What exactly is MTF?

[Joofa]

Some recent threads and messages have explored the notion of MTF. It appears to me that there exists a confusion in many people minds regarding what exactly is an MTF - the celebrated Modulation Transfer Function. For e.g., see the threads below that have many misleading statements, IMHO:

https://www.dpreview.com/forums/post/59557798

https://www.dpreview.com/forums/post/59563398

https://www.dpreview.com/forums/post/59563735

https://www.dpreview.com/forums/post/59619529

Before proceeding, I must say that the optics community is responsible for part of the confusion. By instituting terms such as MTF, which depends upon a certain notion of contrast , sometimes confusingly called modulation contrast (more on it below), and the use of another function called CTF, Contrast Transfer Function, (more on it below also), which is different from MTF. Though they share the words 'modulation' and 'contrast' respectively, and also use the same definition of measuring contrast!

What I write below comes from my memory as an undergrad in electrical engineering a long time ago, so please pardon me for some small technical mistakes or being not very rigorous due to a fact that I don't recall many details now as I haven't dealt with these things regularly for a long time. However, the following shall provide a right template or outline for understanding MTF.

Electrical System Theory:

• A linear shift invariant system (lets call it LSI) has an output that is related to input via a convolution. That is a very important theorem in linear systems.
• That convolution kernel is called impulse response. An LSI system is completely described by its impulse response.
• The collection of the ratio of the Fourier response of the output of an LSI system at each discrete frequency to the corresponding input frequency is called The Transfer Function.
• This transfer function can be shown to be the Fourier transform of the impulse response.
• Complex Exponential Signals are eigenfunctions of an LSI system. When they pass through a LSI the output is still a complex exponential signal but modified by an eigenvalue, which in general is a complex number. This collection of eigenvalues provides a certain transfer function.
• This transfer function can be shown to be the same as the Fourier transfer function mentioned above.
• For a very important class of LSI systems that have a real impulse response, the co-sinusoidal functions are also eigenfunctions.
• With this special LSI system a cosinusodial input yields a cosinusoidal output, possibly shifted in phase and attenuated. This system transforms a real-valued input to a real-valued output.
• The above is also related to the desire to having a linear operator involving the system as hermitian (or self adjoint) - though lets not get into the details of when hermitian and self adjoint are not the same things. That is only for mathematical curiosity. Also fans of Quantum Mechanics would get a kick out of this as such operators yield real eigenvalues that are observable. But lets not invite that delusional bunch here.
• The Fourier transfer function for LSI systems describes all of this.

Optics:

• In Optics a transfer function is commonly used that relates the input and output via their modulation contrast.
• This transfer function, which is called the MTF, is real, as contrast by that definition is real, and so it is a ratio of reals.
• With the application of this transfer function a cosinusodial input yields a cosinusoidal output, possibly attenuated.
• It can be shown that this MTF is the magnitude part of the Fourier transfer function that we found in system theory above.
• The impulse response is now called Point Spread Function, PSF.
• In optics, typically not all impulse responses are called PSFs. Only those that produce observable outputs, and relates physical or real inputs and outputs.
• Hence, the importance of transfer functions that are hermitian and / or the importance of self adjoint operators.

Takeaway:

• MTF is the response of the system to a cosinusoidal input.
• And, is the magnitude of the transfer function, which is the Fourier transform of the impulse response.
• Or equivalently, MTF is the ratio of output and input modulation contrast of a cosinusoid.

MTF and CTF:

• A contrast transfer function or CTF is the response of the system to a square wave input, but otherwise measured with the same definition of contrast as the MTF.
• CTF yields values that are different from MTF, as can been seen easily by decomposing a square wave into Fourier components and applying MTF to each individually and adding up.
• There exists a relationship between MTF and CTF.

 

[J A C S]

Optical transfer function - Wikipedia

en.wikipedia.org

 

[Joofa]

https://en.wikipedia.org/wiki/Optical_transfer_function

Yes, that link is good, but it does not talk about many points that I addressed in the OP.

 

[J A C S]

https://en.wikipedia.org/wiki/Optical_transfer_function

Yes, that link is good, but it does not talk about many points that I addressed in the OP.

The rest follows from the properties of the FT and the convolution.

 

[Joofa]

https://en.wikipedia.org/wiki/Optical_transfer_function

Yes, that link is good, but it does not talk about many points that I addressed in the OP.

The rest follows from the properties of the FT and the convolution.

So. I don't see what you are after? Of course they have to follow from some properties of something. All I have stated is that your link doesn't state them all.

--
Dj Joofa
http://www.djjoofa.com

 

[Jack Hogan]

Thanks Joofa, that's a nice summary.

There is a question that's been nagging at me for while: in the past knowledgeable folks (perhaps you) corrected my assumption that in digital imaging a one-dimensional MTF curve (say obtained through the slanted edge method from the FT of an LSF) be symmetrical about the origin. But why would it not be - given the fact that the input (LSF) is real, so its Fourier Transform should be conjugate symmetric about the origin, hence its spectrum as well?

Jack

 

[Joofa]

Thanks Joofa, that's a nice summary.

There is a question that's been nagging at me for while: in the past knowledgeable folks (perhaps you) corrected my assumption that in digital imaging a one-dimensional MTF curve (say obtained through the slanted edge method from the FT of an LSF) be symmetrical about the origin. But why would it not be - given the fact that the input (LSF) is real, so its Fourier Transform should be conjugate symmetric about the origin, hence its spectrum as well?

The Fourier transform of a real signal will always be symmetric in magnitude. Are you talking about this past discussion that talked about the spectrum 'folding' concept in aliasing instead:

https://www.dpreview.com/forums/post/57782658

--
Dj Joofa
http://www.djjoofa.com

 

[Jack Hogan]

Thanks Joofa, that's a nice summary.

There is a question that's been nagging at me for while: in the past knowledgeable folks (perhaps you) corrected my assumption that in digital imaging a one-dimensional MTF curve (say obtained through the slanted edge method from the FT of an LSF) be symmetrical about the origin. But why would it not be - given the fact that the input (LSF) is real, so its Fourier Transform should be conjugate symmetric about the origin, hence its spectrum as well?

The Fourier transform of a real signal will always be symmetric in magnitude. Are you talking about this past discussion that talked about the spectrum 'folding' concept in aliasing instead:

Right. Isn't 'folding' in a digital imaging context equivalent to adding to the unsampled curve MTF below Nyquist the relative portion of the conjugate MTF shifted to 1 c/p?

 

[Joofa]

Thanks Joofa, that's a nice summary.

There is a question that's been nagging at me for while: in the past knowledgeable folks (perhaps you) corrected my assumption that in digital imaging a one-dimensional MTF curve (say obtained through the slanted edge method from the FT of an LSF) be symmetrical about the origin. But why would it not be - given the fact that the input (LSF) is real, so its Fourier Transform should be conjugate symmetric about the origin, hence its spectrum as well?

The Fourier transform of a real signal will always be symmetric in magnitude. Are you talking about this past discussion that talked about the spectrum 'folding' concept in aliasing instead:

Right. Isn't 'folding' in a digital imaging context equivalent to adding to the unsampled curve MTF below Nyquist the relative portion of the conjugate MTF shifted to 1 c/p?

If you read the chain of messages in that thread linked in my message before then I think you were 'folding' two complexing numbers as abs(a) + abs(b) instead of abs (a + b).

 

[Jack Hogan]

Thanks Joofa, that's a nice summary.

There is a question that's been nagging at me for while: in the past knowledgeable folks (perhaps you) corrected my assumption that in digital imaging a one-dimensional MTF curve (say obtained through the slanted edge method from the FT of an LSF) be symmetrical about the origin. But why would it not be - given the fact that the input (LSF) is real, so its Fourier Transform should be conjugate symmetric about the origin, hence its spectrum as well?

The Fourier transform of a real signal will always be symmetric in magnitude. Are you talking about this past discussion that talked about the spectrum 'folding' concept in aliasing instead:

Right. Isn't 'folding' in a digital imaging context equivalent to adding to the unsampled curve MTF below Nyquist the relative portion of the conjugate MTF shifted to 1 c/p?

If you read the chain of messages in that thread linked in my message before then I think you were 'folding' two complexing numbers as abs(a) + abs(b) instead of abs (a + b).

Right, going through the thread refreshed my memory of your comments there, they still make sense

 

[J A C S]

• Complex Exponential Signals are eigenfunctions of an LSI system. When they pass through a LSI the output is still a complex exponential signal but modified by an eigenvalue, which in general is a complex number. This collection of eigenvalues provides a certain transfer function.

They are actually the functions providing the spectral representation of a convolution operator. Eigenfunctions need to be in the space (here, L^2) to be called eigenfunctions and those complex exponentials are not.

A much simpler version of all that is to say that a convolution acts as a Fourier multiplier (with its FT) in the frequency domain.

Finally, that does not really explain why measuring the contrast with a sinusoidal input gives us the MTF at that frequency. It is a simple calculation but it needs to be done (and it requires some assumptions).

• This transfer function can be shown to be the same as the Fourier transfer function mentioned above.
• For a very important class of LSI systems that have a real impulse response, the co-sinusoidal functions are also eigenfunctions.
• With this special LSI system a cosinusodial input yields a cosinusoidal output, possibly shifted in phase and attenuated. This system transforms a real-valued input to a real-valued output.
• The above is also related to the desire to having a linear operator involving the system as hermitian (or self adjoint) - though lets not get into the details of when hermitian and self adjoint are not the same things. That is only for mathematical curiosity. Also fans of Quantum Mechanics would get a kick out of this as such operators yield real eigenvalues that are observable. But lets not invite that delusional bunch here.
• The Fourier transfer function for LSI systems describes all of this.

 

[Joofa]

• Complex Exponential Signals are eigenfunctions of an LSI system. When they pass through a LSI the output is still a complex exponential signal but modified by an eigenvalue, which in general is a complex number. This collection of eigenvalues provides a certain transfer function.

They are actually the functions providing the spectral representation of a convolution operator. Eigenfunctions need to be in the space (here, L^2) to be called eigenfunctions and those complex exponentials are not.

A much simpler version of all that is to say that a convolution acts as a Fourier multiplier (with its FT) in the frequency domain.

If you are going into details of difference between spectrum of an operator and eigenfunctions, or when they are different, then we are not going into that here - that is not the scope of this discussion. Similarly that complex exponentials, or even cosinudsoidal signals not being strictly L2, and / or the limiting properties of Fourier Transforms, or integrals in general, in such cases, are again not the scope of discussion here.

With the above out of the way, complex exponentials being eigenfunctions of an LSI system, and Fourier Transform as being multiplier in Fourier domain of a convolution in space/time domain, are related facts - one derived using other. This eigenfunction property is an important property with many uses in LSI systems and is not just obvious by stating that Fourier Transform acts as a multiplier, IMHO.

Finally, that does not really explain why measuring the contrast with a sinusoidal input gives us the MTF at that frequency. It is a simple calculation but it needs to be done (and it requires some assumptions).

I mentioned in the OP that when contrast is measured using a specific method called modulation contrast, then the resulting transfer function, which is called MTF in optics, is the same as magnitude of the Fourier Transform of impulse response in systems theory. This can be shown but it is not immediately obvious.

The important fact being that if contrast is measured using a different formula, other than modulation contrast, then the optics community might still call the resulting transfer function as MTF, but, then this MTF may not be the same as the magnitude of the Fourier Transform of the impulse response. It is good to know that.

--
Dj Joofa
http://www.djjoofa.com

 

[J A C S]

• Complex Exponential Signals are eigenfunctions of an LSI system. When they pass through a LSI the output is still a complex exponential signal but modified by an eigenvalue, which in general is a complex number. This collection of eigenvalues provides a certain transfer function.

They are actually the functions providing the spectral representation of a convolution operator. Eigenfunctions need to be in the space (here, L^2) to be called eigenfunctions and those complex exponentials are not.

A much simpler version of all that is to say that a convolution acts as a Fourier multiplier (with its FT) in the frequency domain.

If you are going into details of difference between spectrum of an operator and eigenfunctions, or when they are different, then we are not going into that here - that is not the scope of this discussion.

But you did. Those complex exponentials are not eigenfunctions. A typical convolution operator (with a regular enough kernel) has a continuous spectrum; and no eigenvalues. Some people would still call such objects "eigenfunctions" but they would put that term in quotes.

Complex exponentials being eigenfunctions of an LSI system, and Fourier Transform as being multiplier in Fourier domain of a convolution in space/time domain, are related facts - one derived using other. This eigenfunction property is an important property with many uses in LSI systems and is not just obvious by stating that Fourier Transform acts as a multiplier, IMHO.

In that case (a convolution operator), it is equivalent.

Finally, that does not really explain why measuring the contrast with a sinusoidal input gives us the MTF at that frequency. It is a simple calculation but it needs to be done (and it requires some assumptions).

I mentioned in the OP that when contrast is measured using a specific method called modulation contrast, then the resulting transfer function, which is called MTF in optics, is the same as magnitude of the Fourier Transform of impulse response in systems theory. This can be shown but it is not immediately obvious.

Yes, that was my point, as well.

 

[Joofa]

• Complex Exponential Signals are eigenfunctions of an LSI system. When they pass through a LSI the output is still a complex exponential signal but modified by an eigenvalue, which in general is a complex number. This collection of eigenvalues provides a certain transfer function.

They are actually the functions providing the spectral representation of a convolution operator. Eigenfunctions need to be in the space (here, L^2) to be called eigenfunctions and those complex exponentials are not.

A much simpler version of all that is to say that a convolution acts as a Fourier multiplier (with its FT) in the frequency domain.

If you are going into details of difference between spectrum of an operator and eigenfunctions, or when they are different, then we are not going into that here - that is not the scope of this discussion.

But you did. Those complex exponentials are not eigenfunctions. A typical convolution operator (with a regular enough kernel) has a continuous spectrum; and no eigenvalues. Some people would still call such objects "eigenfunctions" but they would put that term in quotes.

As I said before that without bothering about the differences in spectrum and eigenfunctions, which is a mathematical curiosity, though important and good to know, it is well-known in systems theory to treat complex exponentials as eigenfunctions. For e..g, please see the following link:

http://ptolemy.eecs.berkeley.edu/eecs20/week9/lti.html

--
Dj Joofa
http://www.djjoofa.com

 

[technoid]

Since you guys have a EE background, here's what I use for MTF:

The impulse (at Nyquist) of this 15 bit sequence, when repeated, has special properties: [1 0 0 0 1 0 0 1 1 0 1 0 1 1 1 ]; You then interpolate it using Matlab's interp() function.

It produces an even amplitude spectral pattern at:
1/15, 2/15, 3/15, 4/15, 5/15, 6/15 and 7/15 f.

Think of it as overlapping sine waves at these freqs. of constant magnitude. Best of all, the phase of each is such that the peak to peak magnitude of the pattern is the smallest possible so it makes for a really good pattern that one can place on an wedge, focus in the middle, then scan for max resolution. Since it has no frequencies beyond 7/15th spacially, you don't have to worry about where the camera pixels line up.

Here's the full Matlab script:

% makePR15
g=[1 0 0 0 1 0 0 1 1 0 1 0 1 1 1 ];
gg=repmat(g,1,16);
gg=interpft(gg,360*16);
gg=gg/(max(gg)-min(gg)); gg=gg-min(gg); % scale 0 to 1
gg=gg*.9+.1; % scale .1 to 1.0
gg=reshape([gg gg gg], 1, 360*16, 3);
gg=repmat(gg, 200, 1);
imwrite(imag, 'scale.tiff');

Load it in photoshop then assign a gamma=1 profile to it. Print using Colorimetric Intent w/o BPC.

Capture in raw and extract the green color pixels magnitudes. You can then run an psd on each row and you have 7, evenly spaced points of MTF info.

 

[J A C S]

• Complex Exponential Signals are eigenfunctions of an LSI system. When they pass through a LSI the output is still a complex exponential signal but modified by an eigenvalue, which in general is a complex number. This collection of eigenvalues provides a certain transfer function.

They are actually the functions providing the spectral representation of a convolution operator. Eigenfunctions need to be in the space (here, L^2) to be called eigenfunctions and those complex exponentials are not.

A much simpler version of all that is to say that a convolution acts as a Fourier multiplier (with its FT) in the frequency domain.

If you are going into details of difference between spectrum of an operator and eigenfunctions, or when they are different, then we are not going into that here - that is not the scope of this discussion.

But you did. Those complex exponentials are not eigenfunctions. A typical convolution operator (with a regular enough kernel) has a continuous spectrum; and no eigenvalues. Some people would still call such objects "eigenfunctions" but they would put that term in quotes.

As I said before that without bothering about the differences in spectrum and eigenfunctions,

You mean - spectrum and eigenvalues?

which is a mathematical curiosity,

You seem to be a curious guy. The difference is much more than a "curiosity". It has important implications.

though important and good to know, it is well-known in systems theory to treat complex exponentials as eigenfunctions. For e..g, please see the following link:

http://ptolemy.eecs.berkeley.edu/eecs20/week9/lti.html

Note that there is no underlying Hilbert space in that link; while there is one in your post. You tried to connect that theory to the theory of linear operators in Hilbert spaces, and that connection was unnecessary and incorrectly done. On the other hand, it can be done, and it is not that complicated.

 

[Joofa]

• Complex Exponential Signals are eigenfunctions of an LSI system. When they pass through a LSI the output is still a complex exponential signal but modified by an eigenvalue, which in general is a complex number. This collection of eigenvalues provides a certain transfer function.

They are actually the functions providing the spectral representation of a convolution operator. Eigenfunctions need to be in the space (here, L^2) to be called eigenfunctions and those complex exponentials are not.

A much simpler version of all that is to say that a convolution acts as a Fourier multiplier (with its FT) in the frequency domain.

If you are going into details of difference between spectrum of an operator and eigenfunctions, or when they are different, then we are not going into that here - that is not the scope of this discussion.

But you did. Those complex exponentials are not eigenfunctions. A typical convolution operator (with a regular enough kernel) has a continuous spectrum; and no eigenvalues. Some people would still call such objects "eigenfunctions" but they would put that term in quotes.

As I said before that without bothering about the differences in spectrum and eigenfunctions,

You mean - spectrum and eigenvalues?

which is a mathematical curiosity,

You seem to be a curious guy. The difference is much more than a "curiosity". It has important implications.

though important and good to know, it is well-known in systems theory to treat complex exponentials as eigenfunctions. For e..g, please see the following link:

http://ptolemy.eecs.berkeley.edu/eecs20/week9/lti.html

Note that there is no underlying Hilbert space in that link; while there is one in your post. You tried to connect that theory to the theory of linear operators in Hilbert spaces, and that connection was unnecessary and incorrectly done. On the other hand, it can be done, and it is not that complicated.

I'm sorry, while I can mistakes any time, I tend to think that I didn't make one here. You seem to have a tendency to argue. If you have an issue with taking exponential functions in LSI systems as eigenfunctions then please do a google search and take issue with all the authors including the ones in the link I posted.

--
Dj Joofa
http://www.djjoofa.com

 

[Jack Hogan]

technoid wrote: Since you guys have a EE background, here's what I use for MTF...

OK technoid, I'll bite. Taking a capture of this with a digital camera and good technique produces a fairly accurate estimate of the system's MTF curve in the direction of the edge normal.



How does the MTF curve one gets out of you guys' process look like?

Jack

 

[J A C S]

• Complex Exponential Signals are eigenfunctions of an LSI system. When they pass through a LSI the output is still a complex exponential signal but modified by an eigenvalue, which in general is a complex number. This collection of eigenvalues provides a certain transfer function.

They are actually the functions providing the spectral representation of a convolution operator. Eigenfunctions need to be in the space (here, L^2) to be called eigenfunctions and those complex exponentials are not.

A much simpler version of all that is to say that a convolution acts as a Fourier multiplier (with its FT) in the frequency domain.

If you are going into details of difference between spectrum of an operator and eigenfunctions, or when they are different, then we are not going into that here - that is not the scope of this discussion.

But you did. Those complex exponentials are not eigenfunctions. A typical convolution operator (with a regular enough kernel) has a continuous spectrum; and no eigenvalues. Some people would still call such objects "eigenfunctions" but they would put that term in quotes.

As I said before that without bothering about the differences in spectrum and eigenfunctions,

You mean - spectrum and eigenvalues?

which is a mathematical curiosity,

You seem to be a curious guy. The difference is much more than a "curiosity". It has important implications.

though important and good to know, it is well-known in systems theory to treat complex exponentials as eigenfunctions. For e..g, please see the following link:

http://ptolemy.eecs.berkeley.edu/eecs20/week9/lti.html

Note that there is no underlying Hilbert space in that link; while there is one in your post. You tried to connect that theory to the theory of linear operators in Hilbert spaces, and that connection was unnecessary and incorrectly done. On the other hand, it can be done, and it is not that complicated.

I'm sorry, while I can mistakes any time, I tend to think that I didn't make one here.

It is not just a mistake, it is lack of knowledge. I would not have said anything if you had stuck to an exposition like that in the link but you started throwing remarks about self-adjoint operators. Once you decide to go there, you have to do it right but you lack the background for that.

BTW, your eigenvalues make sense even if the frequency is complex. Does it mean that every complex number is an eigenvalue as well, even of the kernel is symmetric about the origin (and its FT is real-valued)?

You seem to have a tendency to argue.

It takes two to argue.

If you have an issue with taking exponential functions in LSI systems as eigenfunctions then please do a google search and take issue with all the authors including the ones in the link I posted.

Show me authors who talk about self-adjoint operators and LSI systems. Why are you ignoring the point I already made? I guess they exist, and I am curious to see what they say. If you do not specify your space, you can take one containing the complex exponentials and then you do get eigenfunctions. But when you mentioned self-adjoint/Hermitian operators, you made an implicit choice: L^2. Then you have a problem, those complex exponentials do not belong there.

 

[technoid]

technoid wrote: Since you guys have a EE background, here's what I use for MTF...

OK technoid, I'll bite. Taking a capture of this with a digital camera and good technique produces a fairly accurate estimate of the system's MTF curve.



Where's the MTF you get out of you guys' process?

I get, and wanted, 7 points, not a curve. For instance, at 10,20,30,40,50,60,70 l/mm. These have the advantage of being pretty accurate because they are averaged over many horizontal lines, all the same but at the cost of not having a continuous MTF. The results from corner targets with were then used, with another process, to determine the image plane alignment accuracy.

Jack