Dear Petrus, There are a range of flaws in the *Shannons 1949 proof *of the *sampling-theorem* that are not overly simple to understand, and which is the reason it is still seen as a "*Law of Physics*". To understand the details you need to scrutinise my first 2 papers in my series on *symmetric-sampling*. A very *serious methodological flaw* is that Shannon a-priory assumes the input signal to be *deterministically known*. Yet, *deterministic signals/functions *are fundamentally not measurable and *cannot *be exactly represented in a *computer*. That means *the proof of the theorem* is *cyclical* and therefore *false*. There are many other issues with that paper (as well as with the preceding *1948 *one). The *measuring device *is also assumed to be deterministic and thus assumed to have* a Linear Transfer Function* that reaches into *infinity*! So the answer to your legitimate question is *BOTH*. Also, the *Nyquist Sampling sampling theorem* is de-facto only derived for digitizing *1D REAL signals* and cannot be used for 2D aor 3D signals (and even not be used for 1D signals as I have shown it to be untenable). This has led to* scores of erroneous approaches* in *all of Physics* including the resolution criteria considered standard in *Cryo-EM* and in *X-ray crystallography*. Had the problems been obvious, it would not have taken Science more than a century to resolve these issues. One of the culprits in creating *this historical mess* is the *Amplitude and Phase representation* of *complex data* which creates a *singularity at the origin* in *Fourier Space* and which also implicitly assumes the *measured data to be real*. The* Real and Imaginary *representation of complex data does not have that problem. (More in my *Why-o-Why* series in Linked in soon, I hope).
Cheers, Marin On Mon, Oct 6, 2025 at 5:58 PM Petrus Zwart <[email protected]> wrote: > > What is at stake here is that the* Nyquist-Shannon theorem* is > fundamentally *flawed *(Shannon 1949) > > You do mention that in your first paper - is the issue the math or the > assumptions or real world applicability or both? > > P > > > > > On Mon, Oct 6, 2025 at 1:33 PM Marin van Heel < > [email protected]> wrote: > >> *Sorry Tim*... >> The problem is *not limited to Cryo-EM* .... It affects* all forms of >> measurements*, including the measurement of *X-ray diffraction patterns*! >> What is at stake here is that the* Nyquist-Shannon theorem* is >> fundamentally *flawed *(Shannon 1949). In *X Ray Crystallography* you >> unfortunately lose the phases collecting such X Ray diffraction patterns, >> but there is one good thing about that, namely: the *low frequency >> diffraction peaks* disappear behind the central-beam stop! The >> discrete, *symmetric-sampling approach* - for the sake of linearity - >> requires both the *low frequencies* and the* high spatial frequencies* >> to be apodized to zero long before reaching those limits of the sampling >> space. (After all, the zero-order frequency of a deterministic *Continuous >> Fourier Transform*, covers the whole universe in one direction and that >> is not a measurable perspective on the physical world we live in!). >> >> So sorry Tim, you are not giving a valid argument for *ignoring the >> Measurement Problem*! >> But it is great to hear from you again, after so many years! >> >> Cheers, >> Marin >> >> >> On Mon, Oct 6, 2025 at 10:46 AM Tim Gruene <[email protected]> >> wrote: >> >>> I am so happy I do diffraction, not microscopy >>> Tim >>> >>> On Sat, 4 Oct 2025 19:13:10 -0300 >>> Marin van Heel <[email protected]> wrote: >>> >>> > Dear All, >>> > >>> > Electron counting cameras in Cryo-EM have an unexpected direct >>> > connection to the Measurement Problem in Quantum Mechanics. In fact, >>> > the measurement problem already existed in Classical Newtonian >>> > Physics due to its Deterministic Mathematical Axioms and is not >>> > specifically tied to Quantum Mechanics. The Measurement Problem also >>> > existed in Shannon’s sampling theorem. All measurements, and not only >>> > those in Electron Microscopy must adhere to symmetric-sampling rules. >>> > Check the link to a series of papers: >>> > >>> > >>> https://www.linkedin.com/feed/update/urn:li:activity:7379515419424022528/ >>> > >>> > Sorry, I know it is tough stuff… >>> > >>> > Marin >>> > >>> > >>> ######################################################################## >>> > >>> > To unsubscribe from the CCP4BB list, click the following link: >>> > https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1 >>> > >>> > This message was issued to members of www.jiscmail.ac.uk/CCP4BB, a >>> > mailing list hosted by www.jiscmail.ac.uk, terms & conditions are >>> > available at https://www.jiscmail.ac.uk/policyandsecurity/ >>> >>> >>> >>> -- >>> -- >>> Tim Gruene >>> Head of the Core Facility Crystal Structure Analysis >>> Faculty of Chemistry >>> University of Vienna >>> >>> Phone: +43-1-4277-70202 >>> >>> https://ccsa.univie.ac.at >>> >>> GPG Key ID = A46BEE1A >>> >> >> ------------------------------ >> >> To unsubscribe from the CCP4BB list, click the following link: >> https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1 >> > > > -- > > ------------------------------------------------------------------------------------------ > Peter Zwart > Staff Scientist, Molecular Biophysics and Integrated Bioimaging > Berkeley Synchrotron Infrared Structural Biology > Biosciences Lead, Center for Advanced Mathematics for Energy Research > Applications > Lawrence Berkeley National Laboratories > 1 Cyclotron Road, Berkeley, CA-94703, USA > Cell: 510 289 9246 > > ------------------------------------------------------------------------------------------ > > ######################################################################## To unsubscribe from the CCP4BB list, click the following link: https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1 This message was issued to members of www.jiscmail.ac.uk/CCP4BB, a mailing list hosted by www.jiscmail.ac.uk, terms & conditions are available at https://www.jiscmail.ac.uk/policyandsecurity/
