MTF of a telescope

[vlaiv]

In another thread we touched upon concept of telescope MTF and instead of further discussion in that particular thread - we decided to open a new one - dedicated to understanding of:

MTF and how it relates to resolution, detail, sharpness and contrast.

These terms are often used to convey telescope performance and for that reason I think it is important that we are all on the same page when using these terms.

I'll kick off discussion by saying that above terms are somewhat synonymous when we talk about telescope performance and are all "encoded" in MTF. 

Discussion such as these inevitably rely on some math and I hope that whilst being formally correct - we can still make content understandable to most by means of analogy. First step in understanding all of that is to grasp concept of image as 2d function. We need to think of image as 2d function, but how?

2d function is defined with values it has in points in 2d plane - we take some coordinates X and Y and we get some value in that location. Image is 2d function if we take value to be light intensity in some point along height (Y coordinate) and width (X coordinate) of an image.

Above we see an image and its 2d representation as a function plotted against XY. You can see that values go from 0 to 255 (in Z direction - direction of value of function) - which is common for 8bit computer images.

Next we need to know that functions can be represented as sums of simple wave functions - sine and cosine. This is called Fourier transform. In fact - some functions can only be represented as infinite sum of sines and cosines while others only need finite number of terms.

I have found a few animated gifs to explain this phenomena better thru animated sequences:

This one shows how square wave function can be represented with sum of 4 sin functions each with higher frequency.

same with even more frequency components. Note that these are all 1d functions, but same applies for 2d functions as well.

In the end, Fourier transform or our function is just another way to write down our function.

What does this have to do with MTF?

Well - MTF also has its "analog" - or another way to "write it down" - it is PSF or point spread function - or image of a single star in telescope terms. PSF describes how our image is blurred when observed thru a telescope. This involves rather complex mathematical operation of convolution.

MTF is analog of that operation - just much more simple. It also operates on our image - only on its analog in Fourier domain - by simple multiplication. This is what makes it much more understandable - it is simple multiplication.

So here is how MTF looks like:

But I have to say - that is actually not MTF - it is only "cross section" of MTF. Actual MTF is 2d function and looks like this:

above graph is made when you plot values on a single line from center of the cone outward.

There is special class of functions that have this circular symmetry and you can use cross section to represent whole 2d function. MTF is one of those in some cases.

What does height of MTF stand for? Since MTF multiplies 2d Fourier transform of our image - it is just number but since it is in range from 0-1 we can say that it represents - attenuation of particular frequency. If number is say 0.2 - then that particular frequency is left at only 20% of its original value.

MTF has highest frequency - after which value of the function falls to 0 - that is cutoff point and image at focal plane of telescope can never contain frequencies higher than this cutoff frequency.

If we look at shape of MTF - we see that it has tendency to attenuate higher frequency - higher the frequency higher attenuation of it - until at one point we reach cut off and all frequencies above that one are completely removed (multiplied with 0). Now if we look at our animated gif above of how rectangle is formed - you'll notice something interesting.

Even sine wave to some extent approximates rectangle pulse train. It is not good approximation - but it is still approximation. We need higher frequencies to make that sine wave look more like rectangle - "to sharpen its edges". What would reverse of that process be  - well if we remove high frequency components - we remove "sharpness" of the edges in the image.

Telescope that "kills off frequencies faster" - is less sharp telescope.

This does not mean that it such telescope won't show feature - we will see later when "features disappear" and why - it means it will show it less sharp.

We can directly relate to this by examining star image in two telescopes - one with small aperture and one with large aperture. Here we are strictly speaking without influence of atmosphere - so imagine night of excellent seeing. You are observing a star with two telescopes - one with smaller aperture that "kills off frequencies faster" and one with larger aperture - that "kills off frequencies slower".

Smaller telescope will show star as larger "blob" or known by other name - Airy disk, than larger aperture telescope - smaller telescope is less sharp - or star image is more blurred.

This is how MTF impacts sharpness - higher frequencies are needed for things to be sharp and MTF attenuates and kills of high frequencies and makes things blurry.

 

[jetstream]

13 minutes ago, vlaiv said:

We need higher frequencies to make that sine wave look more like rectangle - "to sharpen its edges".

I might just have understood some of this. I know of rectangle waves in industrial applications. Can you go through how the higher frequencies make the squarewave again please?

Am I correct to say that the sharp edges of the rectangle wave relate to sharpness within the view or image? And that the softer, more prolonged transition of the sine wave softens the view?

[vlaiv]

I'd like now to address another thing in MTF light and give an example for that.

When we talk about aperture and telescopes - we get the notion that some small features can only be seen with large aperture telescope. Is that correct and in which cases.

It is both "wrong and correct" at the same time - and I'll give you two examples.

First one is obvious. We see stars with our telescopes - a lot of them. Diameter of those stars is many orders of magnitude smaller than resolving capability of aperture - yet we see them. That would suggest that no matter how small - feature can be seen in a telescope.

But - point telescope to the Moon and try to see very small crater - you won't be able to. Take small telescope and try to see very small detain in Jovian atmosphere on the edge of the belt. You won't be able to do it. What is up with that?

Here we encounter contrast and its impact.

I'm going to show you two types of features and how they are rendered by ever decreasing aperture - with one important thing. I'm not going to normalize images - I'm going to try to make them look like images we see at the eyepiece. If things get darker - I'll leave them darker - if they get brighter - I'll leave them like that.

Here is set of images without impact of telescope aperture. Left is a star and right is what we could call "gap in Saturn's ring system". Let's look at profile plot of these two features:

Now we are going to look at those features thru a first telescope:

Note that both features are still there but two important things happened:

1. sharpness is lost - look at profile plots - features are getting "fatter" and "softer"

2. contrast is lost - features are no longer so distinct - star is fading into background and gap is no longer black it is "fading into foreground"

Let's use twice smaller aperture to see what will happen next.

We are continuing to see the same trend - things get lower contrast and get less sharp. I'll jump to x8 smaller aperture now to really show effect:

Here we no longer can see the star - and we are virtually at the edge of seeing gap - but it is much wider than original and very low contrast.

On the other hand - math plots suggest that both features are still there - they are now really blurred and low contrast - we can see very familiar Airy pattern in our star now and something similar in gap.

Conclusion is - details don't really disappear - they just loose sharpness and contrast. We see stars because their initial contrast is extraordinary - stars are very bright but most observers know that seeing is particularly poor - they can't go as deep as on night of a good seeing - that is because seeing is again type of blur. When star is blurred - it's contrast is reduced.

Planetary features on the other hand are very low contrast to begin with and it does not take much of contrast reduction to get them to disappear beyond detectability.

Increasing magnification reduces brightness and therefore contrast - want to see detail - don't use too much magnification as you'll further reduce contrast that is already reduced by telescope aperture.

[vlaiv]

27 minutes ago, jetstream said:

I might just have understood some of this. I know of rectangle waves in industrial applications. Can you go through how the higher frequencies make the squarewave again please?

Am I correct to say that the sharp edges of the rectangle wave relate to sharpness within the view or image? And that the softer, more prolonged transition of the sine wave softens the view?

I guess that without getting to mathematical about this - best way to describe how higher components create sharpness is to carefully watch this animated gif again:

Note how first few sine components have sloped edge versus rectangle pulse train that has vertical edge. Higher the frequency faster sine wave goes from -1 to 1 and back again - we could say that it faster goes up and down and that it has steeper curve when it goes thru 0.

By adding higher and higher frequencies - you make sum behave the same - look how it is getting steeper and steeper - in fact you need to infinite frequency component as slope is infinite with pulse train - so this sum goes to infinity.

Yes, sharp edge of rectangle wave is what is sharp edge in image - very fast contrast change - in instant we go from no signal to signal - no light to light in pixel - that is steep slope. Loosing sharpness means that our slopes are no longer steep.

 

[John]

MTF ?

Modulation Transfer Function

I think that is all that I can contribute to this

 

 

[Robindonne]

17 minutes ago, John said:

 

Edited February 3, 2021 by Robindonne

[Robindonne]

4 minutes ago, Robindonne said:

Edited February 3, 2021 by Robindonne

[John]

Speak up Robin, we can't hear you ! 

[vlaiv]

To expand a bit further on topic and to present some "real life" examples, here are a few simulations that will help understand - sharpness, contrast and detail and resolution:

This is set of features on planet X. Largest of these features is about 5.5 arc seconds in size. Others are progressively 15/16th, 14/16th ... 1/16th of the size of original - so the smallest one is ~0.345" in "size".

Note that features consists of "smaller parts" - that are roughly 1/3rd to 1/5th of original size (length and width of spike).

We observe planet X with 8" of clear aperture telescope in extraordinary seeing conditions, what do we see?

Such telescope has airy disk diameter of ~1.26" and highest frequency component of ~1.94 cycles per arc seconds or equivalent wavelength of ~0.5156" (x2.44 times smaller than airy disk diameter, or x1.22 times smaller than airy disk radius). This is all for 500nm wavelength.

We see that first few features exhibit what we would call - sharpness loss primarily. Second row also, but last feature in second row starts to show contrast loss as well. In third row dominant thing that we notice is contrast loss - but we still can recognize that thing is pentagonal in its structure.

In last row - we completely loose resolution, contrast is severely impacted and last feature is almost invisible - we can sort of tell that maybe something is there.

We can now start to get the idea - when feature size is about x5 that of cutoff wavelength  (2-2.5 size of airy disk) - we start to see contrast loss. When feature is about the size of airy disk - or roughly x2.5 size of cutoff frequency - we start to loose all resolving capability - we cannot longer tell anything about the shape of the feature.

How does smaller telescope compare to this? Here is perfect 4" clear aperture telescope aimed at the same target (take above stats and multiply by two - airy disk size will be 2.52", cutoff wavelength at about 1.03" and so on).

Here we jump straight into contrast loss part - even first row is experiencing that. That sort of makes sense now since largest feature size is 5.5" and airy disk diameter is now 2.52" - so again roughly x2-x2.5. Third row is no longer resolved and fourth shows "disappearing" feature - where first two can be said to have something there - even third - but last is gone completely.

Mind you - with contrast and disappearing features - base contrast plays major role - we can still see features that are beyond resolving power if they are very high contrast to start with - but their size - can't be smaller than airy disk (it can in one direction - for example rile or gap in rings can be long - but width is limited to roughly airy disk diameter).

 

[andrew s]

@vlaiv nice series of post with clear demonstrations Regards  Andrew 

PS If you feel like it you  might like to compare a bark line on a light background with a light line on a dark back ground. 

 

[vlaiv]

12 minutes ago, andrew s said:

@vlaiv nice series of post with clear demonstrations Regards  Andrew 

PS If you feel like it you  might like to compare a bark line on a light background with a light line on a dark back ground. 

 

I guess to some extent there is already something similar in dark line on white background vs single star above?

[andrew s]

9 minutes ago, vlaiv said:

I guess to some extent there is already something similar in dark line on white background vs single star above?

Yes, similar but "old" observations comment on linear v "point" features and visibility differences between l on d and d on l linear features. I felt it would complete the set.

Clearly,  it's up to you. You have already made a significant contribution so don't  feel obliged to do it.

Regards Andrew 

[vlaiv]

1 minute ago, andrew s said:

Yes, similar but "old" observations comment on linear v "point" features and visibility differences between l on d and d on l linear features. I felt it would complete the set.

Clearly,  it's up to you. You have already made a significant contribution so don't  feel obliged to do it.

Regards Andrew 

If you feel it will contribute, I'll certainly put an effort to demonstrate it.

I wanted first to address JND (just noticeable difference) and MTF attenuation part as well as to give a bit more understanding on what high and low frequencies contribute to image composition.

[Captain Scarlet]

Thank you vlaiv. Just two things, one suggestion, one general observation from my reading around this topic.

Suggestion: You start off by assuming everyone knows what MTF actually is ... perhaps elaborate exactly what MTF is, even mathematically perhaps, I think the formula for it isn't too complicated for your readers.

General observation: whenever I come across MTF plots in the literature, almost without exception the frequency x-axis is scaled from 0 to 1, i.e. the proportion of maximum possible frequency for the scope in question. This is for comparing different configurations of the same scope, perhaps deformed mirror vs perfect mirror for example.

Almost never do you see an MTF plot of one aperture scope vs another aperture, i.e. with two different "maximum possible frequencies", for which the x-axis would need to be in absolute frequency-units rather than "relative to the ideal for one scope" units. Plotting vs absolute frequencies then allows one to see quite how much better a larger aperture scope is than a smaller one (for contrast and resolution). Even that a badly abberrated large scope can outperform a better-engineered small scope.

Great thread, Magnus

[alex_stars]

Hi all and @vlaiv thanks for putting in so much work in this thread already.

I just wanted to add a slightly different perspective to the discussion. Let us for the sake of simplicity stick to 1 dimension ideas and I add the two dimensional counterparts in brackets.

Let us imagine we observe a bright light source, but cover that source with a black screen halfway:

(this image is taken from https://en.wikipedia.org/wiki/Optical_transfer_function#/media/File:MTF_knife-edge_target.jpg)

then we would in an ideal world with a perfect telescope observe a sharp edge with the possible strongest contrast there is. Bright on one side of the edge and completely dark on the other side. Now in reality we will never manage that as the optics, no matter how good, will blur the image. So our sharp edge gets blurred.

if we now plot the intensity of the pixels across that blurred edge:

(this image is taken from https://www.researchgate.net/figure/Computation-of-the-Modulation-Transfer-Function-using-the-knife-edge-target_fig2_237077760)

we get a plot like the one on the left. This is what is called an edge spread function or ESF. If we apply the mathematical tool of differentiation (don't worry what that actually does), we get the line spread function or LSF (in 2D this would be the well known point spread function). The LSF tells us how much a perfectly thin line will be spread out due to the optics we use. Large aperture optics will spread out the line less than small aperture instruments.

Now we want a more overview like, comparative measure of this spread, so we apply another mathematical tool called Fourier Transform (as @vlaiv described above). This tool now allows us to plot a graph that describes how well different signals with different spatial extents are transferred through our optical system, a graph we call Modulation Transfer Function or MTF.

Here you can see an example for a 180 mm Maksutov and different unobstructed APOs. As @Captain Magenta pointed out, we can compare different optics if we keep the units on the X-axis (cycles per arcsec in my case).

To the left we have the large spatial frequencies. As an example I plotted lines in the MTF graph for different sizes in arcsecs and included well known solar system objects in brackets for comparison. As we go towards the right of the graph, we get to smaller spatial scales and see how the transfer through the system decreases.

An easy way to understand the MTF graph is with lines (and I know I simplify here). Imagine you would observe a bight line though your scope that extends all the way across your eyepiece with the width of 50 arcsecs. That bright line would be perfectly well transported through all telescopes in the MTF graph above (the MTF value is very close to 1).

Lets make our bright line thinner, say 8 arcsecs. First off you see that all compared scopes will transfer that line less good than the 50 arcsecs wide line, the MTF value has dropped. And you start to see differences between the scopes. However that does not mean that the 8 arcsecs wide line somewhat disappears. No it is just less well transfered through the optical system, and we often describe this as being less "sharp".

An interesting line width is the 2 arcsecs line above, where a 125 mm aperture scope is slightly better than a 172 mm aperture scope with an central obstruction. At smaller line widths (towards the right) you see the 102 mm and 125 mm scopes steadily decreasing in their capability of transferring narrow lines. The Maksutov however has a region of flattened decrease, which is caused by the central obstruction. And you might notice that how flat that region is, is controlled by how large the central obstruction is (green vs magenta line).

At a line width of 0.93 arcsec, the 125 mm aperture scope reached its Dawes' limit, meaning that lines smaller than that do not get transferred at all through the system. They would be so blurred that we could not well perceive them anymore (given that they have similar brightness than their surroundings and hence low contrast). The large aperture scope (the Maksutov) continues to transfer small features reasonable well through its optical system until that scope also reaches its limit.

One other thing to notice is the region around 1 (cycles/arcsec) on the X axis. This is where 1 arcsec wide lines would be placed. I mention this because if we talk about the seeing being around 1 arcsec, then we mean that a infinitely narrow line would be smeared out to about 1 arcsec width by atmospheric turbulence. For us observers on the ground this means that the atmosphere blurs everything on that scale and that we can not make use of our fine telescope capabilities (the region right of that limit where the Maksutov outperforms the 125 mm APO) on that particular night.

However this discussion does not directly relate to imaging, it is only valid for visual observation. When we image and utilize lucky-imaging techniques, we can "break through" the seeing induced blur and harvest the capabilities of our large aperture scope. In practical terms that's the reason why I prefer the 5" Apo for visual and would pick up a C14 for imaging, given that I would have a large budget and a beach house in Barbados (both of which I don't!) 😉

Hope this helps a bit in the understanding of MTFs

Edited February 3, 2021 by alex_stars

[vlaiv]

7 minutes ago, alex_stars said:

At a line width of 0.93 arcsec, the 125 mm aperture scope reached its Dawes' limit, meaning that lines smaller than that do not get transferred at all through the system. They would be so blurred that we could not well perceive them anymore. The large aperture scope (the Maksutov) continues to transfer small features reasonable well through its optical system until that scope also reaches its limit.

How come that we see the stars then?

Stars are much smaller features than 0.93 arc seconds - they are in micro arc second range and smaller than that.

[vlaiv]

While this is valid way to compare two telescopes:

And here we see MTF plot of 125mm APO with perfect figure and 172mm Mak with 58mm CO and again perfect figure

(I did not calibrate graph - it is still in pixels since I used FFT to generate these curves).

We can use concept of JND to really compare two telescopes. Just noticeable difference is idea that we only perceive difference in some stimuli if it is above percentage of that actual stimulus value.

In another words - we will see difference in intensity of the light if difference in intensity is about 10% - or 0.1. For stronger light we need stronger change before we see any difference.

I'm going to propose plotting above graph slightly differently - let's take ratio of MTF curve between two scopes APO / MAK and see what it looks like

If above line goes above 1.1 - there will be noticeable difference where APO wins (more than 10% of brightness on that frequency). When graph falls below ~ 0.91 (1/1.1 - or 10% or more brightness advantage for mak on given frequency)  - Mak wins.

From this graph we can see that APO never provides advantage that we can actually see - no frequency component will be more than 10% brighter in APO than in Mak - while Mak will show frequencies above certain frequency to be more than 10% brighter than APO.

[Louis D]

15 hours ago, vlaiv said:

we could say that it faster goes up and down and that it has steeper curve when it goes thru 0.

In digital electronics, we generally refer to this as the slew rate.  The higher the slew rate, the steeper the curve transitioning from 0 to 1 or 1 to 0.

[alex_stars]

 

28 minutes ago, vlaiv said:

How come that we see the stars then?

Stars are much smaller features than 0.93 arc seconds - they are in micro arc second range and smaller than that.

I'm surprised you ask again, did you not answer that yourself in an earlier post:

16 hours ago, vlaiv said:

Conclusion is - details don't really disappear - they just loose sharpness and contrast. We see stars because their initial contrast is extraordinary - stars are very bright but most observers know that seeing is particularly poor - they can't go as deep as on night of a good seeing - that is because seeing is again type of blur. When star is blurred - it's contrast is reduced.

Planetary features on the other hand are very low contrast to begin with and it does not take much of contrast reduction to get them to disappear beyond detectability.

Increasing magnification reduces brightness and therefore contrast - want to see detail - don't use too much magnification as you'll further reduce contrast that is already reduced by telescope aperture.

I assumed that people had already read through your posts and we were all on the same page that we talk about transfer of "features" of comparable contrast. 😉

I will correct my post

[alex_stars]

18 minutes ago, vlaiv said:

From this graph we can see that APO never provides advantage that we can actually see - no frequency component will be more than 10% brighter in APO than in Mak - while Mak will show frequencies above certain frequency to be more than 10% brighter than APO.

I have never claimed that a 125 mm APO would provide a perceivable better optical performance (better modulation transfer) than a 172 mm obstructed Mak. I know that you know what I mean and I think that you wanted to make it clear to the readers that there is no advantage in optical performance of the 125 mm APO to the 172 mm Mak. I agree.

However, do people really interpret MTF plots like that?

I would summarize the comparison in different terms: "A 125 mm aperture unobstructed scope is equal in performance to a 172 mm aperture scope with a 58.5 mm CO down to a spatial frequency limit of about 2 arcsecs. For finer resolution features the 172 mm scope is better". However since we are close to the seeing limits people experience, one can further state: "A 125 mm aperture unobstructed scope might be preferable to the 172 mm aperture scope with a 58.5 mm CO in situations where the seeing is regularly around 1-1.5 arcsecs and the 125 mm scope has other advantageous properties like faster cooling time, easier handling etc. besides its optical performance".

[jetstream]

@vlaiv keep any and all info on this coming please, it is much appreciated.

[vlaiv]

19 minutes ago, alex_stars said:

I have never claimed that a 125 mm APO would provide a perceivable better optical performance (better modulation transfer) than a 172 mm obstructed Mak. I know that you know what I mean and I think that you wanted to make it clear to the readers that there is no advantage in optical performance of the 125 mm APO to the 172 mm Mak. I agree.

I know you did not claim that - I was sort of building up "case" in order to start discussion on some long held beliefs.

One of those - smaller aperture apo will have better contrast than larger telescope with central obstruction.

It might well be that 4" APO indeed has better contrast than say 6" Newtonian - but I don't think that is due to central obstruction. It could be due to much more effective baffling and mirror light scatter and such - but I don't think it is due to central obstruction.

I wanted next to compare clear aperture to 25% CO in the fashion that I proposed. I have a feeling that difference will hover at or just above 10%.

In fact - let's do it now.

Here are graphs of clear aperture and 25% CO

[JeremyS]

Ok, so this applies to standard optics and the everyday world. But what about Takahashi?

 

 

🤣
 

Edited February 3, 2021 by JeremyS

[Deadlake]

@vlaiv @alex_stars

How would the quality of the scope effect the MTF functions?
How would poor seeing (say atmospheric turbulence) in combination with optics quality effect the MTF, presume poor optics would propagate errors introduced by atmospheric turbulence?
Thanks

 

 

[Deadlake]

1 minute ago, JeremyS said:

Ok, so this applies to standard optics and the everyday world. But what about Takahashi?
 

I don't see any discussion of optical quality, hence the question.