value (b/c in reality it always is). To get more time and thus less photon noise, we are forced to use more subs. There will eventually come a point at which the linearly increasing read noise starts to dominate the sublinearly decreasing photon noise, and adding more subs won't make a lick of difference. Obviously this point is going to depend on many things like sensor performance and temperature, but I imagine that it is quite large for well calibrated images and low temperatures. The only way you could assert that there is never a point of diminishing returns is that if you could show how to build a sensor with absolutely 0 read noise (if you can do this, I would be interested in going into business with you), that's the only way to make the 0(n) term go away.
There is another point of diminishing returns, and that is when we've collected enough photons for even the dimmest part of the image to have such a high photon SNR that noise is not perceivable. It might still be possible to improve SNR, but the noise would be so small that it would not be even distinguishable on something like the 8 bit computer displays we use to view our pretty pictures on (that would be a SNR that is smaller than the number of discrete brightness values or 256:1 which would be about 50dB, and would require collection of 65,536 photons  let's just say: good luck with that  such an image would be *perfectly* noise free, though I'm sure *acceptably* noise free is a lot sooner than that).

Samir Kharusi
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Re: How much difference do more subs make?
[Re: dmilligan]
#6272928  12/25/13 11:45 PM

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The equations involved for SNR are well understood and quoted here. In particular, I would refer to Line 3. You may note that the Read Noise, R, is not multiplied by N, number of subs. Additionally, provided the subs are long enough (20% of BoC Histogram or more), the dominant term in the denominator is the Skyfog signal, Sky. It is overwhelmingly important to make sure that the Skyfog signal in each sub is overwhelmingly larger than the Read Noise, R, otherwise one gets lousy Stacking Efficiency. For dim objects, those very faint tendrils, the signal from the Object itself, Obj, can also be small compared to Sky. Basically, provided the subs are long enough, the dominant term in the denominator is the the Skyfog signal, Sky. Of course the larger Sky, the larger its quantum statistical noise. In brief, there is NO upper limit as to how many subs will improve the SNR, and while the photographer may be happy with a SNR of, say, 30 on the brighter parts of his nebula, the fainter parts may have a SNR of 3 (barely acceptable). That's why some people would say that x hours is sufficient for a bright nebula, but one has to go longer for fainter stuff. IMHO, all nebulae ought to be treated as super faint and dim. They all have very faint tendrils, and it is only the photographer himself who can determine how deep he wishes to go on that object.
In conclusion please note the statement in Line 8. of the referenced URL:
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8. The curve predicted from equation 6 has an asymptote that can be easily calculated as subexposure time approaches infinity. Specifically, as subexposure time approaches infinity, the contribution of R becomes negligible, and this factor can be ignored in determining the asymptote. The equation becomes: SNR asymptote = sqrt[K]*(Obj) / sqrt[(Sky+Obj)]. Thus, in photon noise limited conditions, the SNR is minimally affected by the read noise or subexposure duration (i.e., assuming that the subexposure duration is sufficiently long).
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If Steve Cannistra has his equations wrong, and they have stood the test of time, I would appreciate hearing where he went wrong. Certainly in the time I spent digesting all this I could not find any faults. But science progresses and what was obvious yesterday may be proven wrong tomorrow.

dmilligan
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Reged: 11/21/13

Re: How much difference do more subs make?
[Re: Samir Kharusi]
#6273661  12/26/13 01:26 PM

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You're quote is for calculating the effect of increasing subexposure time. Not the effect of increasing the number of subs (which is actually doing the opposite, decreasing subexposure time). It also clearly states the equations are 'ignoring the contribution from dark noise'
Obviously, if the subexposure time approaches infinity, the read noise component goes away (that actually agrees with what I said), but that is not at all what we are talking about. We have short limited sub exposure times due to our *BLEEP* equipment (we cannot 'assume the subexposure duration is sufficiently long'). In those situations, read noise is very important.
All this page really asserts is that sky background also puts a limit on the subexposure duration. Which is true and is one of the limitations I stated in my post (simply as 'LP') in addition to many other limitations such as tracking error. But it doesn't have much to do with what we are talking about here.
I would love to see where those equations came from and how he got from step 1 to step 2, but there are no citations or references listed on that page, and he does not derive them himself.

Footbag
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Re: How much difference do more subs make?
[Re: dmilligan]
#6273892  12/26/13 03:47 PM

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You're quote is for calculating the effect of increasing subexposure time. Not the effect of increasing the number of subs (which is actually doing the opposite, decreasing subexposure time). It also clearly states the equations are 'ignoring the contribution from dark noise'
I think you are talking about two separate things. The original post is about adding subs to reduce noise. It's not about dividing the total exposure time into as many subs as possible. In that case, yes, more subs would mean more noise.
But, if you are asking how much does an additional [or extra] sub help, SNR increases proportionally to the square root of the number of subs. So, it depends on whether it's your second sub or your hundredth. If it's your second sub, it helps a lot. If it's your hundredth sub, it helps a little.

Jerry Lodriguss
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Re: How much difference do more subs make?
[Re: dmilligan]
#6274093  12/26/13 05:42 PM

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We have short limited sub exposure times due to our *BLEEP* equipment (we cannot 'assume the subexposure duration is sufficiently long'). In those situations, read noise is very important.
This is exactly what we are NOT talking about.
We are talking about using subs that are skynoise limited, not readnoise limited.
Jerry

dmilligan
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Reged: 11/21/13

Re: How much difference do more subs make?
[Re: Footbag]
#6274605  12/26/13 11:02 PM

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But, if you are asking how much does an additional [or extra] sub help, SNR increases proportionally to the square root of the number of subs. So, it depends on whether it's your second sub or your hundredth. If it's your second sub, it helps a lot. If it's your hundredth sub, it helps a little.
No, SNR increases proportionally to the square root of the amount of time, not the number of subs. These are close, but not the same thing. Adding more subs adds more time, but at a price: read noise. This means that SNR increases almost to the square root of the number of subs, but not quite.
You did agree that more subs for a given amount of total exposure time increases noise:
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In that case, yes, more subs would mean more noise.
The reason I brought that up was to prove that adding more subs does not quite improve SNR at the rate of sqrt(N), but by some amount slightly less.
Example:
9min exposure has 3x the SNR of a 1 min exposure 9x1min exposure has slightly more noise than a 9 min exposure.
Therefore if we are limited to 1 min exposures, going from 1 mins to 9 mins by adding 8 more subs (rather than by adding time, since we can't) means we will not improve the SNR by fully 3x, since we already agreed that 9x1min exposure has more noise than a single 9 minute one, therefore we will only improve it by something slightly less than 3x.
Since the contribution of read noise remains the same per sub (and the error keeps compounding the more subs we take), while the advantage of more time decreases per sub the more subs we take:
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If it's your second sub, it helps a lot. If it's your hundredth sub, it helps a little.
This means there will be a point of diminishing returns: The effect of photon noise will continue to decrease while the effect of read noise will remain constant, eventually the image will go from being photon noise dominated to read noise dominated in the deepest shadows. No matter how many more subs we add, we won't get rid of the read noise which is now the dominant source of noise, because we can't. Stacking does not eliminate read noise (if it did then 9x1min would have a better SNR than 1x9min, but we already agreed it doesn't).

Samir Kharusi
Carpal Tunnel
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Re: How much difference do more subs make?
[Re: dmilligan]
#6274804  12/27/13 03:23 AM

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Cannistra's equations are so straightforward that I never thought he needed to provide references. It's almost like providing a reference to 2+2=4. But of course what is obvious to one person may not be to another. His entire presentation there is aimed at quantifying Stacking Efficiency, basically an answer to your query as to how well One exposure of 9 minutes is replicated by 9 subs of one minute. But I feel that part of it is somewhat above the level of a very basic discussion we had here above. Nevertheless I tried to answer that particular aspect in a separate post here as related to DSLRs. I feel we have flogged this current horse to death. Some people will be happy with shooting a very bright nebulae, like M42, with an integraion time of mere seconds (me too!), others will prefer to go much deeper to capture the fainter tendrils and use an integration time of 20 hours using many different filters. But personally I would still not dare say that the latter wasted his time or that he might as well have stopped at ten hours. Here is a 20 sec integration on M42 (5x4sec):
and a very festive M13 (29x10sec):
A very happy New Year to all, full of dark, still and clear skies!

Footbag
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Re: How much difference do more subs make?
[Re: Samir Kharusi]
#6275124  12/27/13 10:33 AM

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Quote:
This means there will be a point of diminishing returns: The effect of photon noise will continue to decrease while the effect of read noise will remain constant, eventually the image will go from being photon noise dominated to read noise dominated in the deepest shadows. No matter how many more subs we add, we won't get rid of the read noise which is now the dominant source of noise, because we can't. Stacking does not eliminate read noise (if it did then 9x1min would have a better SNR than 1x9min, but we already agreed it doesn't).
To bring the theory into practice, you are saying that at some point in stacking; read noise becomes the dominant source of noise. Are you suggesting the SNR stops improving at this point? Not just that it follows the curve asymptotically(diminishing returns)?
Nearly everyone that commented in this thread qualified their statements saying "...given a properly exposed image". It is important that you overcome read noise with brute force.
So at what point does read noise become an issue? Given a specific setup, is it the 1,000th sub? 10,000th sub? 1,000,000th sub? Or is it a nonissue, because given a properly exposed image, with enough time, read noise wouldn't be significant? It would've been overcome with the combined signal. In other words, having read noise dominat a short exposure is a bad thing. But having read noise dominate an extremely long exposure is a sign that all of the other noises have been removed, and you probably have a very nice image at that point.
Also, if we use dark frames, we are removing the read noise. Does this further push back the point where the read noise becomes significant?
Also, if we have the read noise within the SNR formula, then why wouldn't a sub be the exact same thing as an interval of time?
Edited by Footbag (12/27/13 10:41 AM)

tazer
sage
Reged: 12/22/11
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Re: How much difference do more subs make?
[Re: whwang]
#6275171  12/27/13 11:01 AM

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In other words, S/N improves as we collect more photons. This is exactly why we want longer total integration time, and also exactly why we want more subs. They are the same thing.
SNR improves by either reducing N via stacking or increasing S by exposing for longer. Mathematically the ratio is the same, but stacking discards S if it's at or below the threshold of N (varies by stacking algorithm of course.) A stack of 20 5minute subs will retain more S than a stack of 100 1minute subs.

dmilligan
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Reged: 11/21/13

Re: How much difference do more subs make?
[Re: Footbag]
#6275382  12/27/13 01:25 PM

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To bring the theory into practice, you are saying that at some point in stacking; read noise becomes the dominant source of noise. Are you suggesting the SNR stops improving at this point? Not just that it follows the curve asymptotically(diminishing returns)?
Technically it will be asymptotic. When one term dominates another term doesn't mean that the dominated term suddenly vanishes at a certain point, but it will continue to get smaller and smaller (basically that is the definition of an asymptote).
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So at what point does read noise become an issue?
It depends, I'm not trying to calculate it here, I'm just trying to show that there is such a point, as many ppl have implied that there is not such a point. Surely this point will vary and will depend on quite a number of things: sensor performance, temperature, light pollution, etc. I imagine it is quite large (many hundreds if not thousands of subs) for a decent setup and dark skies. It's also very likely that most ppl will stop well before ever reaching this point, esp. with bright objects, b/c they would have deemed their result sufficiently noise free (Even the extreme guys who take days worth of total integration time).
Most counter arguments have been something like "well this guy's image is 40 hours". But all of those amazing images, while really long, typically have rather reasonable number of subs usually < 100 (per filter of course, when collecting more subs with a different filter, you are collecting a different signal, you don't stack images with different filters anyway, you merge them in different ways, like by assigning them to different color channels). They just have incredibly long subexposures (on the order of 30 minutes).
I'm not saying there is a point of diminishing returns on total integration time, I'm saying there's a point of diminishing returns on increasing the number of subs to get more total integration time. At some point the only way to get more useful total integration time is by increasing subexposure length, not by taking more subs.
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Also, if we use dark frames, we are removing the read noise. Does this further push back the point where the read noise becomes significant?
Yes it does, but it doesn't eliminate it completely, because it is impossible to eliminate read noise completely (we can do a very good job, esp. with modern sensor with really low read noise, but perfection is impossible). So there will still be a point somewhere where it does happen, it may be many hundreds of subs later, but it will surely happen.
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Also, if we have the read noise within the SNR formula, then why wouldn't a sub be the exact same thing as an interval of time?
If a sub were exactly the same as an interval of time, we could divide a given total integration time into as many subs as we wanted and never loose any quality > I could take many thousands of 1/10 sec subs and achieve the same result as an equivalent single exposure. Clearly I can't do that. That is the proof that more subs is not exactly the same as more integration time. It's close, but it's not exactly, and the difference is important, the difference is what puts an upper bound on the number of subs.

Footbag
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Re: How much difference do more subs make?
[Re: dmilligan]
#6275536  12/27/13 03:01 PM

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It depends, I'm not trying to calculate it here, I'm just trying to show that there is such a point, as many ppl have implied that there is not such a point. Surely this point will vary and will depend on quite a number of things: sensor performance, temperature, light pollution, etc. I imagine it is quite large (many hundreds if not thousands of subs) for a decent setup and dark skies. It's also very likely that most ppl will stop well before ever reaching this point, esp. with bright objects, b/c they would have deemed their result sufficiently noise free (Even the extreme guys who take days worth of total integration time).
At this point, I'd really like to see the math.
You seem to be falling back on the idea that the fact that you cannot divide an exposure time into further exposures without increasing noise. That is true and obvious. But, with multiple properly exposed image, where read noise is overcome by signal; I just don't understand how read noise could build relative to the signal. I agree that at some point read noise becomes dominant over other types of noise.
You seem to be arguing that subexposures aren't as effective at removing noise as additional integration time. In a sky limited image, which is what we are all dealing with, what is the alternative? How do you add more time without taking more subs?

tazer
sage
Reged: 12/22/11
Loc: Central North Carolina

Re: How much difference do more subs make?
[Re: Footbag]
#6276267  12/27/13 10:41 PM

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f a sub were exactly the same as an interval of time, we could divide a given total integration time into as many subs as we wanted and never loose any quality > I could take many thousands of 1/10 sec subs and achieve the same result as an equivalent single exposure. Clearly I can't do that.
Stacking reduces random noise at the cost of any signal that's below the noise threshold of whichever stacking algorithm you're using. Since random noise remains at a constant level, increasing exposure allows you to bring fainter signal levels above the random noise threshold thus preserving it in the stacking process.
At least that's my understanding of why 1 1sec sub is not equal to 10 1/10sec subs.

whwang
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Re: How much difference do more subs make?
[Re: tazer]
#6276358  12/27/13 11:36 PM

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At least that's my understanding of why 1 1sec sub is not equal to 10 1/10sec subs.
However, once the sub is long enough such that readout noise is no longer important, 1 30min sub is equal to 10 3min subs.

dmilligan
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Reged: 11/21/13

Re: How much difference do more subs make?
[Re: whwang]
#6278318  12/29/13 12:20 AM

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But, with multiple properly exposed image, where read noise is overcome by signal
It's not like read noise just goes away once you get to a high enough signal, sure the ratio gets better, but read noise is still there, think of read noise as simply an error in the measurement (which is what it is). the ratio goes down as the signal goes up, but the amount stays constant.
A signal of 2 e on a sensor with a read noise of 2 e is completely 'lost in the noise', you're measuring 2±2 e. When you measure a signal of 10 e, the ratio is much better (5:1), but you're measurement is still 10±2 e
So lets say you have a 'well exposed' sky background 'signal' of 100 e per frame, and some signal, above that at 200 e per frame with read noise of 2 e. With one frame we'd have 200±2e. But the 200e signal is going to have photon noise of sqrt(200) ~= 14. So really we'd be measuring 200±16e. The photon noise is way more dominant than our read error, and the SNR is 200:16 = 12.5. So we have a lot of room for improvement. But there's going to be a limit, we can keep removing photon noise with more subs, and get that 14 down really small, but there's still going to be that 2e of measurement noise. Our SNR is not going to get better than 200:2 no matter how many more subs we add.
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I just don't understand how read noise could build relative to the signal. I agree that at some point read noise becomes dominant over other types of noise.
It doesn't build relative to signal, it remains constant relative to the signal. Since other types of noise diminish thanks to stacking, it becomes the limiting factor. Since more subs won't eliminate it, the only thing that would eliminate it is longer subs, to push up the difference between the signal level and the read noise level.
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You seem to be arguing that subexposures aren't as effective at removing noise as additional integration time. In a sky limited image, which is what we are all dealing with, what is the alternative? How do you add more time without taking more subs?
You don't, you can't, nothing you can do can improve your image any further. Does it not make sense that this should be true? That in an image with a sky background, there is some theoretical limit to the maximum amount of information you can get, some limit to how deep you can go? If this weren't true, then I should be able to do astrophotography during the day given enough subs. It's pretty easy to get a 'well exposed' image of the sky during the day, do you think I should just be able to combine 10 billion 1/100s 'well exposed' photos of the sky during the day and get an image of M42? What is the difference between daytime skyglow and nighttime skyglow other than intensity?
If we aren't limited by skyglow, does it not make sense that there is still some maximum limit to what we can do that is based on the limitations of the sensor?
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However, once the sub is long enough such that readout noise is no longer important, 1 30min sub is equal to 10 3min subs.
Nope, that never happens, they will be really really close, but because you read the sensor 3 times instead of one, you repeated some tiny error three times instead of just once, and the SNR will be ever so slightly lower. Read noise might not be "important", but it doesn't just vanish. If you just compared the two you'd be hard pressed to see the difference, but small as it may be it has an important effect, and that is that it puts an upper bound on the maximum number of useful subs. It makes the best SNR you can get asymptotic to some value.

Footbag
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Re: How much difference do more subs make?
[Re: dmilligan]
#6278668  12/29/13 08:33 AM

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The SNR continues to improve in your example even if read noise stays constant relative to signal. This is due to the other forms of noise infinitely approaching 0 as the read noise remains constant relative to it. I think everyone must have a different definition of diminishing returns. Perhaps we are asymptotically coming to an agreement.
Again, at this point, you have a pretty darn nice image and wouldn't be looking at a formula to improve it. At this point you speak of, read noise WOULD be insignificant. It would be buried in strong signal. What you may call very close, others would call insignificant.
You make a good point about read noise, though. Proper calibration is very important to remove the noise that doesn't stack out. By limiting the read noise, you are really getting a jump on the SNR. And proper exposure is important for getting the signal up above that read noise. By sticking to a few rules, you can mitigate the read noise.
Edited by Footbag (12/29/13 08:41 AM)

tazer
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Reged: 12/22/11
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Re: How much difference do more subs make?
[Re: whwang]
#6278704  12/29/13 09:02 AM

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At least that's my understanding of why 1 1sec sub is not equal to 10 1/10sec subs.
However, once the sub is long enough such that readout noise is no longer important, 1 30min sub is equal to 10 3min subs.
Not exactly. In a 30minute sub you've got 30minutes of signal. In 10 unprocessed 3minute subs you have 30minutes of signal, but when you stack them you discard a portion of that signal for the sake of noise reduction. If all of the signal in your 3minute sub is above the noise threshold, then yes, 10 3minute subs are equal to 1 30minute sub.

Footbag
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Re: How much difference do more subs make?
[Re: tazer]
#6279586  12/29/13 05:25 PM

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At least that's my understanding of why 1 1sec sub is not equal to 10 1/10sec subs.
However, once the sub is long enough such that readout noise is no longer important, 1 30min sub is equal to 10 3min subs.
Not exactly. In a 30minute sub you've got 30minutes of signal. In 10 unprocessed 3minute subs you have 30minutes of signal, but when you stack them you discard a portion of that signal for the sake of noise reduction. If all of the signal in your 3minute sub is above the noise threshold, then yes, 10 3minute subs are equal to 1 30minute sub.
I belive that theoretically the stacking would add all the pixel values and divide by the number of subframes. This is what the forulas for SNR seem to assume.
Perhaps average or median stacking method or others would dump some of the pixel values. Visually, it seems to help. I wouldn't know hor to carry the SNR ratio through these methods of stacking.

scottk
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Re: How much difference do more subs make?
[Re: Footbag]
#6279715  12/29/13 06:34 PM

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I'm very curious about this subject. I have about 3 hours of 104 second subs on m33. I'm in a red/white l.p. zone. The outer arms are still pretty faint.
I wonder if I go up to like 20 hrs of 104 second subs what will happen.

tazer
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Re: How much difference do more subs make?
[Re: Footbag]
#6279741  12/29/13 06:53 PM

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Perhaps average or median stacking method or others would dump some of the pixel values. Visually, it seems to help.
Visually it does help but I guess my point is that, short of a perfect QE/noiseless sensor, you can't "go deeper" just by adding more subs. I think some of the higherend, low noise, and cooled CCD's go a long way towards achieving this goal but not in the DSLR world.
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I wouldn't know hor to carry the SNR ratio through these methods of stacking.
Good question. I hadn't actually thought through that, but it does appear to be calculated as others have done in this thread (i.e., sqrt(# of subs)). The DSS site is in agreement:
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The signal to noise ration in increasing with the square root of the number of combined frames regardless of the exposure time of each frame. This is true with all the combining methods (average, median, kappasigma clipping, autoadaptive weighted average, ...) except entropy weighted average since this one in using the entropy to weight each pixel and thus is increasing the noise that is a big entropy contributor.

whwang
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Re: How much difference do more subs make?
[Re: scottk]
#6280314  12/30/13 12:46 AM

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I'm very curious about this subject. I have about 3 hours of 104 second subs on m33. I'm in a red/white l.p. zone. The outer arms are still pretty faint.
I wonder if I go up to like 20 hrs of 104 second subs what will happen.
You will see improvement. The theoretical maximum improvement you may see is a factor of 2.6x better in S/N. To reach this, your flat field, bias, and dark need to be very well calibrated, otherwise errors in the calibration will "eat" the S/N improvement.
Edited by whwang (12/30/13 09:10 PM)


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