Dear all,
This is my first posting to CCPEM :) .
It sounds as if the matters discussed bear some relationhip to
the investigation of errors introduced into the Fourier spectrum of a
bandlimited function by linear interpolation from the values of that
function on a grid. I had the opportunity to look into this question
in the Appendix at the end of the following paper:
https://journals.iucr.org/a/issues/1976/05/00/a12981/a12981.pdf
when implementing phase improvement by noncrystallographic symmetry
through iterative map averaging. My final conclusion (the key step in
reaching it being illustrated by Figure 6) was that unless the said
symmetry was of very high order, it was necessary to use a sampling
interval of 1/6 or 1/5 of the "resolution", i.e. at 3.0 or 2.5 times
the Nyquist frequency, to keep interpolation errors within acceptable
limits. I used this sampling rate in my own phasing calculations on
the Tobacco Mosaic Virus coat protein disk and the Tomato Bushy Stunt
Virus. A competing group who at the time was both algorithmically and
CPUresourcewise challenged resisted this conclusion, claiming that
such a sampling rate was unnecessarily high and that the "traditional"
(one third of the resolution) rate was enough. A few years later, when
computing resources were no longer so limiting, their landmark Nature
paper on another virus structure simply stated that "All averaging
calculations were performed at a sampling interval of one fifth of the
resolution"  considering it as obvious.
Steven underlines the fact that it takes 4 points to completely
and unambiguously specify a sinusoidal oscillation, so one might
wonder why one would need 5 or 6 points. The fact is that an input
sinusoid liearly sampled between 4 points, then Fourier transformed,
would come back with an attenuation factor; and that the rms power
that is shaved off that term in this attenuation reappears as noise
through the "sidebands" as shown in Figure 6.
In any case, this is just to point out that the damage done in
Fourier space by linear interpolation in real space should never be
overlooked. It has caused confusion that this "damage" has been much
underestimated by computing a correlation coefficient between *maps*
and concluding that the sampling rate mattered little. However this is
because map correlation coefficients are dominated by the large terms
at low spatial frequencies  since, by Parseval's theorem, they are
equal to a FSC computed in a single shell containing all the Fourier
data  and are therefore quite insensitive to the degradation of the
smaller highfrequency terms that is caused by linear interpolation.
I hope this connection to another instance of a similar problem
may be helpful.
With best wishes,
Gerard.

On Mon, Sep 03, 2018 at 04:37:37PM +0000, Ludtke, Steven J wrote:
> Again, I am NOT arguing that FFTs are inconsistent in some way, or that aliasing will prevent you from achieving FSC curves giving you "resolution" past 2/3 Nyquist. It is absolutely possible to perform iterative refinements which extend beyond 2/3 Nyquist, and always has been (ie  this isn't something new with "modern software").
>
> The point is that the real space representation of signal between 1/2 Nyquist and Nyquist has significant artifacts because the sine waves with frequencies in this range do not have complete information in the original image. In realspace, you must have 4 pixels (1/2 Nyquist), not 2 pixels (Nyquist) to completely and unambiguously specify a sinusoidal oscillation. Between 2 pixels and 4 pixels you have partial information. This does not mean you cannot achieve Fourier space reconstructions which are self consistent to Nyquist, it means that there are artifacts in the realspace representation.
>
> When you do Xray crystallography you are sampling directly in Fourier space, and as Pawel said (assuming you have the right phases), you can oversample the results in realspace as much as you like to produce nice smooth densities, the details of which will be limited by the highest order reflection you use.
>
> In CryoEM, we are making measurements in realspace, meaning the information between 1/2 Nyquist and Nyquist is incomplete at the time the data is measured. I used the +1,1,+1,1 example because it is the easiest case for people to picture. That is, it is clear that if you try to measure a pattern with exactly Nyquist periodicity, if you see a signal with some amplitude, you cannot tell if the amplitude you observe is correct, with zero phase, or if it is a sampling of a phaseshifted signal with much higher amplitude. This ambiguity extends partially all the way to 1/2 Nyquist, with odd spatially localized patterns. At 1/2 Nyquist periodicity, full information is present.
>
> So, the argument is that beyond 1/2 Nyquist, you will have realspace artifacts which can lead to misinterpretation when doing model building and other tasks, but that to ~2/3 Nyquist the effect is pretty minimal.
>
> 
> Steven Ludtke, Ph.D. <[log in to unmask]<mailto:[log in to unmask]>> Baylor College of Medicine
> Charles C. Bell Jr., Professor of Structural Biology
> Dept. of Biochemistry and Molecular Biology (www.bcm.edu/biochem<http://www.bcm.edu/biochem>)
> Academic Director, CryoEM Core (cryoem.bcm.edu<http://cryoem.bcm.edu>)
> CoDirector CIBR Center (www.bcm.edu/research/cibr<http://www.bcm.edu/research/cibr>)
>
>
>
> On Sep 3, 2018, at 10:46 AM, Dimitry Tegunov <[log in to unmask]<mailto:[log in to unmask]>> wrote:
>
> Dear Steven,
>
> thank you for the examples.
>
> However, I'm not sure the Nyquist sine wave is the best example of aliasing. It is one extreme case valid only for the FFT of evensized, realvalued signals. To circumvent this behavior of the FFT without breaking any of your initial conditions, please consider this experiment: Fourierpad the signal by a factor of 2 to make space for the original Nyquist frequency component's Friedel buddy, shift back and forth by 0.5*2, Fouriercrop back to original size, find no changes in the original pattern. For the opposite, fill an evensized window with noise, shift back and forth by a noninteger value, find the Nyquist frequency component corrupted. FFTbased noninteger shifts in evensized windows are lossless up to, but not including, Nyquist.
>
> The PDB example, indeed, illustrates the aliasing in a single undersampled observation. Now let's consider a pipeline where the only undersampled observation of the signal in real space is made at the image acquisition stage. All subsequent resampling is performed in Fourier space with sufficient padding in real space. The result is an average of many independently aliased observations of the underlying non bandlimited signal. Sure, the aliasing corrupts each initial observation (and not only its Nyquist frequency), but this noise will be independent between the halfmaps and thus won't artificially increase the FSC. As far as I can tell, it will also be 0mean – resulting in perfectly fine maps beyond 2/3 Nyquist. Am I missing something?
>
> Cheers,
> Dimitry
>
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