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Hi Steffan,
I've always worked under the assumption that the 2.3 (sometimes 2.35) was an empirical factor to take the possibility of under-estimating the cutoff frequency into account. It should be noted that this factor is also often used when choosing a sampling frequency in electronics, audio, and radio applications, so it's not specific to some resolution metric. If I had to hazard a guess, it looks awfully like the factor that you multiply the std. deviation of a Gaussian by to get the FWHM, it could equally be related to the the typical frequency response of anti-aliasing filters.
For true Nyquist sampling in microscopy you should theoretically use the band limit, rather than the resolution (ie the less than half the Abbe resolution formula for widefield, and half that again for confocal) - resulting in confocal pixel sizes on the order of 45 nm regardless of pinhole size. This does not seem necessary in practice, and might well be overkill. Does anyone know of a rigorous analysis of how low the OTF must be for it to effectively be considered as zero? 
cheers,David 

     On Monday, 19 January 2015 8:52 AM, Zdenek Svindrych <[log in to unmask]> wrote:
   

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Dear all,
Steffen's argument is simply not right, sorry to say that. It is without 
doubt that in the fourier-transformed image there is more headroom in the 
diagonal directions, so higher frequencies can be encoded with the same 
sampling. In real space it's illustrated here:
https://drive.google.com/file/d/0B5vWyBYrDvcJckZNbE5lRVIyaUk/view?usp=
sharing
The Nyquist is, however, somewhat puzzling. The following notes may not 
clarify the issue much:
(1) There is a hard limit of the OTF, it's the 'lambda/2NA' criterion 
(properly adjusted for confocal/SIM/STED/...). But usually the magnitude of 
the OTF rolls off quickly and you are left with nothing but noise even at 
frequencies well below the hard limit.
(2) Review what Nyquist says: any band-limited signal sampled at frequency 
at least twice the bandwidth can be restored exactly. But his sampling was 
very different, he considered sampling of 1D (e.g. electrical) signal at 
discrete time points (I call it 'sampling with delta-functions'). In 
widefield microscopy each pixel integrates all pixels hitting the area of 
that pixel ('sampling with box functions'). This sampling attenuates the 
highest frequencies (compare fourier-transforms of a delta-function to that 
of a box function), i.e. the frequiencies that are so weak and precious in 
microscopy...
(3) Most of the time our photon budget is limited and the associated poisson
noise is critical for the final resolution. Maybe the simple OTF concept is 
not appropriate and should be replaced by something like 'Stochastic 
Transfer Function' (Somekh et al).
(4) Also there are other effects, such as limited modulation transfer 
function of camera chips, that further attenuate the highest frequencies, 
calling for finer sampling.
Bottom line? I think 2.3 x the_ultimate_frequency_limit is sufficient 
sampling.
Best, zdenek
-- 
Zdenek Svindrych, Ph.D.
W.M. Keck Center for Cellular Imaging (PLSB 003)
University of Virginia, Charlottesville, USA
http://www.kcci.virginia.edu/workshop/index.php



---------- Původní zpráva ----------
Od: Christian Soeller <[log in to unmask]>
Komu: [log in to unmask]
Datum: 19. 1. 2015 7:47:31
Předmět: Re: Nyquist and the factor 2.3

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Hi Steffen,

First, I guess we may be seeing a long thread as Nyquist always seems to 
resonate with microscopists.

I have heard about and read the 'diagonal argument' and I think it is 
flawed, at least as presented. My reason for that is as follows: IF a 2D 
signal is bandwidth limited at a frequency f, i.e. if all Fourier components
outside a circle with radius f are truly zero, then it can be mathematically
rigorously shown that sampling it on a 2D grid with spacing < 1/2f is 
sufficient to fully reconstruct the original signal. This makes no reference
to the "orientation of features etc" and argues that the diagonal argument 
cannot be strictly speaking correct. (There are some issues with a signal of
limited spatial extent being incompatible with finite bandwidth).

It may well be that the 'diagonal argument' leads to a result that is sort 
of "the right result" but I think does that by incorrect reasoning for 
reasons as above. At the very least I would like to see how those who think 
the diagonal argument is ok deal with the rigorous Fourier result which can 
be found in the literature.

One issue is the term "The Resolution". The frequency response of a 
microscope (OTF) rolls of in a characteristic way and it is not always clear
how this can be properly captured by one number ("The Resolution"). Choices 
are, for example, the FWHM of the lateral PSF, some XX dB rolloff of the 
frequency response (and then using the inverse of that) etc. For example, 
the FWHM "resolution" r_fwhm does not mean that the frequency response 
outside the circle with radius 1/r_fwhm is zero. Therefore sampling with r_
fwhm/2 is generally not enough. In that sense a factor of >2 may be thought 
of as a safety factor to take care of the fact that "The Resolution" may be 
underestimating the true finest detail where a frequency response is still 
distinguishable from 0.

In that sense before thinking about the "right factor" it may be more 
important to clarify how "resolution" may be properly defined (and 
measured).

If you look at the Nyquist calculators for deconvolution these generally 
recommend what might appear to be quite fine sampling.

Happy to hear other's perspective.

Cheers,
Christian



On Monday, 19 January 2015 at 11:32 AM, Steffen Dietzel wrote:

> *****
> To join, leave or search the confocal microscopy listserv, go to:
> http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
> Post images on http://www.imgur.com and include the link in your posting.
> *****
> 
> Hi all,
> 
> Nyquist again: I gather that the actual Nyquist criterion says that
> pixel size must be smaller than 1/2 the physical resolution. In the
> literature, I also find the factor 1/2.3 and I wonder where the 2.3
> comes from. Is this just one interpretation of <1/2 or is this the
> result of some calculation of which I could not find the source?
> 
> (I am aware that if the structure of interest is oriented diagonally to
> the pixel pattern, an additional factor of 1.41 comes into play, see
> discussion on this list in April 2012 or chapter 4 in the Handbook, so
> that it could be argued the factor should rather be <1/2.8 or 1/3.2, but
> my question is about the origin of the 2.3).
> 
> Steffen
> 
> 
> -- 
> ------------------------------------------------------------
> Steffen Dietzel, PD Dr. rer. nat
> Ludwig-Maximilians-Universität München
> Walter-Brendel-Zentrum für experimentelle Medizin (WBex)
> Head of light microscopy
> 
> Marchioninistr. 27
> D-81377 München
> Germany
> 
> 
>"