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CCPEM  September 2018

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Subject:

Re: [3dem] GroEL best resolution map. (FSC RESOLUTION ??)

From:

Dimitry Tegunov <[log in to unmask]>

Reply-To:

Dimitry Tegunov <[log in to unmask]>

Date:

Mon, 3 Sep 2018 16:46:08 +0100

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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

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