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<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif"><span> </span></p>
<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif">Dear
Gerard,<span></span></p>
<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif"><span> </span></p>
<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif">Sorry
for this late reaction but this discussion erupted at a time I was fully
occupied with the Brazil school (<a href="http://www.brazil-school.org">www.brazil-school.org</a>). <br></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif"><br><span></span></p>
<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif"><span> Reacting to your posting:<br></span></p>
<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif">“<i>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 CPU-resource-wise 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.</i>”<span></span></p>
<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif"><span> </span></p>
<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif">What we
suggested many years ago in EM is to (at least) use what you call the <i>"traditional" (one third of the
resolution) sampling rate.</i> <i><span> </span></i>What the abusers of this #twothirdsNyquist
rule are suggesting is that you can use up #fullNyquist and still get 'good results'. I agree with you that the traditional 2/3 Nyquist sampling frequency rule may in many cases even not
be sufficient and that realisation is supported by our calculations (paper
in preparation). (This may even go beyond the <span class="gmail-im">'4 pixels' ('1/2 Nyquist')</span> that Steve Ludke was discussing.) The argument that the #fullNyquist proponents are using for interpreting
the FSC beyond its validity range is that the results ‘look good’.
Good looking results are in the eye of the beholder but the objective 1982/1986
FRC/FSC metrics should not be used beyond their validity range
(<a href="https://en.wikipedia.org/wiki/Fourier_shell_correlation">https://en.wikipedia.org/wiki/Fourier_shell_correlation</a>). Those who use such metrics beyond their definition range are in fact creating a new metric that they themselves must justify. I suggest the name "My Shell Correlation" (MSC) for such validity extensions (he-he).
<span></span></p>
<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif"><br>
“<i>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
high-frequency terms that is caused by linear interpolation. I hope this
connection to another instance of a similar problem may be helpful</i>.”<span> </span><span></span></p>
<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif"><span> </span></p>
<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif">The connection is helpful and I fully
agree that “correlation coefficients are dominated by the large terms at low
spatial frequencies”.<span> </span>We even wrote a
separate paper about that topic <span style="font-family:"Segoe UI Emoji",sans-serif">😉</span>
(“Correlation function revisited”)
<a href="https://doi.org/10.1016/0304-3991(92)90021-B">https://doi.org/10.1016/0304-3991(92)90021-B</a>
. The considerations that all spatial frequencies should be weighted correctly were at the base of our earlier FRC/FSC proposals and the normalisation we chose for them. (Details in <a href="https://www.biorxiv.org/content/early/2017/11/24/224402">https://www.biorxiv.org/content/early/2017/11/24/224402</a>)<br></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif"> <br></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif">
<span></span></p>
<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif">Merry
Xmas<span></span></p>
<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:"Times New Roman",serif">Marin<br>
<br>
<br>
<span></span></p>
</div></div><br><div class="gmail_quote"><div dir="ltr">On Mon, Sep 3, 2018 at 2:23 PM Gerard Bricogne <<a href="mailto:gb10@globalphasing.com">gb10@globalphasing.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">Dear all,<br>
<br>
This is my first posting to CCPEM :-) .<br>
<br>
It sounds as if the matters discussed bear some relationhip to<br>
the investigation of errors introduced into the Fourier spectrum of a<br>
band-limited function by linear interpolation from the values of that<br>
function on a grid. I had the opportunity to look into this question<br>
in the Appendix at the end of the following paper:<br>
<br>
<a href="https://journals.iucr.org/a/issues/1976/05/00/a12981/a12981.pdf" rel="noreferrer" target="_blank">https://journals.iucr.org/a/issues/1976/05/00/a12981/a12981.pdf</a><br>
<br>
when implementing phase improvement by non-crystallographic symmetry<br>
through iterative map averaging. My final conclusion (the key step in<br>
reaching it being illustrated by Figure 6) was that unless the said<br>
symmetry was of very high order, it was necessary to use a sampling<br>
interval of 1/6 or 1/5 of the "resolution", i.e. at 3.0 or 2.5 times<br>
the Nyquist frequency, to keep interpolation errors within acceptable<br>
limits. I used this sampling rate in my own phasing calculations on<br>
the Tobacco Mosaic Virus coat protein disk and the Tomato Bushy Stunt<br>
Virus. A competing group who at the time was both algorithmically and<br>
CPU-resource-wise challenged resisted this conclusion, claiming that<br>
such a sampling rate was unnecessarily high and that the "traditional"<br>
(one third of the resolution) rate was enough. A few years later, when<br>
computing resources were no longer so limiting, their landmark Nature<br>
paper on another virus structure simply stated that "All averaging<br>
calculations were performed at a sampling interval of one fifth of the<br>
resolution" - considering it as obvious.<br>
<br>
Steven underlines the fact that it takes 4 points to completely<br>
and unambiguously specify a sinusoidal oscillation, so one might<br>
wonder why one would need 5 or 6 points. The fact is that an input<br>
sinusoid liearly sampled between 4 points, then Fourier transformed,<br>
would come back with an attenuation factor; and that the rms power<br>
that is shaved off that term in this attenuation reappears as noise<br>
through the "side-bands" as shown in Figure 6.<br>
<br>
In any case, this is just to point out that the damage done in<br>
Fourier space by linear interpolation in real space should never be<br>
overlooked. It has caused confusion that this "damage" has been much<br>
underestimated by computing a correlation coefficient between *maps*<br>
and concluding that the sampling rate mattered little. However this is<br>
because map correlation coefficients are dominated by the large terms<br>
at low spatial frequencies - since, by Parseval's theorem, they are<br>
equal to a FSC computed in a single shell containing all the Fourier<br>
data - and are therefore quite insensitive to the degradation of the<br>
smaller high-frequency terms that is caused by linear interpolation.<br>
<br>
I hope this connection to another instance of a similar problem<br>
may be helpful.<br>
<br>
<br>
With best wishes,<br>
<br>
Gerard.<br>
<br>
--<br>
On Mon, Sep 03, 2018 at 04:37:37PM +0000, Ludtke, Steven J wrote:<br>
> 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").<br>
> <br>
> 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 real-space, 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 real-space representation.<br>
> <br>
> When you do X-ray crystallography you are sampling directly in Fourier space, and as Pawel said (assuming you have the right phases), you can oversample the results in real-space as much as you like to produce nice smooth densities, the details of which will be limited by the highest order reflection you use.<br>
> <br>
> In CryoEM, we are making measurements in real-space, 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 phase-shifted 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.<br>
> <br>
> So, the argument is that beyond 1/2 Nyquist, you will have real-space artifacts which can lead to misinterpretation when doing model building and other tasks, but that to ~2/3 Nyquist the effect is pretty minimal.<br>
> <br>
> --------------------------------------------------------------------------------------<br>
> Steven Ludtke, Ph.D. <<a href="mailto:sludtke@bcm.edu" target="_blank">sludtke@bcm.edu</a><mailto:<a href="mailto:sludtke@bcm.edu" target="_blank">sludtke@bcm.edu</a>>> Baylor College of Medicine<br>
> Charles C. Bell Jr., Professor of Structural Biology<br>
> Dept. of Biochemistry and Molecular Biology (<a href="http://www.bcm.edu/biochem" rel="noreferrer" target="_blank">www.bcm.edu/biochem</a><<a href="http://www.bcm.edu/biochem" rel="noreferrer" target="_blank">http://www.bcm.edu/biochem</a>>)<br>
> Academic Director, CryoEM Core (<a href="http://cryoem.bcm.edu" rel="noreferrer" target="_blank">cryoem.bcm.edu</a><<a href="http://cryoem.bcm.edu" rel="noreferrer" target="_blank">http://cryoem.bcm.edu</a>>)<br>
> Co-Director CIBR Center (<a href="http://www.bcm.edu/research/cibr" rel="noreferrer" target="_blank">www.bcm.edu/research/cibr</a><<a href="http://www.bcm.edu/research/cibr" rel="noreferrer" target="_blank">http://www.bcm.edu/research/cibr</a>>)<br>
> <br>
> <br>
> <br>
> On Sep 3, 2018, at 10:46 AM, Dimitry Tegunov <<a href="mailto:tegunov@GMAIL.COM" target="_blank">tegunov@GMAIL.COM</a><mailto:<a href="mailto:tegunov@GMAIL.COM" target="_blank">tegunov@GMAIL.COM</a>>> wrote:<br>
> <br>
> Dear Steven,<br>
> <br>
> thank you for the examples.<br>
> <br>
> 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 even-sized, real-valued signals. To circumvent this behavior of the FFT without breaking any of your initial conditions, please consider this experiment: Fourier-pad 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, Fourier-crop back to original size, find no changes in the original pattern. For the opposite, fill an even-sized window with noise, shift back and forth by a non-integer value, find the Nyquist frequency component corrupted. FFT-based non-integer shifts in even-sized windows are lossless up to, but not including, Nyquist.<br>
> <br>
> The PDB example, indeed, illustrates the aliasing in a single under-sampled observation. Now let's consider a pipeline where the only under-sampled 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 band-limited signal. Sure, the aliasing corrupts each initial observation (and not only its Nyquist frequency), but this noise will be independent between the half-maps and thus won't artificially increase the FSC. As far as I can tell, it will also be 0-mean – resulting in perfectly fine maps beyond 2/3 Nyquist. Am I missing something?<br>
> <br>
> Cheers,<br>
> Dimitry<br>
> <br>
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