Re: Nyquist sampling advice for a short talk

Posted by James D. Manton-2 on
URL: http://confocal-microscopy-list.275.s1.nabble.com/Nyquist-sampling-advice-for-a-short-talk-tp7591495p7591497.html

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Hi Dale,

I'm going to make some suggestions, of which I think many on the list
will find at least a few objectionable. I'll list them all first, then
provide justification below.

== Suggestions ==

1. Don't bother mentioning the Airy disk, Rayleigh and Sparrow criteria,
etc..

2. Use the Abbe criterion for determining theoretically achievable
resolution, both laterally and axially, with the sampling being at twice
this rate.

3. Don't overemphasise the difference between a point scanner and a
camera-based system. The point scanner lets you place your pixels where
you want them, but the job being performed, and hence the sampling
conditions (modulo confocal pinhole effects), are the same.

4. Measuring the resolution of an image is, in many cases, more useful
than measuring the resolution of an instrument. While many people know
how to measure PSFs, few seem to know how to achieve the former. For
this, I recommend the Demmerle et al. article at
https://doi.org/10.1016/j.ymeth.2015.07.001. Fourier Ring Correlation
should be avoided if at all possible.

5. If you expect your audience to know Fourier transforms, it's worth
talking about the OTF as the Fourier dual of the PSF. This lets you show
how the noise floor alters the resolution limit given by Abbe, and hence
the sampling condition, with a new cutoff of the Fourier support.

6. If you don't expect your audience to know Fourier transforms, don't
try and include anything Fourier-related (including point 4). 15 minutes
is not enough time.


== Justifications ==

1. The Rayleigh and Sparrow criteria were developed in a background of
naked eye stellar astronomy, where the question was whether an observer
could distinguish a double star from a single star. These observations
are very different from the vast majority of microscopical observations
for which the image is much more complicated. Only the Abbe criterion is
based on fundamental physical law and is the only one which is readily
combined with the understanding of signal theory that was developed
later, in the 20th century (i.e. Abbe + Nyquist-Shannon makes sense,
while Rayleigh + Nyquist-Shannon does not).

2. The Abbe criterion gives the lateral resolution as wavelength / (2 *
numerical aperture), and the axial resolution as wavelength /
(refractive index * (1 - cos(arcsin(numerical aperture / refractive
index))). These can be derived from a consideration of the Ewald sphere,
but I think that would be too much for a 15 minute video.
Nyquist-Shannon shows that the sampling rate should be at half those
distances. Extra factors of sqrt(2), or whatever, that people like to
throw in because we have 'square pixels' are erroneous and should
perhaps not be discussed for risk of confusion.

3. This falls over when we get into 'exotic' point scanning schemes like
Lissajous patterns and the corresponding Chebyshev polynomial sampling
theory, but hardly anyone in the world needs to worry about that.

4. The 'Radial Profile Extended' ImageJ plugin is quite useful for a
basic version of the Demmerle et al. Fourier Spectral Analysis
(https://imagej.nih.gov/ij/plugins/radial-profile-ext.html). Fourier
Ring Correlation relies on arbitrary thresholds with no underlying
physical justification and is harder to calculate. Showing a radially
averaged Fourier spectrum is more informative and more clearly shows the
difference between instruments where the absolute resolution limit is
the same but the intermediate transfer is different (e.g. ISM/Airyscan
vs SIM).

5. Both dominant noise types in microscopy (Poisson and Gaussian) are
spectrally white and hence contribute a flat noise floor in the Fourier
domain. Other sources of noise, such as the correlated noise from
different sub-detectors in an Airyscan, are much harder to reason about,
but they are infrequently as important.

6. True understanding of resolution limits and appropriate sampling
comes when people are confident with autocorrelating (complex) functions
and converting between real space and the Fourier domain. In my
experience, even most physics graduate students, who have already been
subjected to introductory courses on Fourier transforms, struggle with
this for much longer than 15 minutes.

I look forward to hearing people's arguments against these suggestions.

Good luck with the video,
James


--
James Manton
MRC Laboratory of Molecular Biology
Cambridge CB2 0QH, UK
[phone]