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I have a picture of a hexagonally close-packed lattice and I took the FFT of the image using ImageJ. Below are the results. I expected the FFT to also be a lattice with reciprocal lattice spacing, I can see the lattice spacing, but I also see a large ring. Does this ring have significance?

HCP lattice image

FFT of HCP lattice

I have some background knowledge of how Fourier Transforms work but I have little experience with them in practice so I'm not familiar with the different types of artifacts.

Thanks,

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  • $\begingroup$ I don't actually have a clue, but here's a question: why is one of the dots on the original picture blue instead of black? Is this on purpose? Does the FFT change much when this dot is black like the others? $\endgroup$
    – Dan Shved
    Mar 21, 2016 at 15:12
  • $\begingroup$ I don't know why the centre point is blue, this image is not mine, I just wanted to practice using FFTs. I changed it to black and the FFT image is not significantly changed. $\endgroup$
    – user668074
    Mar 21, 2016 at 15:20

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The ring appears, because your input is not a perfect Dirac comb, its size should increase if you use a higher resolution image. The mathematical description, using the convolution theorem of the Fourier transform would be the following: Your input is a Dirac comb convoluted with the shape of you dots. For that reason the FFT of your input gets multiplied with the FFT of the shape of your dot. The smaller the dot, the larger the ring. That's the reciprocity in Fourier space. Additionally you have the cross artifacts form the finite size of your input, they can be explained the other way around.

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