f-stop limits for full sensor resolution

[Ray]

There's a much easier and faster option:

The attached images are crops of a 1Ds macro shot with the 35-350L at f/32, which I sharpened and processed normally, saved the first crop, downsized to 2MP (1154x1734), upsized back to original size, and saved the second crop. I'll let you all compare for yourselves and come to your own conclusions.
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Jonathan,
That's quite interesting that you should interpret Myhrvolds's article in this way. I simply don't read it this way. There's no doubt in my mind that an 11mp shot at f22, downsized to 2mp then upsized back to 11mp, will have lower resolution than the original 11mp image. I wouldn't even bother to do the experiment to confirm this, although I recognise that the scientific method requires that one test the obvious. Sometimes there are surprises.

My interpretation of Myhrvold's  assertions are that the higher MTF response of a lens which is diffraction limited at f8 will give the same resolution with a 2mp camera as f22 will produce with a 12 or 16mp camera; same sensor size.

Below are the relevant quotes from his article. I agree that clarity could be improved.

Camera Max f/stop Max practical f/stop

Canon 5D f/8.6 f/9
Canon 1Ds f/9.3 f/9
Canon 1Ds Mark II f/7.6 f/8
Nikon D2X f/5.8 f/5.6
Canon 20D f/6.76 f/7.1
Canon G7 f/2.06 f/2

Note that in each case this is the maximum f-stop to get the full resolution of the camera. It is perfectly OK to stop the lens down further than this, but know that when you do, you will be getting less than the full resolution. This may or may not matter – lots of people obsess about resolution pointlessly (pun intended).

So, for example, if you take a Canon EOS 1Ds Mark II and stop it down to f/9, you are going to get greater depth of field, but you will not get the full 16 million pixel resolution – instead you’ll get resolution more like a 5D or 1Ds. That is still plenty good for many purposes. In fact, if you want the greater depth of field, then it may well be worth it. So, I still stop down to f/16 or even f/22 on occasion, but only when I decide that depth of field is more important that resolution – everywhere. When I know that I want to make a very large print, I stay at or below the maximum f-stop for diffraction limited resolution.
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Now, I don’t think anybody would be very excited about turning their EOS 1Ds Mark II, or Canon 5D or other full frame camera into a 2 megapixel camera. It sounds pretty drastic, but that is exactly what you do when you stop down to f/22 – the diffraction limit imposes this condition. If you shoot with a full frame 24 x 36 sensor at f/22 you are throwing away a lot of resolution. There is no getting around this – it is fundamental in the physics of light.

This comes as a shock to many photographers I have talked to, who assume that f/22 is the way to get the sharpest possible prints. Well, it just ain’t so. Diffraction limits your resolution at high f-numbers.
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Note that this is not quite the same as saying that it is equal to a 2 megapixel camera – that would only be true if the 2 megapixel camera was exposed with an f-stop that was within its diffraction limit.

I would add in reference to his comment,

This comes as a shock to many photographers I have talked to, who assume that f/22 is the way to get the sharpest possible prints

, that I think there would be few photographers who think that f22 will get them the sharpest print. I don't know who Myhrvold has been talking to, but everyone who knows anything about 35mm photography surely knows that f8 is more often than not the f stop that gives them the sharpest results, but not of course the greatest DoF.

However, there's something in the mathematics that might lead one to suppose that a lens which is truly diffraction limited at f8 is capabale of rendering the same detail with a 2mp camera as the same lens at f22 will render with a 16mp camera.

The problem is, I don't believe there are any 35mm lenses available which are diffraction limited at f8. Or to put it another way, I don't believe there are any 35mm lenses which have double the resolution at f8 that they have at f16.

[Jonathan Wienke]

I don't think anyone is arguing that diffraction does not have a degrading effect on image quality at f/22, but Myhrvold's statement that diffraction's effect when shooting at f/22 turns a "full frame camera into a 2 megapixel camera" is clearly wrong, as my example shows. 7-10MP, maybe, but certainly not 2. Even with the lens aberrations (which the 35-350 has plenty), it's definitely more than 2MP. Even given your interpretation of his statement, he's still way off. He's correct that diffraction decreases resolution as aperture diameter decreases, but way off with regard to how much.

[John Sheehy]

There's a much easier and faster option:

1. Shoot a detailed subject at  f/22 or smaller.

2. Downsize the the shot to 2MP.

3. Upsize the shot back to the original pixel dimensions.

4. Compare the original and the downsized-upsized images. If Myhrvold's assertions are correct, there won't be much difference between the two. But if the original is significantly better, he misplaced some decimal points or something.
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One caveat here; downsampling and sampling back to original resolution is not the same thing as having an image at the lower resolution.  WHen you downsample, you wind up with a better image MTF and color resolution than you ever could have had with a lower-MP original version from the camera.  Of course, this helps your case even more, since a real 2MP image would have had poorer MTF, and less pixel-meaningful color.  IOW, your worst image would have looked even worse with a real 2MP camera.

[Jonathan Wienke]

One could also argue that the double resampling process introduces some artifacts of its own, but in general I'd say your point is valid.

If theory and reality correlate poorly, it's generally not reality's fault.

[bjanes]

There's a much easier and faster option:
I'll let you all compare for yourselves and come to your own conclusions.
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I examined the images that Jonathan posted, and IMO the first has greater detail. However, I think that further analysis is needed before one can draw definite conclusions. Conclusions without any numeric quantification and analysis are always subjective, and Lord Kelvin (Sir William Thomson, 19th century British scientist, whose temperature scale we use in white balance) has summed up the matter:

"I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the state of Science, whatever the matter may be." [PLA, vol. 1, "Electrical Units of Measurement", 1883-05-03]

Drs. Johnson and Myhrvold have published brilliant essays that explain their basic assumptions very well, but they have caused controversy when they extend their conclusions. Johnson states that "I conclude that the Canon 1Ds, Mark II, with a pixel pitch of 7.2 microns, can use all of its resolution to describe an image at f/22", which is somewhat cryptic. Myhrvold states that stopping down to f/22 causes his 16 mp camera to have 2 MP effective resolution.

The Brian Wadell et al paper from Stanford relates resolving power to the Nyquist limits of digital sensors for both monochrome and color work with a Bayer array imager. For monochrome, maximal resolution is achieved when the diffraction spot radius is equal to the pixel pitch, but for a Bayer array the effective pixel size is the 2 by 2 pixel kernel. The math becomes quite complex and an observational approach is more understandable for most of us. Myhrvold derives his 2 MP figure from the size of the Airy disc at f/22 using an assumed relationship with resolution.

[a href=\"http://www.photozone.de/8Reviews/lenses/canon_60_28/index.htm]Photozone[/url] has published resolution data for the Canon EF-S 60 mm f/2.8 macro and EOS 350D using Imatest. The results are expressed in line pairs/picture height, and this can be converted to lp/mm. This camera has a 22.4 x 14.8 mm sensor with pixel dimensions of 3458 x 2304 pixels, giving a pixel pitch of 6.4 microns and MP value of 8. The Nyquist limit for this sensor is 78 lp/mm. The data are summarized in here in tabular form:

[attachment=1879:attachment]

In modern optical theory using MTF, one must always state the contrast for a given resolution. Resolution without reference to contrast is ambiguous. Roger Clark has published a table relating resolution to contrast and lens aperture for green light and the data are summarized in this graph. When one stops down to f/22, the Rayleigh resolution (about 9% MTF) is 75 lp/mm, only slightly below Nyquist as Dr. Johnson states, but this level of contrast does not yield a good image.

[attachment=1869:attachment]

Perceived sharpness in an image is related to the resolution at 50% contrast (MTF 50), and these values have been determined observationally by Klaus at Photozone. In the following table, I have interpolated the results for f/11 and f/22 and also calculate an effective megapixel resolution using the MTF 50 resolution for the 22.4 by 14.8 mm sensor.

[attachment=1881:attachment]

It is apparent that maximal MTF 50 is achieved at about f/4-f/5.6 when the Airy disc size is about equal to the pixel spacing. As one stops down further, the Airy disc expands and the MTF 50 decreases. At f/22, the effective resolution falls to 2.9 MP for this 8 MP camera.

Now perhaps Jonathan can repeat his experiment for 2.9 MP  . And it would be useful to have additional comments by the original expert authors on these matters.

Bill

[BJL]

It has great significance in the context of Myhrvold's claim that using a lens at f22 with a 16mp camera such as the 1Ds2, is equivalent to using a 2mp camera of the same pixel pitch at f8, a claim that is widely disputed in this thread.
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But Ray, you were talking about lenses not being diffraction limited at f/4, and comparing lens sharpness at f/4 to f/8, and resolutions limits at those relatively large apertures are not relevant to Myhrvold's claim that when you stop down to f/22 you turn [a] full frame camera into a 2 megapixel camera (and he uses the word "exactly".) Performance at f/4 is only relevant to this sort of claim if one moves to pixel size of about 4 microns or less, or 54MP territory for 35mm format.

I would hope that all my 35mm format lenses are largely diffraction limited by f/22, and my FourThirds lenses by f/11.  (Because otherwise they are "Coke bottles".)

[Ray]

But Ray, you were talking about lenses not being diffraction limited at f/4, and comparing lens sharpness at f/4 to f/8....

BJL,
Just as an aside in case some readers confuse the aperture at which a lens is sharpest as the aperture at which it is diffraction limited.

I would hope that all my 35mm format lenses are largely diffraction limited by f/22, and my FourThirds lenses by f/11.  (Because otherwise they are "Coke bottles".)

Perhaps you could do some experiments with your Olympus E1 and finest prime Zuiko lens at f22 and f8 and tell us how many pixels at f8 are equivalent to 5mp at f22.

These could be your first images posted on this site   .

[Jonathan Wienke]

It is apparent that maximal MTF 50 is achieved at about f/4-f/5.6 when the Airy disc size is about equal to the pixel spacing. As one stops down further, the Airy disc expands and the MTF 50 decreases. At f/22, the effective resolution falls to 2.9 MP for this 8 MP camera.

I disagree. My images prove that at f/32, the Airy disc size is not significantly larger than 1 pixel on a 1Ds. I'm seeing resolution approaching single-pixel level with a reasonably good level of contrast. The formulas are simply not correct; they do not accurately predict real-world results. As I said before, when reality and theory diverge, it is not reality's problem. I may not be able to exactly quantify what I'm seeing, but it is obvious that the real-world results do not match what the formulas (at least those used by Myhrvold) predict.

Another thing. Bayer sensors are capable of resolving much better than a 2x2 pixel kernel. The AA filter prevents the sensor from resolving all the way to single-pixel detail, but resolution much better than 4:1 is certainly possible.

[Ray]

I don't think anyone is arguing that diffraction does not have a degrading effect on image quality at f/22, but Myhrvold's statement that diffraction's effect when shooting at f/22 turns a "full frame camera into a 2 megapixel camera" is clearly wrong, as my example shows. 7-10MP, maybe, but certainly not 2. Even with the lens aberrations (which the 35-350 has plenty), it's definitely more than 2MP. Even given your interpretation of his statement, he's still way off. He's correct that diffraction decreases resolution as aperture diameter decreases, but way off with regard to how much.
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He's certainly way off by your interpretation and he's also way off by my interpretation, although clearly not by as much.

I'm simply making the point that the major reason he is way off (by my interpretation) is due to some confusion as to what is meant by 'diffraction limitation'. As BJL has pointed out, there is a state of transition over a range of f/stops from the first f/stop (when stopping down) that begins to show the first hint of diffraction, to the f /stop that shows not the slightest hint of any of the other aberrations, just the effects of diffraction.

Myhrvold has made the following statements that I repeat from his article, which you appear to have missed. The implication here, I would suggest, is that Myhrvold is not saying what you attribute to him.

Camera Max f/stop Max practical f/stop
Canon 1Ds Mark II f/7.6 f/8

In other words, if you want the sharpest result s with the 1Ds2, do not stop down below f8 (although the theoretical limit is f7.6).

Note that this is not quite the same as saying that it is equal to a 2 megapixel camera – that would only be true if the 2 megapixel camera was exposed with an f-stop that was within its diffraction limit.

I understand by 'an f-stop within its diffraction limit' to mean an f-stop at which the lens is fully diffraction limited.

[John Sheehy]

I understand by 'an f-stop within its diffraction limit' to mean an f-stop at which the lens is fully diffraction limited.
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What exactly would that mean, though?

In my mind, unless optical effects of diffraction cause the entire sensor to recieve the same light, there will always be *some* contrast between neighboring pixels, concerning a real-world high-contrast edge or point.  Just like the AA filter knocks down the possible contrast between adjacent pixels, so does diffraction.  And like the AA filter, the effects of diffraction are easily dealt with in digital, where math is at your command.

[Jonathan Wienke]

Myhrvold has made the following statements that I repeat from his article, which you appear to have missed. The implication here, I would suggest, is that Myhrvold is not saying what you attribute to him.

Bull***t. The statement about a full-frame DSLR turning into a 2MP camera at f/22 is a direct quote from Mhyrvold. And it is clearly incorrect.

[Ray]

What exactly would that mean, though?
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The degree to which sharpening can compensate for the softening of 35mm images at f16 and f22 is another issue. As you know, whole books have been devoted to nothing but sharpening routines.

Generally, my impression is that at f22, some low contrast detail is completely lost and cannot be recovered by any amount of skillful sharpening. At f8, that same low contrast detail will probably be captured. Furthermore, such detail can then be enhanced with appropriate sharpening.

[Ray]

Bull***t. The statement about a full-frame DSLR turning into a 2MP camera at f/22 is a direct quote from Mhyrvold. And it is clearly incorrect.
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Jonathan,
I'm surprised at your sense of logic. Someone with a name so difficult to spell (and remember) as Myhrvold, must know something.  

[Jonathan Wienke]

Generally, my impression is that at f22, some low contrast detail is completely lost and cannot be recovered by any amount of skillful sharpening. At f8, that same low contrast detail will probably be captured. Furthermore, such detail can then be enhanced with appropriate sharpening.

Nobody is disputing that diffraction has a negative effect on resolution at small apertures like f/22. The image I posted required more local contrast enhancement and more aggressive sharpening than a typical f/8 image, no question. I think a more realistic estimate of full-frame diffraction-limited resolution is something like 8MP at f/32, 32MP at f/16, and 128MP at f/8. It's certainly better than 2MP @ f/22.

[bjanes]

I disagree. My images prove that at f/32, the Airy disc size is not significantly larger than 1 pixel on a 1Ds. I'm seeing resolution approaching single-pixel level with a reasonably good level of contrast. The formulas are simply not correct; they do not accurately predict real-world results. As I said before, when reality and theory diverge, it is not reality's problem. I may not be able to exactly quantify what I'm seeing, but it is obvious that the real-world results do not match what the formulas (at least those used by Myhrvold) predict.
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Your disagreement is preposterous. The pixel spacing of the 1Ds MII is 7.21 microns and the Nyquist frequency is 69 lp/mm. For green light (500 nm), the Airy disc is 36 microns in diameter. At this aperture, the MTF 50 is 24 lp/mm, the resolution at Rayleigh (about 9% contrast) is 51 lp/mm and the resolution at Dawes (0% contrast) is 63 lp/mm. At 9% contrast, you will be able to resolve very high contrast objects such as star pairs in the night sky (the raison d'etre of the Rayleigh criterion), but this level of contrast is not very useful for practical photography of average scenes. You can recover some contrast with proper sharpening, but aliasing artifacts will also be accentuated. The mathematical derivation of the size of the Airy disc is given [a href=\"http://cnx.org/content/m13097/latest/]here[/url], along with nice 3 dimensional graphs of the discs at Rayleigh. I don't think your pictures at f/32 prove anything in a scientific way.

Myhrvold is a very smart guy, but I agree that the 2MP figure is off and he should clarify his derivation of that figure.

Another thing. Bayer sensors are capable of resolving much better than a 2x2 pixel kernel. The AA filter prevents the sensor from resolving all the way to single-pixel detail, but resolution much better than 4:1 is certainly possible.
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The 2x2 kernel has more to do with aliasing than actual resolution, according to the paper by [a href=\"http://white.stanford.edu/~brian/papers/ise/CMOSRoadmap-2005-SPIE.pdf]Wandell[/url] from the electrical engineering department at Stanford University. The anti-alaising filter is not completely effective, and if you look at the resolution tests at Dpreview.com near Nyquist, you will see plenty of detail, but much of it is false detail: aliasing. The MTF 50 of your system (in terms of lp/mm) would be slightly inferior to that of the D350 as shown on Photozone. However, the MTF 50 in terms of picture height would be better.

The mathematical modeling of Wandell et al is quite sophisticated. Just as nuclear scientists can test the design of an atom bomb with super computers, obviating the need for an actual detonation, Wandell could enter the parameters into his computer and end this discussion.

Bill

[Jonathan Wienke]

Your disagreement is preposterous. The pixel spacing of the 1Ds MII is 7.21 microns and the Nyquist frequency is 69 lp/mm. For green light (500 nm), the Airy disc is 36 microns in diameter. At this aperture, the MTF 50 is 24 lp/mm, the resolution at Rayleigh (about 9% contrast) is 51 lp/mm and the resolution at Dawes (0% contrast) is 63 lp/mm. At 9% contrast, you will be able to resolve very high contrast objects such as star pairs in the night sky (the raison d'etre of the Rayleigh criterion), but this level of contrast is not very useful for practical photography of average scenes. You can recover some contrast with proper sharpening, but aliasing artifacts will also be accentuated. The mathematical derivation of the size of the Airy disc is given here, along with nice 3 dimensional graphs of the discs at Rayleigh. I don't think your pictures at f/32 prove anything in a scientific way.

They certainly prove that the 1Ds with a decent, but not excellent, lens at f/32 can still capture 11MP of image data with an MTF somewhere on the sunny side of 50%. Your MTF 50 figures are what is preposterous; if they were correct, the smallest details in my spider image would be 3-4 pixels in size, and the hairs in the first image would be indistinguishable form those in the second image. Since single-pixel-wide details are obviously present in the first image (the body/leg hairs), and are equally obviously NOT aliasing artifacts, there is an obvious error in the formulas being used to predict MTF-50 resolution for a given aperture and a diffraction-limited lens. I don't know exactly what the error is, or where it is, but there certainly is an error somewhere.

[01af]

Myhrvold is obviously not an expert on the question of optical diffraction effects on digital sensors.

Obviously not. He seems to believe the resolution resulting from a chain of two imaging devices (lens and sensor) where each has its own resolution limit was equal to the lower resolution of the two. I'd call this the "weakest-link theory." And this theory is wrong.

Actually the resulting resolution R[span style=\'font-size:8pt;line-height:100%\']res[/span] depends on the sequence of two input resolutions R[span style=\'font-size:8pt;line-height:100%\']1[/span] and R[span style=\'font-size:8pt;line-height:100%\']2[/span] like this:

1/R[span style=\'font-size:8pt;line-height:100%\']res[/span] = 1/R[span style=\'font-size:8pt;line-height:100%\']1[/span] + 1/R[span style=\'font-size:8pt;line-height:100%\']2[/span]

If this formula looks familiar to you---yes, it's the same that also describes the resulting resistance of two parallel resistors.

What does this mean? Let's say we have a sensor that due to pixel pitch can resolve up to 40 lp/mm. And we have three lenses that, for a given subject contrast, can resolve 40 lp/mm, 60 lp/mm, and 80 lp/mm respectively. Now, when using the 40 lp/mm lens on the 40 lp/mm sensor this seems like a good match, doesn't it? And using the better lenses on that sensor seems like a waste of resolving power as the poor sensor cannot exploit it, right? Wrong! Actually on the 40 lp/mm sensor, the 60 lp/mm lens will yield a sharper image than the 40 lp/mm lens, and the 80 lp/mm lens a sharper image still (albeit not twice as sharp as the 40 lp/mm lens).

Of course this works out the same the other way around. When using a lens that can resolve, say, 40 lp/mm, then a 60 lp/mm sensor will yield a sharper image than a 40 lp/mm sensor, and an 80 lp/mm sensor will yield a still sharper image (albeit not twice as sharp as the 40 lp/mm sensor).

So, the Myhrvold threshold of "f-stop equal to pixel spacing in microns" is a completely wrong concept---even when augmented by a corrective factor or two. Generally, the optimal f-stop roughly correlates to image size (among other things, the most important being lens quality) ... but not to pixel count or pixel pitch. With all other things equal, a higher pixel count will establish a higher overall resolution level and will make the degradation more obvious---but it will occur at the same aperture.

... observations of several users of the Nikon D2X, with 5.5 micron pixel spacing, say that diffraction starts to limit resolution at somewhere between f/8 and f/11.

With good lenses, this matches my own observations exactly. And my D-SLR camera has only half the D2X's pixel count (that is, 6 MP) and consequently, a pixel pitch of 7.8 microns which is 1.4× the pixel pitch of the D2X. So according to Myhrvold I should see diffraction setting in at apertures one stop smaller than D2X owners. But as a matter of fact I am seeing it at the same aperture as D2X owners do. And the only thing my camera has in common with the Nikon D2X is the image size which is APS-C (form factor 1.5×, relative to 35-mm format).

With good or very good (but not exceptional) lenses on APS-C format (form factor 1.5× or 1.6×), diffraction starts to become visible---upon very close inspection!---at apertures between f/8 and f/11 typically ... no matter what the pixel count is. With 35-mm-format cameras, the limiting f-stop is somewhere between f/11 and f/16. With medium-format cameras, it's around f/22. And so on.

With exceptionally good lenses, the limit is reached at apertures one or maybe even two stops larger (i. e. smaller f-stop numbers).

However, don't let these facts keep you from stopping down beyond these limits whenever you need the depth-of-field! Overall image quality does not so much depend on highest resolution at the plane of focus. If a composition depends on DOF then give it all the DOF it needs (but not more)!

-- Olaf

[BJL]

Diffraction does not smear light into a clearly defined disk with an unambiguous diameter. Instead, intensity falls of gradually from the center out to zero at a certain radius (the edge of the Airy disk), with further weak rings of light continuing beyond that. In principle, the rings continue out to infinity, but no one infers from that that diffraction reduces resolution to zero; instead, one has to ask what fraction of light is smeared by a certain distance, and look at questions like what MTF one achieves at a given length scale of number on line pairs per mm.

According to Norm Koren at http://www.normankoren.com/Tutorials/MTF6.html#Diffraction diffraction alone give 50% MTF at 0.38/( N *W ) lp/mm where N is aperture ratio, W is the wavelength of light, and this corresponds to line pairs of width 1.078 times the Airy disk diameter of 2.44*N *W. That is, close enough to line pair width equal to the Airy disk diameter. By the usual estimate that it takes between two and three Bayer interpolated pixels to resolve a line pair, this suggest that for a sensor to extract all the "50% MTF detail" allowed by diffraction would need pixel spacing P between 1/2 and 1/3 of the Airy disk diameter, a factor of two or three smaller than the pixel size beyond which Myhrvold claims that results will be _exactly_ the same, due to resolution being determined by diffraction effects.

This suggests that for any given aperture ratio N, pixel size reduction can continue to improve resolution at least down to somewhere in the range 1.3*N*W to 0.88*N*W. Using wavelength W=555nm as Johnson does, the range is about P=0.7*N to P=0.5*N.

Turning this around, at a given pixel size P, one can probably increase aperture ratio at up to the range from N=1.4*P to N=2*P, and go to even higher aperture ratios if one can settle for significantly less than 50% MTF in the finest details of interest.

Honestly, I did not pre-plan the calculations to fit so nicely to my previous observation-based rule of thumb of aperture ratio one or two stops higher than pixel spacing in microns.

[01af]

Honestly, I did not pre-plan the calculations to fit so nicely to my previous observation-based rule of thumb of aperture ratio one or two stops higher than pixel spacing in microns.

And still this seemingly perfect fit is just pure coincidence ... as I explained in my previous post. Actually the aperture where diffraction will set in visibly does not I repeat: NOT depend on the pixel pitch. In several cameras with different pixel counts but equal image sizes (and with the same lenses) you will always observe the same limiting f-stop.

-- Olaf

[bjanes]

Actually the resulting resolution R[span style=\'font-size:8pt;line-height:100%\']res[/span] depends on the sequence of two input resolutions R[span style=\'font-size:8pt;line-height:100%\']1[/span] and R[span style=\'font-size:8pt;line-height:100%\']2[/span] like this:

1/R[span style=\'font-size:8pt;line-height:100%\']res[/span] = 1/R[span style=\'font-size:8pt;line-height:100%\']1[/span] + 1/R[span style=\'font-size:8pt;line-height:100%\']2[/span]
If this formula looks familiar to you---yes, it's the same that also describes the resulting resistance of two parallel resistors.
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Your formula applies only for MTF around the Rayleigh resolution criterion for contrast of 9%. For a more realistic MTF of 50% one must convert the MTFs into the frequency domain via a Fourier transform, multiply the frequency components, and then perform an inverse transform back to the spatial domain via a complicated process called convolution. This is explained by [a href=\"http://www.normankoren.com/Tutorials/MTF.html]Norman Koren[/url] on his web site.

Bill