Question about FFT

I have an image of hexagonal close-packed particles and I wish to find the distance between the centers of a particle and its closest neighbor particle. Using the manual counting method as explained here, I found that distance to be around 0.81um.

Line selection for plot profile

Plot profile
manual

Using FFT, I have an image with concentric rings about the centre. The radial distance of the innermost ring, which value should be the distance I wish to know, is 0.71um/cycle instead.

Area selection for FFT

FFT image

Why is there a discrepency? Am I missing something?

Image file

Good day,

I guess you are looking at the wrong Fourier-spectral components. The second spectral “ring” is what you want.

I get a radius of about 163 pel which gives a Fourier-frequency of about 1.23/µm which results in a periode of 0.81µm.

Now it’s your turn to find out why this is so. Hint: “height of an equilateral triangle”

Regards

Herbie

3 Likes

Hi Herbie,

Thanks! I think I know why I should look at the second ring instead. There are 2 periodic patterns; 1 along the line connecting the centers of nearest neighbor particles and the other connecting the particles along the height of the equilateral triangle formed by 3 close-packed particles. The latter of which have a lower frequency as particles are spaced further apart. Due to this, it corresponds to the ring nearer to the center.

However, I still don’t get a radius of 163 pel for the second ring. From the FFT image, I tried measuring the radius with the line tool and got ~325 pel. Am I doing it incorrectly?

Also, I don’t know how to convert the radius measured to Fourier-frequency. Any hints?

Regards,
Joey

Dear @anon96376101,

I am somewhat confused by your assessment.

  1. How do you calculate your formula of 163 pel > 1.23/um? What size FFT do you have? I would like to compare this to the results of the calibrated FFT which should give already the number of um per cycle on calibrated images. (https://imagej.nih.gov/ij/docs/examples/tem/)

  2. If I take the FFT and remove all frequencies except the innermost ring, take the inverse FFT, I get the features @joeyong is interested in.



  3. If i take the second ring


    I get the small gaps between the particles,

So is it just a problem with the calibration in the FFT plugin?

Good day Olivier,

of course I took the biggest square-sized 2^x window possible (same as the original poster), i.e. 1024x1024, and I hate to work with scales. Scales can be converted at the end of the processing if absolutely necessary, but this is my personal opinion.

With 1024x1024 the Nyquist frequency (1024pel in the frequency plane) is 7.725/µm which, for 163pel gives a frequency of 1.23/µm. Consequently the period is 1µm/1.23 = 0.813µm.

Does this make sense?

Best

Herbie

Dear Joey,

here is an example from a smaller excerpt of your image and its log Fourier power spectrum:


I’ve indicated the corresponding distances by colored lines.

Now here is the center of the 1024x1024-sized Fourier spectrum:


and the red line has a length of about 163pel – no?

Also, I don’t know how to convert the radius measured to Fourier-frequency. Any hints?

Please make sure that you understand Fourier-calculus and see my reply to Olivier.

Regards

Herbie

2 Likes

Dear @anon96376101,
Thank you for the explanations and details!