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Expected FWHM (or HFD) Values and Spatial Resolution

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#1 rgsalinger

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Posted 14 August 2017 - 10:16 AM

I've become more critical over the past couple of years about focus. I'm wondering is there is a formula lurking out there that relates the focal length of a telescope, the pixel size on the chip, the guiding error (rms) and the seeing to give a theoretical limit for FWHM or HFD. I've found things online that get me some information but I keep wondering what to expect as a theoretical minimum FWHM (in arc seconds) from an particular imaging system.

 

I originally thought that as long as my guiding was well under the seeing, that error term would be irrelevant. I also thought that the Nyquist sampling equation meant that the minimum I FWHM I could expect was around 3 pixels no matter what the scope was doing. And, finally I thought that meant that as long as 3 times the image scale was a lot more than the seeing, my FWHM is determined by the image scale and nothing else. 

 

Now I'm not sure at all that I understand how all these things go together. So, if anyone has some insights, I'd love to get them. I just put a new camera (ASI1600) on my PW 12.5 and I immediately got (apparently) around 1.5 arc seconds for a star in focus. The old camera (STF8300) with larger pixels was giving around 2.2 arc seconds with short focused exposures. This is confusing to me because the image scale in both cases if really small .31 for the new camera and .44 for the old camera and both (x3) are well under the seeing where the scope is located. 

 

 



#2 Goofi

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Posted 14 August 2017 - 02:51 PM

I'm not sure if I've got something like this in my notes, but in the dark cobwebs of my memory I think I do. I'll check later today and see if I can dig them up.

 

But, as for your recent star test.  We normally have pretty good seeing - but even still, a few nights ago was exceptional here.  I noticed my FWHM were about 10% less than normal ... I chalked it up to seeing since nothing else in my setup or environment changed.  I'm pretty well undersampled (your setup means you're oversampled),  I'm not sure if that helps or hurts in this situation but it's something to be aware of.



#3 jhayes_tucson

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Posted 14 August 2017 - 02:56 PM

I'll assume that your PW system is a F/8 system with a efl of 2,540 mm.  With that aperture, seeing conditions will almost always determine the size of the blur disk for star images in the image plane.  With 1 arc-sec conditions, you can expect a blur size of around 12 microns in the focal plane.  With 3.8 micron pixels, you'll have roughly three samples across the blur diameter.  That's a reasonable sampling rate although the signal will be a bit low with pixels that small.  Still, it should work just fine.

 

In general, as long as your pixels are smaller by roughly a factor of two that the seeing blur disk, the size of the pixels will not be a significant factor for FWHM values that you measure.  The atmospheric conditions will be the main factor.  I generally use about 1 arc-second seeing as the best that I might expect; however, it may be smaller at a really good site.  Having said that, most folks experience conditions that are roughly twice that bad.  I think that the reason that you measured better results with the new camera is likely because either the seeing was much better--or perhaps because your focus might have been better.

 

John



#4 freestar8n

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Posted 14 August 2017 - 03:53 PM

I've become more critical over the past couple of years about focus. I'm wondering is there is a formula lurking out there that relates the focal length of a telescope, the pixel size on the chip, the guiding error (rms) and the seeing to give a theoretical limit for FWHM or HFD. I've found things online that get me some information but I keep wondering what to expect as a theoretical minimum FWHM (in arc seconds) from an particular imaging system.

 

I originally thought that as long as my guiding was well under the seeing, that error term would be irrelevant. I also thought that the Nyquist sampling equation meant that the minimum I FWHM I could expect was around 3 pixels no matter what the scope was doing. And, finally I thought that meant that as long as 3 times the image scale was a lot more than the seeing, my FWHM is determined by the image scale and nothing else. 

 

Now I'm not sure at all that I understand how all these things go together. So, if anyone has some insights, I'd love to get them. I just put a new camera (ASI1600) on my PW 12.5 and I immediately got (apparently) around 1.5 arc seconds for a star in focus. The old camera (STF8300) with larger pixels was giving around 2.2 arc seconds with short focused exposures. This is confusing to me because the image scale in both cases if really small .31 for the new camera and .44 for the old camera and both (x3) are well under the seeing where the scope is located. 

This is a classic case of actually doing the experiment to see how things really work - and if simple models and assumptions apply rigorously.  They usually don't.

 

The pixels aren't truly isolated from each other and Nyquist is a mathematical theorem that only applies if the requirements of the theorem are met - and they never are in practice.  Nyquist refers to sampling a band-limited signal - and then filtering the result - and reconstructing the full signal.  In this case you are simply sampling the signal if blocky pixels - and then analyzing those blocky pixels.  It's completely different.

 

Then there are details of how exactly the software calculates fwhm.  Depending on how it is done, the larger pixels will themselves somewhat swell the star spot.

 

And finally there are indirect benefits of smaller pixels - such as the ability to recognize in-focus better - so that your focus is itself better with the smaller pixels.

 

I realized this stuff many years ago when trying to image the diffraction pattern of stars at very fine scale.  Theoretically it should be visible with good contrast at a certain pixel size - but I always found additional enlargement was needed.  This is consistent with planetary imagers  over-sampling planets beyond what you would expect based on diffraction and nyquist alone.

 

So what I have been saying about this issue is - smaller pixels have improved benefit beyond simple nyquist assumptions and there is no sudden point where that benefit stops.  If you use smaller pixels you will likely see continued improvement in fwhm - and a rough limit would be about 1/6 fwhm.  I am at 0.28" per pixel and my fwhm's are in the low 1" - so I am under-sampled.

 

This all refers to looking at a single sub-exposure.  If you align and stack many exposures, there is even more benefit from finer sampling.

 

Frank



#5 rgsalinger

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Posted 14 August 2017 - 04:50 PM

 I like both of these answers. There's the theoretical model and the idea that things might not be the same in actual practice.

 

What I'm getting from both, though, is that I might get a bit under the seeing under the right sampling conditions, but I certainly ought to be able to resolve at F8 and 2540mm down to the level of the seeing. If I'm not,  then I'm not focusing well enough or I've got a (collimation?) error or some other problem. There seemed to be very little difference (measured with the old camera) between long and short exposures. My guiding (not my talent, my equipment) is usually around .3 arc seconds in each axis. That can't be much of a contributor to what I keep thinking is FWHM a little higher than the seeing with the original camera.

 

I was hoping for a formula, but I guess that the point is that eventually it's just seeing limited once you go long enough in focal length with small enough pixels. 

 

I sometimes get to use a 25" F8 RC scope. With that scope and my STF8300 I was able to get about 15 percent under what the seeing monitor said was happening at our dark sky site. I've never been able to do that with the larger camera we have on the mount now. I can get down to right around the seeing (using Focusmax) but I've never seen anything less than the 2" we generally have.

 

Rgrds-Ross



#6 jhayes_tucson

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Posted 14 August 2017 - 05:01 PM

Frank,

Whenever you claim that theory disagrees with the real world, I become concerned about how that contributes to growing skepticism surrounding the foundations of both science and engineering and I feel compelled to respond.  Science is built its ability to test theory.  If the predictions of a theory are wrong, then the theory is no good and it gets thrown out.  With one very famous and notable exception, most theories are built on math because that makes them testable.  If your measurements don't match theory, then either A) the theory is wrong, or B) you don't understand the theory.  Simply claiming that the real world is too complicated for the theory doesn't really cut it either way.  Regardless, I actually think that you understand this stuff, but that's not completely clear from what you wrote (or at least from the way that I read it) so I'll just make two points.

 

1)  You are indeed working with a band limited signal.  The cutoff frequency occurs at 1/(lamda*F), where F= F/#...determined solely by the optical system.

 

2)  In order to get the right result, you have to account for the transfer function of the detector itself and it's effect on the imaging chain.  The detector transfer function is determined by the pixel size and shape but also involves things like the diffusion transfer function caused by charge leakage between adjacent pixels and charge transfer function caused by the transfer efficiency present in CCD sensors, and maybe a few other effects; but the net result is a single transfer function for the whole sensor.  In x,y space, the sensor transfer function acts to convolves the pixel shape (slightly modified by the CFT, CTE, and other things) with the image before sampling occurs.  The net result is that if you model the imaging chain correctly, you can indeed accurately predict the results and set limits on the required sampling rates to avoid aliasing and to find an optimum sampling rate with respect to reconstruction.  Optimum sampling is often at odds with other requirements (such as signal strength) so the answer isn't always one "best" number; but, that doesn't mean that real imaging systems can't be accurately modeled to a very high degree of accuracy.  So, your description of the sampling process as being done by a bunch of "blocky pixels" is incomplete at best.  As you know, blocky pixels are not delta functions and that is not how the imaging chain is modeled if you want to get the right result regarding optimum sampling.

 

 

John



#7 dayglow

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Posted 14 August 2017 - 06:21 PM

Here are the mathematics I use to estimate telescope-mount-camera resolution capability.

Estimating_Resolution

 

-- David F.



#8 rgsalinger

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Posted 14 August 2017 - 09:44 PM

That's the formula that I was after. Using it I found that my results with the PW12.5 are actually right on the money given 1.5 arc second seeing. Very nice, thank you.

Rgrds-Ross



#9 freestar8n

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Posted 15 August 2017 - 04:38 AM

 I like both of these answers. There's the theoretical model and the idea that things might not be the same in actual practice.

 

What I'm getting from both, though, is that I might get a bit under the seeing under the right sampling conditions, but I certainly ought to be able to resolve at F8 and 2540mm down to the level of the seeing. If I'm not,  then I'm not focusing well enough or I've got a (collimation?) error or some other problem. There seemed to be very little difference (measured with the old camera) between long and short exposures. My guiding (not my talent, my equipment) is usually around .3 arc seconds in each axis. That can't be much of a contributor to what I keep thinking is FWHM a little higher than the seeing with the original camera.

 

I was hoping for a formula, but I guess that the point is that eventually it's just seeing limited once you go long enough in focal length with small enough pixels. 

 

I sometimes get to use a 25" F8 RC scope. With that scope and my STF8300 I was able to get about 15 percent under what the seeing monitor said was happening at our dark sky site. I've never been able to do that with the larger camera we have on the mount now. I can get down to right around the seeing (using Focusmax) but I've never seen anything less than the 2" we generally have.

 

Rgrds-Ross

Hi Ross-

 

Yes - theory is fine - but it almost always involves assumptions about what is actually happening - and an important factor may be missed.  But CN is a good place where people can share experiences on how actual measurements depart from simple theoretical predictions - and even provide new insight into other factors that are important but had not been considered.

 

I have not studied Dayglow's formulation - but it appears to include empirical factors that are important, so that is a good step.  

 

When it comes to true diffraction limited imaging at f/8, you will need very small pixels.  And the rms error from the guider is not a true indication of what is really happening.  So when you say you were surprised that the fwhm improved so much when you switched to much smaller pixels - that is exactly what I would have predicted.  Reality often throws simple theory out the window.  It doesn't mean the theory is bad or wrong - it just means reality had other stuff going on that was not part of the simple theory - and its associated assumptions.

 

Frank



#10 RDBeck

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Posted 15 August 2017 - 08:39 AM

Do I understand the original post correctly, in that the image scales are 0.31"/pixel and 0.44"/pixel?  If so, it's interesting that the FWHM values calculate to 5 pixels for both cases (5*0.31=1.5, 5*0.44=2.2).



#11 rgsalinger

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Posted 15 August 2017 - 09:39 AM

Yes my old camera had 5.4 micron pixels and the new camera has 3.8 micron pixels. And that's actually what I saw when changing cameras. Last night I was getting around 1.5 arc seconds where before I was getting around 2.2 arc seconds.  

 

However, when I used the formula with my 663 refractor and either of those cameras, I got strange results - that the FWHM would be even smaller in terms of arc seconds. This makes no sense to me at all. I have the feeling I'm going something wrong in my spreadsheet. 

 

The refractor is 127mm, and so the image scale is around 1.68 arc seconds per pixel using 5.4 micron pixels. I used a seeing value of 1 arc second. That gave me 2.4 pixels for star. OK so far. Then, dividing that number by the 1.68 scale I end up with 1.4 pixels for the star diameter in pixels. I would expect larger stars as the scale goes up. So, that seems wrong. When I then divide by 3 (sampling) I end up with .48 arc seconds for the size of a star which I know is dead wrong. I can't figure out why the formula works for my long scope but not for my short scope.

 

Help working this example greatly appreciated. 

 

Rgrds-Ross



#12 freestar8n

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Posted 15 August 2017 - 04:50 PM

Hmm - ok that does sound odd.

 

If you can put fits files somewhere I would be happy to take a look.

 

Frank



#13 dayglow

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Posted 15 August 2017 - 11:02 PM

My fault:  I put in the wrong equation at the end.  This should be the correct computation for star width

 

W (arc-sec) = MAX (Q * Scale, MSDas)

 

I apologize for the error.

-- David F.



#14 rgsalinger

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Posted 15 August 2017 - 11:40 PM

Thank you, thank you, thank you. I had decided to give up astro-photography because I couldn't even put some equations into a spreadsheet. Now I get the correct answer of around 3 arc seconds which is actuall what I've been getting! Amazing.




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