mjkzz wrote:My head hurts now with all these PSF and MFT, fourier transform, convolution of PSF, etc, etc.
I sympathize. It has taken me years to become comfortable with this material. But I have
spent those years -- including experimental validation -- and I am
comfortable with the material. So I do not mind admitting that I feel some slight annoyance when somebody who does not
understand the material apparently tries to argue with me about it.
That said, let me now thank you for prompting me to simplify the presentation.
First, please note that the method I'm using is exactly the same as what Nikon uses on their Matching Camera to Microscope Resolution
So if we're going to appeal to authority, then I'm appealing to Nikon and you're appealing to cambridgeincolour. I know which one I would trust more regarding scientific issues, but if you feel differently, I will not try to change your mind.
I would, however, like to make clear what the technical issues are. For that, I'm going to use math first, then follow up with a bit of experimental validation.
Assuming a circular aperture and an aberration-free lens, here are the governing equations:
(Note: I'm using the "quote" tag here only for its formatting, not to imply that I'm copying letter for letter material that appears someplace else. If the forum provided an [indent] tag, I would use that.)
lambda: wavelength of the light we're working with
Feff: effective F-number = 1/(2*NA) in the same focus plane
Airy disk diameter = 2.44*lambda*Feff
cutoff frequency nu_0 = 1/(lambda*Feff)
cutoff wavelength = 1/nu_0 = lambda*Feff
Nyquist sampling interval at cutoff frequency = cutoff wavelength / 2 , that is, two samples per cycle.
Now just do the algebra:
cutoff wavelength = lambda*Feff = Airy disk diameter / 2.44
Nyquist sampling interval at cutoff = cutoff wavelength / 2 = (Airy disk diameter / 2.44) / 2 = Airy disk diamater / 4.88
The conclusion, in English, is that you need at least 4.88 pixels per Airy disk diameter to meet the Nyquist criterion for capturing the finest detail that is present in the optical image.
Any sampling rate that is less dense will
result in losing information, and will likely result in aliasing some of the high frequency information so that it looks like lower frequency information.
Barring typos, these statements are facts of physics. If you disagree with the statements, then please point out the errors in my text.
To understand the effects of magnification, we need one more simple fact: mag/Feff is constant at all focus planes in the system.
Combining that fact with the equation relating NA and Feff in the same focus plane, we get the relationship for a microscope that
Feff_sensor = magnification / (2*NA_subject)
We now have enough formulas in hand to step through the calculations on the Nikon web site.
Example: for magnification 10 and NA 0.25, Nikon quotes a required pixel size of 5.5 microns. Stepping through the calculation...
Feff_sensor = 10/(2*0.25) = 20
Airy disk diameter = 2.44 * lambda * Feff = 2.44 * 0.55 * 20 = 26.84 microns.
Cutoff wavelength = Airy disk diameter / 2.44 = 11 microns
Nyquist sampling interval at cutoff = cutoff wavelength / 2 = 11/2 = 5.5 microns
To summarize: Nikon's table says 5.5 microns is the minimum required pixel size; the formulas say that 5.5 microns is Nyquist limit, two pixels per cycle, at the diffraction-limited cutoff frequency.
You can repeat the calculation for yourself, for any other row in Nikon's table. The numbers all say the same thing: the required pixel size is 4.88 pixels per Airy disk diameter, to capture all the information that is present in the optical image.
Now, going back to cambridgeincolour's text, what their words say is that "an airy disk can have a diameter of about 2-3 pixels before diffraction limits resolution
Clearly there is a discrepancy between 2-3 pixels per Airy disk (cambridgeincolour) and 4.88 pixels per Airy disk (Nikon).
The challenge is how to understand that discrepancy.
The way I make sense of the discrepancy is that cambridgeincolour and Nikon are simply addressing different questions:
- Nikon is addressing the question of what sensor do you need, to capture all the information in the optical image for the aperture you have in hand.
- Cambridgeincolour is addressing the question of what aperture do you need, before something bad happens to the image captured by the sensor you have in hand.
mjkzz wrote:The bottom line is, I do NOT agree with the analysis using spatial cutoff frequency (as Rik did) because ultimately, it is the convolution factor of PSF that limits maximum resolution
I see some confusion in this statement, given that you're arguing against my application of Nikon's method. For an aberration-free lens, the PSF is
just the Airy disk, and it is
convolution with that PSF which establishes the cutoff frequency. So on the one hand it sounds like you believe in the relevance of convolution with the PSF, but on the other hand you're arguing against the conclusions that pop out of that analysis.
If you want experimental confirmation of the numbers, go to http://www.photomacrography.net/forum/v ... 164#101164
, look at the f/11 images, and read the accompanying text.
Reproducing the images here, on the left is a blowup of an f/11 optical image, and on the right is that optical image captured by direct projection onto a camera sensor:
The accompanying text reads as follows:
rjlittlefield wrote:What we have here is an f/11 image that is clearly not fully captured by a 15 mp APS sensor (pixel size 4.7 microns).
The Airy disk at f/11 is 14.7 microns. Using Nikon's method, an f/11 image requires pixel size 3 microns, so the experimental result at 4.7 microns is no surprise. In comparison, using cambridgeincolour's guideline of 2-3 pixels per Airy disk diameter would suggest a pixel size around 4.9 to 7.4 microns. That criterion is met by the particular camera that I'm using, and clearly it does not capture all the line pairs that are present in the optical image.
The conclusion seems pretty clear: Nikon's method gives a good number for Nikon's question, while cambridgeincolour's method does not. Presumably cambridgeincolour's method does
give a good number for whatever issue it is that they're addressing. It's just not clear exactly what that issue is.
[Edits: tweak wordings about aliasing & what question is being addressed by each model. ]