The maths of CMOS, read noise, and sub length

[Placidus (Mike and Trish)]

Introduction

We continue to see claims that with the arrival of the CMOS chip, the days of the 1 hour sub are over, and that we can start routinely doing subs of a few seconds only.

There can be compelling reason for really short subs, such as poor polar alignment, dodgy guiding, shallow quantum wells, light pollution, wind buffet, airplanes, etc. Let us assume that we have an arbitrarily dark site, an observatory dome not under a flight path, and permanent polar alignment.

We will derive an easy formula for signal to noise ratio as a function of quantum efficiency, source brightness, readout noise, sub duration, and total number of subs.

We will see that increasing the quantum efficiency alone cannot produce a truly huge improvement over say a 16803 chip.

We will see that for a sufficiently faint target, ten 1-hour subs will yield a 600% better signal to noise ratio than the same exposure time in 1-second subs.

We will also see that it is not a good idea to be testing really short subs on ultra-bright easy targets (say M42) if we are later on hoping to photograph truly faint things.

The results will bear out and put in context comments made by Peter Ward, Rick Stevenson, myself and several others in other forums.

Some easy mathematics

Let the quantum efficiency be Q, the photon flux at a given pixel be P photons per second, the duration of a single sub be T seconds, the number of subs be N.

The signal, or total number of photo-electrons captured in a single sub will be QPT. The higher the quantum efficiency, the brighter the object, and the longer the subs, the more signal we will have.

There are two components to the noise that we are interested in here. The first is shot noise. If we have a very faint source of photo-electrons appearing at random intervals, then during any short interval, we may or may not get a photo-electron, so for some subs our count may be zero, in others it may be 1, in others 2, etc, but not always the same.

The actual number captured in a single sub will follow a Poisson distribution, with an expected value (population mean) for any single sub of QPT as mentioned already, a variance also of QPT, and a standard deviation of root(QPT).

The second source of noise that we are interested in here is read noise. Let the read noise in a single sub have a standard deviation of R electrons.

Over a set of N subs, the expected total signal will be NQPT. That makes sense: the more subs, the higher the quantum efficiency, the brighter the target, and the longer each sub, the more photons we will collect. Dim subjects will have poor signal.

At a single pixel, the two sources of noise under discussion (shot noise and read noise) are uncorrelated with each other and uncorrelated from frame to frame. The variances (the squares of the standard deviations) add linearly.

The total variance is therefore N x (shot noise variance + read noise variance) = N ( QPT + RR ), and the total noise is the square root of that.

(Because we can't type superscripts here, I've written the read noise variance R x R as RR.)

The signal to noise ratio is therefore NQPT / root [ N (QPT + RR)].

We will now examine some important and interesting limiting cases.

The first limiting case is for God's Own Camera, where Q = 1, and R = 0. This does not yet exist. The signal to noise ratio is snr = NPT / root [ NPT] = root(NPT). Since P is a constant set by the Almighty, the signal to noise ratio is proportional to root(NT). With the perfect camera, we can choose to do one ten-hour sub, or ten one-hour subs, and get the same result.

The second is an arbitrarily bright source. Think lunar photography. In the limit as P goes to infinity, QPT + RR approximates QPT as closely as we like. The signal to noise ratio becomes root(NQPT). Since Q is set by the manufacturer and P by the choice of target, the signal to noise ratio is proportional to root(NT). Thus again we can choose to do 1 ten second sub, or ten 1 second subs, and the snr will be the same.


The real world of very faint targets

Our third limiting case is the big one. We will imagine an incredibly faint source, where P (the number of photons per second arriving at a given pixel) is as small as we choose. Recall that the signal to noise ratio is NQPT / root [ N (QPT + RR)]. As we choose ever fainter sources, QPT becomes tiny compared with RR, and in the limit, the signal to noise ratio becomes

snr = NQPT / root [NRR] = QPT root(N) / R.

What does this mean?

(1) Traditional CCD chips have a quantum efficiency around 0.3 to 0.7. Even a perfect chip cannot have Q > 1, so if it takes tens of hours to photograph the outer chevrons of the helix using particular scope with a Q of 0.5, no camera in creation can do it in less than half that time. The advantage of a CMOS chip has little to do with efficiency or sensitivity.

(2) The big advantage of a CMOS chip is the greatly reduced read noise. If, all else being equal, we can reduce the read noise R by a factor of ten, then on a sufficiently faint object, we will improve the signal to noise ratio by a factor of ten. That is astonishing and is a good reason for hoping that Santa will produce a 4000x4000 chip with 100Ke quantum wells.

(3) All else being equal, then for the very faintest targets, the snr is proportional to exposure time T, but proportional only to the square root of the number of subs N. That means that for very faint targets, it will be hugely better to do ten 3600 second subs than 36,000 one-second subs. It will be better by a ratio of 3600 root(10) to 1 root(36000), or a ratio of six to one. Long subs rule.

The caveats are "very faint targets" and "all else being equal". Thus if your mount can't do a 1-hour sub, or there are intermittent clouds about, or aeroplanes, or wind buffet, or your quantum wells are too small, or you are overwhelmed by sky glow, or you only want to photograph the moon, shorter subs make sense. If you don't have those limitations, longer subs make sense.

(4) Test your new CMOS camera on very faint targets. Under super-bright conditions, the read noise is, as already discussed, almost irrelevant. It is not so informative to be doing your first test shots on easy targets like M42 or the Lagoon, unless that is all you are ever going to photograph. Have a crack at the outer chevrons of the Helix, or a crack at the faint OIII super-bubble in NGC 602.

(5) We will always want to do enough subs to be able to do good data rejection to get rid of satellite trails and cosmic rays, and to dither so that we can also handle hot pixels and bad columns using statistical data rejection. However, ten 1-hour subs is adequate for good data rejection. 36,000 1 second subs will not add any important protection and is kind of unmanageable.

Someone else might like to write an equation for signal to noise ratio that includes sky glow.

[glend (Glen)]

Mike n Trish, my initial reaction was to just 'let this go through to the keeper', but hidden in your thesis is the attitude we so often hear. While your theory is of course sound, the 'caveats' and 'all things being equal' are real gotcha's in many peoples lives, and i would add another one - budget. In a perfect world of a very dark site in an observatory with a good pier, a high end mount perfectly aligned and corrected, no wind, perfect seeing, etc etc your long sub ccd nirvana camera will out perform on SNR. However, many of us lack the financial ability ( through lack of funds or other priorities), to build the nirvana system. We have to deal with the short comings of our particular situations, and thus compromises are required in order to produce and enjoy imaging. What the new, low entry cost, very low noise, reasonable QE (say 60%) cmos cameras provide to us humble masses, is the ability to achieve more than we possibly could in the past with our obvious constraints.

[Shiraz (Ray)]

I doubt that anyone has seriously proposed 36,000* 1 second subs. However, let's look at something far more practical to see what is going on.

With a camera having a RN of 10e and using 1 hour subs, the total RN over a 16 hour period will be SQRT(16)*10 = 40. If you have a camera with a RN of 2e and take 1 hour subs you will get a total RN of SQRT(16)*2 = 8. the total number of photons detected will be the same in each case as will the shot noise, so the low RN camera will provide higher SNR.

Now try shorter subs on the low read noise camera. If you take 96* 10 minute subs, the read noise is SQRT(96)*2 = 19.6. ie with subs of 10 minutes, the total read noise is still less than that half that of the camera with 10e RN and 1 hour subs. The camera is still exposed for 16 hours so the number of photons detected and the shot noise will both be the same as before. Even with 10 minute subs, the low RN camera will provide better SNR than the 10e camera with 1 hour subs.

Clearly there will be a crossover point where total RN of the 2 cameras is the same. That will be when you have 25x as many subs with the 2e RN camera as with the 10e RN camera. If you have the same total exposure time, you will still have the same signal and shot noise and the 2e RN camera will produce identical SNR to the 10e RN camera - but with subs that are 1/25 as long.

Low read noise does not allow ridiculous results and it certainly does not reduce the total exposure time by orders of magnitude, but it does allow very much shorter subs to be used than previously. The situation for sky limited performance is even more striking and very short subs can be used in some circumstances. However, there is no gain in SNR available with properly exposed sky-limited subs, just the ability to use very short subs.

edit: in your terms, from the dim target equation and the same total exposure, a camera with 2e read noise and one with 10e read noise will have equivalent SNR when:

QP*(T2)*(sqrt(N2))/(R2) = QP*(T10)*(sqrt(N10))/(R10)

now (N2)*(T2) = (N10)*(T10) for the same total exposure, so

(T2)/(R2*R2) = (T10)/(R10*R10) or

T2 = (T10)*(R2*R2)/(R10*R10) = (T10) /25

ie, for equivalent SNR and the same total exposure, the sub lengths can be 1/25 as long with the 2e RN camera

[lazjen (Chris)]

So my takeaway from this is:

* I'll need a lot more subs to compare to the deep single subs (ignoring the extreme 1sec comparison being used for the match, but more realistic exposure durations)
* I'll need to be able to process a lot of subs (I am getting used to it already)
* I won't be as concerned about losing an hour sub versus a 10 or 5 min sub
* And I need to pick appropriate targets for my system and location and not expect Hubble deep field imaging from an inexpensive setup (that cost probably much less than 10-20% of the system that can do 1 hour subs).

Seems reasonable to me.

Q: Although truly extreme, would it be possible to process those 36000 x 1 sec subs? Could they be grouped into batches of say 100, integrated, then those results integrated, etc until down to the final single integrated result?

[Placidus (Mike and Trish)]

Originally Posted by glend

Mike n Trish, my initial reaction was to just 'let this go through to the keeper', but hidden in your thesis is the attitude we so often hear. While your theory is of course sound, the 'caveats' and 'all things being equal' are real gotcha's in many peoples lives, and i would add another one - budget. In a perfect world of a very dark site in an observatory with a good pier, a high end mount perfectly aligned and corrected, no wind, perfect seeing, etc etc your long sub ccd nirvana camera will out perform on SNR. However, many of us lack the financial ability ( through lack of funds or other priorities), to build the nirvana system. We have to deal with the short comings of our particular situations, and thus compromises are required in order to produce and enjoy imaging. What the new, low entry cost, very low noise, reasonable QE (say 60%) cmos cameras provide to us humble masses, is the ability to achieve more than we possibly could in the past with our obvious constraints.

Excellently put, Glen. My very first astrocamera was a Canon EOS 20Da, which had a cmos chip. I was a very good boy for mentioning the caveats and all-else-being-equals.

Originally Posted by Shiraz

I doubt that anyone has seriously proposed 36,000* 1 second subs. However, let's look at something far more practical to see what is going on.

With a camera having a RN of 10e and using 1 hour subs, the total RN over a 16 hour period will be SQRT(16)*10 = 40. If you have a camera with a RN of 2e and take 1 hour subs you will get a total RN of SQRT(16)*2 = 8. the total number of photons detected will be the same in each case as will the shot noise, so the low RN camera will provide higher SNR.

Now try shorter subs on the low read noise camera. If you take 96* 10 minute subs, the read noise is SQRT(96)*2 = 19.6. ie with subs of 10 minutes, the total read noise is still less than that half that of the camera with 10e RN and 1 hour subs. The camera is still exposed for 16 hours so the number of photons detected and the shot noise will both be the same as before. Even with 10 minute subs, the low RN camera will provide better SNR than the 10e camera with 1 hour subs.

Clearly there will be a crossover point where total RN of the 2 cameras is the same. That will be when you have 25x as many subs with the 2e RN camera as with the 10e RN camera. If you have the same total exposure time, you will still have the same signal and shot noise and the 2e RN camera will produce identical SNR to the 10e RN camera - but with subs that are 1/25 as long.

Low read noise does not allow ridiculous results, but it does allow very much shorter subs to be used than previously. The situation for sky limited performance is even more striking and very short subs can be used in some circumstances.

Cheers, Ray. We continue to be in total agreement. You've done a real-world comparison between a real CMOS and a real CCD, and shown that you can do shorter subs with the CMOS and get a better result. True !!! But it is still the case that if you can go a bit longer, your result is even better again !

I'm currently looking around for a 16 megapixel CMOS chip with a good software development kit. Until I can find one that I can fully control myself using C sharp, I'm stuck with my existing Apogee 16803, which comes with a really good DLL interface for back-yard developers such as myself.

Best,
Mike

[Placidus (Mike and Trish)]

Originally Posted by lazjen

So my takeaway from this is:

* I'll need a lot more subs to compare to the deep single subs (ignoring the extreme 1sec comparison being used for the match, but more realistic exposure durations)
* I'll need to be able to process a lot of subs (I am getting used to it already)
* I won't be as concerned about losing an hour sub versus a 10 or 5 min sub
* And I need to pick appropriate targets for my system and location and not expect Hubble deep field imaging from an inexpensive setup (that cost probably much less than 10-20% of the system that can do 1 hour subs).

Seems reasonable to me.

Q: Although truly extreme, would it be possible to process those 36000 x 1 sec subs? Could they be grouped into batches of say 100, integrated, then those results integrated, etc until down to the final single integrated result?

Hi, Chris,

I think your comments, and Ray's and Glen's, help focus my mind on what it is I'm really trying to say. It's:

(a) When doing your testing, try some faint things. The big advantage of the CMOS chip's low read noise shows up with really faint things.
(b) Once you've got your CMOS chip, it is always better to do the longest subs that are practical given all the real-world issues like wind buffet, clouds, aeroplanes, guiding and tracking, etc.

Huge stacks: Suppose we went for something more reasonable like 600 one minute subs. My own software, GoodLook 64, would have to do that in say 10 batches of say 60 subs (depending on chip size), producing 10 partial 32 bit FITS stacks. (It needs to hold all the subs in memory for good reasons unrelated to the task at hand). But that set of 10 partial subs would be very easy to stack to a final image. I already used that technique for a big mosaic on the Tarantula.

Cheers,
Mike

[lazjen (Chris)]

Thanks Mike. I think I might try doing an image sometime in the future with a large number of subs per channel ( > 200) to see what the processing challenge is like.

I will try to do faint(er) objects from my site for testing but I need to work out what's the appropriate settings I need to use - exposure time, gain setting, etc.

Even with my slightly haphazard or adhoc testing I've done with the camera so far, I achieved far better results for LRGB from my backyard than I've ever been able to do so in the past with other cameras. In fact, I had basically given up on LRGB and was expecting to really get NB only. For the price, the ASI1600 is a real game changer.

BTW, when you get your desired CMOS system, send your "discarded" 16803 camera my way for, er, "comparative testing purposes".

[Shiraz (Ray)]

I suppose that the other thing that could be said is that, if you can get the same (or better) result from much shorter subs, there is no need for deep wells.

M&T, the other issue that should be noted is that current CMOS chips have small pixels - they do not match well with long fl scopes, so let's hope that someone eventually brings out a mono CMOS full frame camera with large pixels.

[SimmoW (SIMON)]

Excellent, mature discussion chaps. I hope Ray is right, there has to be an effective middle ground. 1 second subs are extreme, Mike's explanation proves that. 10 min ones are far more real world and attainable.

[Shiraz (Ray)]

Originally Posted by Placidus

Introduction


Someone else might like to write an equation for signal to noise ratio that includes sky glow.

FWIW, suggest that the equation for sky limited imaging is:

SNR = N*Q*T*P / sqrt(N*Q*T*P + N*Q*T*S + N*T*D + N*RN*RN)

where S is the sky signal and D the dark current

the generally agreed setting for sub length is to make N*Q*T*S the dominant term in the denominator, at which point the SNR is independent of RN (ie there is no gain in SNR from using short subs).
Normally, subs will be long enough that N*Q*T*S >~ 10*N*RN*RN - or something similar. ie, N cancels out and the required sub length T scales directly with RN*RN, the same as for narrowband. Since sky-limited exposures with normal cameras come in at 10-20 minutes for a 10e RN camera, an equivalent system with a 2e RN camera could do the same job with 24-48 second subs.

[Atmos (Colin)]

Here is a camera for you MnT
A little over 2e- RN, 350k e- wells and pixel sizes that really suit your long FL

It is a great discussion in this thread btw

[SimmoW (SIMON)]

Hmm, I bet only God could afford that camera!!

[Slawomir (Suavi)]

Originally Posted by Atmos

Here is a camera for you MnT
A little over 2e- RN, 350k e- wells and pixel sizes that really suit your long FL

It is a great discussion in this thread btw

Sensor diagonal = 87mm...now you are talking! But it weighs 12 kg...

I nearly pulled a trigger but then I noticed that 2e read noise is only for 100kHz readout...16mp would probably take 2 minutes to download! I think I will have to wait for an improved model...

[Placidus (Mike and Trish)]

Thanks everyone for sensible and helpful comments extensions and advice. I'm learning.

Originally Posted by Shiraz

I suppose that the other thing that could be said is that, if you can get the same (or better) result from much shorter subs, there is no need for deep wells.

That had vaguely occurred to me too. A good point.

Originally Posted by Shiraz

M&T, the other issue that should be noted is that current CMOS chips have small pixels - they do not match well with long fl scopes, so let's hope that someone eventually brings out a mono CMOS full frame camera with large pixels.

Another very good point, not just for CMOS but in general.

Originally Posted by Shiraz

FWIW, suggest that the equation for sky limited imaging is:

SNR = N*Q*T*P / sqrt(N*Q*T*P + N*Q*T*S + N*T*D + N*RN*RN)

where S is the sky signal and D the dark current ... ... ...

Thanks, Ray, that's excellent. You've written the equation in a way that is clear unambiguous and intuitive. It'll take me a while to scrape together the relevant numbers and plug them in.

Originally Posted by Atmos

Here is a camera for you MnT
A little over 2e- RN, 350k e- wells and pixel sizes that really suit your long FL

It is a great discussion in this thread btw

Wow! That is a big beast. Some of the 60x60 mm real estate would be wasted on us mere mortals with 50 mm filters.

[lazjen (Chris)]

Originally Posted by Shiraz

FWIW, suggest that the equation for sky limited imaging is:

SNR = N*Q*T*P / sqrt(N*Q*T*P + N*Q*T*S + N*T*D + N*RN*RN)

where S is the sky signal and D the dark current

the generally agreed setting for sub length is to make N*Q*T*S the dominant term in the denominator, at which point the SNR is independent of RN (ie there is no gain in SNR from using short subs).
Normally, subs will be long enough that N*Q*T*S >~ 10*N*RN*RN - or something similar. ie, N cancels out and the required sub length T scales directly with RN*RN, the same as for narrowband. Since sky-limited exposures with normal cameras come in at 10-20 minutes for a 10e RN camera, an equivalent system with a 2e RN camera could do the same job with 24-48 second subs.

How do I measure the sky signal for this equation?
I assume the Q figure is different for each channel or narrowband being used?
And for the ASI1600, the gain setting determines the read noise, right?

If I know these 3 things, I can get a good value for T? Then effectively after that, increasing N should increase SNR at nearly the best possible "rate" I can theoretically achieve (for the sky, camera, etc)?

Is this right?

[Camelopardalis (Dunk)]

Originally Posted by lazjen

How do I measure the sky signal for this equation?

Can't you just point it a patch of sky near your target and measure the background values in between stars?

[Placidus (Mike and Trish)]

Originally Posted by Shiraz

You must have read my mind. Attached quick test on the edge of the Helix - not in any way comparable with your effort, but nonetheless, there is clearly something beginning to show in there. This is with 7.5 hours of 10 minute subs with the ASI1600 on a 10 inch f4 scope, software binned, but no noise reduction applied. Taking everything into account, my system would have about 1/2 the geometric sensitivity cf yours and I exposed for about 0.4x as long, so this possibly looks like a confirmation of sorts - the theory may just work after all

I don't normally like to post an image in someone else's thread, so if you find it annoying, will happily remove the post.

Wow! Your image is highly relevant and much appreciated! We had to do 2x2 on-chip binning to get anything like that result.

Originally Posted by lazjen

How do I measure the sky signal for this equation?
I assume the Q figure is different for each channel or narrowband being used?
And for the ASI1600, the gain setting determines the read noise, right?

If I know these 3 things, I can get a good value for T? Then effectively after that, increasing N should increase SNR at nearly the best possible "rate" I can theoretically achieve (for the sky, camera, etc)?

Is this right?

I just had a crack at measuring the sky signal. I looked at a single 1 hour dark and flat corrected frame near the south galactic pole (in Sculptor) taken around midnight. The brightness of the starless background was 1650 ADU.

That very much surprised me. It is a high number, much higher than our dark current or the read noise, expressed in consistent units.

With a gain of 1.5 e-/ADU, 1650 ADU is 2,475 electrons. Taking Q = 0.5, that's a photon flux of 4950 photons per pixel per hour, or 1.375 photon per pixel per second.

Is that typical for rural sky? To tell, we need to divide by the unobstructed aperture, and also divide by the angular view of the sky per pixel.

1.375 photons per pixel per second / 0.14 sq meters / 0.3 square arcsec view per pixel = 32.7 photons per square meter of unobstructed aperture per square arcsec of sky.

From that, one can work out the sky brightness in mag/square arcsec, but the calculations are horrific, requiring knowledge of sky absorption, reflectance of primary and secondary mirrors, bandpass of filters, coatings on correctors and ccd chamber, etc, etc, etc. I end up with a value somewhere between 21 and 22 mag/sq arcsec, which is pretty pleasing, but there are so many assumptions that all I can say is yes, 1650 ADU per hour is actually pretty much to be expected .

The really good news is that for working out signal to noise ratio, you don't actually need to do any of that. You just need your actual raw measurement of photons per pixel per hour, and plug it into the formula for snr.

[RickS (Rick)]

I'd like to play but I'm sitting in a hotel room in Singapore preparing slides for a conference presentation. See if you can keep the discussion going for a couple more days please

[lazjen (Chris)]

Should be easy Rick - it'll take me a while to understand things enough...

Ok, we're aiming to solve T given we know or choose other parts of the equation. The main, difficult variable is S. Also, for practical purposes (darks), we're going to limit T to various values (60 sec, 600 sec, etc).

The sky signal will vary based on the time of the night, phase of the moon, etc. Is it best for T to be such that you're under sky-limited exposure length, or is going over ok (but "wasting" time)?

[glend (Glen)]

Originally Posted by lazjen

Should be easy Rick - it'll take me a while to understand things enough...

Ok, we're aiming to solve T given we know or choose other parts of the equation. The main, difficult variable is S. Also, for practical purposes (darks), we're going to limit T to various values (60 sec, 600 sec, etc).

The sky signal will vary based on the time of the night, phase of the moon, etc. Is it best for T to be such that you're under sky-limited exposure length, or is going over ok (but "wasting" time)?

Chris, Ray has produced a very useful chart that shows optimal broadband sub times for differing Gain settings , for three different f ratios and two different levels of darkness (SQM), it is published in the ASI1600 thread here.

I would suggest eliminating variables such as moon effect, and time of night light effects ( like nearby sports fields, neighbors, etc).
Of course narrowband is not as suscepable to 'external influences' imho.