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Dear All, Sorry to resurrect this FSC skirmish, but I was finding this one very educational compared to previous tussles, but would like to understand better. (btw I think using Nyquist period e.g. “2/3 nyquist” is confusing so I’m using Nyquist frequency, so “3/2” or “1.5 Nyquist") Is aliasing between 2x Nyquist frequency and Nyquist "2-1N" in the individual particles a problem after reconstruction? I imagine that the many different observations of the 2-1N frequencies effectively samples them, but I am not sure how they are recovered and preserved. Steven, isn’t Dimitry right that 2-1N aliasing in the experimental data is not critical since alignment based on lower freqs will recover signal for the real waveform and the aliasing will contribute noise? Won’t alignment of increasing experimental data reinforce signal of the ‘real’ frequencies before they were undersampled? (Are these recovered frequencies preserved in Fourier space without loss, in order that resampling is lossless, Dimitry?) Or are aliasing artefacts in this range also reinforced by reconstruction? I hope I’m not the only person with an appetite to understand the debate better - thanks for sharing expertise and views! All the best Teige On 3 Sep 2018, at 19:35, Dimitry Tegunov <[log in to unmask]<mailto:[log in to unmask]>> wrote: Wait, is this about what the features of a map like EMD-0144 (Who got 0143?) would look like in Chimera? I don't think anyone would look at that map without first resampling it on a grid at least 2x as fine. In case it isn't, I'd like to reiterate: the aliasing caused by under-sampling the raw data will be just additional 0-mean noise when averaging everything in 3D. A map reconstructed from 0.8 A/px data won't be different from one reconstructed from 0.5 A/px data if both are filtered identically and resampled on the same grid. As can be seen with EMD-0144 and EMD-9599. On Mon, Sep 3, 2018 at 6:37 PM Ludtke, Steven J <[log in to unmask]<mailto:[log in to unmask]>> 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 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. 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. 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. 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. -------------------------------------------------------------------------------------- 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/>) Co-Director 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 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. 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? Cheers, Dimitry ######################################################################## To unsubscribe from the CCPEM list, click the following link: https://urldefense.proofpoint.com/v2/url?u=https-3A__www.jiscmail.ac.uk_cgi-2Dbin_webadmin-3FSUBED1-3DCCPEM-26A-3D1&d=DwIFaQ&c=ZQs-KZ8oxEw0p81sqgiaRA&r=GWA2IF6nkq8sZMXHpp1Xpg&m=dzG4INJqk9Niq7zSShsu9mZb3Q1kqybUVPwN7eRH234&s=ii43w7MbFj1V2eYlRIviZzRoDp_YB4aH_eXMFpympEw&e= ________________________________ To unsubscribe from the CCPEM list, click the following link: https://www.jiscmail.ac.uk/cgi-bin/webadmin?SUBED1=CCPEM&A=1 The Francis Crick Institute Limited is a registered charity in England and Wales no. 1140062 and a company registered in England and Wales no. 06885462, with its registered office at 1 Midland Road London NW1 1AT ######################################################################## To unsubscribe from the CCPEM list, click the following link: https://www.jiscmail.ac.uk/cgi-bin/webadmin?SUBED1=CCPEM&A=1