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I apologize first for some questions that might be dumb. Unfortunately, I am only a bachelor student (writing my bachelor thesis in physics) and have almost zero image processing experience. And I cannot receive much help offline in uni.
I summarized my questions first, then wrote the details with questions in each phase.


Edit 1:
Sample characterisation
characterisation
Taken by another microscope. Each element consists of one stab and two circles. The whole sample consists of four arrays, which each is formed by elements facing different direction. The length of the stab is about 425nm.

The raw tif image is larger than limitation. Uploaded to OneDrive.


Original:

Software: FIJI
Questions:
1. What is the relationship between spatial cutoff frequency and effective resolution in the following practical scene? How could they be converted to each other?
2. Can the effective resolution be directly read out by distance between 2D FFT function cutoffs?
3. What do vertical and horizontal axis of FFT result represent? Do they represent vertical and horizontal resolution?
4. How am I supposed to acquire vertical and horizontal resolution from my images?

Detailed situation:

About question 1: Horizontal ROI on FFT Vertical ROI on FFT Here are two images in my series as example. They are my current state of work. The ROI on raw image is FFTed. And the line profile of chosen area on FFT is plotted on the right side.
Based on instruction I received, I should get the cutoffs from 2D FFT function in frequency space, and the resolution at the end by getting cutoff. Now I am stuck and confused with what I should do. And I have found the following image online, which confused me too.
Source:https://opg.optica.org/directpdfaccess/90fe16f1-b0b4-4b36-97481908ad1720f8_380617/optica-5-1-32.pdf?da=1&id=380617&seq=0&mobile=no
My FFT result is similar to the upper left corner of Fig. 1. Here comes the first question:
1. What is the relationship between spatial cutoff frequency and resolution on Fig. 1 and on my FFT result? How could they be converted to each other?
I thought first that the author of Fig. 1 has spatial frequency as original unit of x-axis in FFT, but apparently not? It seems like he/she also has length unit as original unit on FFT image. So he/she did not convert spatial frequency to resolution on Fig. 1, but the opposite conversion?
I also know that the bright spot in centre on FFT image is zero frequency, the further spots are higher frequencies.
So if we take first image in the post as example.
Horizontal ROI on FFT
On the right side (Plot of FFT), the distance between two cutoffs are about 280nm. And there are 12 spikes (except zero frequency) in total on line profile on right side (Plot of FFT). Does that mean that spatial cutoff frequency is 12/280 nm^-1? Is this conversion correct?

About question 2:
2. Based on my considerations above about question 1, am I able to read out the resolution simply by distance between cutoffs? E.g. I mentioned the distance 280nm, which means the effective resolution is 140nm? Is this correct?

About question 3:
3. What do vertical and horizontal axis of FFT result represent? Do they represent vertical and horizontal resolution?
(I don't believe so.)
FFT_1FFT_2

About question 4:
4. How am I supposed to acquire vertical and horizontal resolution from my images? Horizontal ROI on FFT
The raw image is on the left side. There are probes heading 4 different directions in total. Am I supposed to choose 4 separate ROI, which each include one probe heading one direction? And do FFT, read cutoff, and so on?

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  • $\begingroup$ Please edit your question to cite your sources, including links -- especially links to things that aren't behind paywalls. Posting your raw image, without a surrounding window or any ROI selector graphics, would be helpful too -- sometimes someone will do computations on posted images, but only when they're clean. $\endgroup$
    – TimWescott
    Mar 9 at 23:52
  • $\begingroup$ "There are probes heading 4 different directions in total..." I'm seeing six or seven groups of blobs that could each be four little blobs with uneven spacing or three blobs of varying size. Please edit your question to clarify. $\endgroup$
    – TimWescott
    Mar 10 at 0:02
  • $\begingroup$ I apologize for that and raw image of the example here is added. Also a clearer image for characterisation is added too. $\endgroup$ Mar 10 at 13:42

1 Answer 1

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  1. What is the relationship between spatial cutoff frequency and effective resolution in the following practical scene? How could they be converted to each other?

Spatial frequencies are just that -- the frequency of occurrences of features in space. "100 dots per $\mu \mathrm m$" is another way of saying "dots spaced every 10nm".

The cutoff frequency is a measure of how much smearing there is. For identically-shaped spectra, if one spectrum is 1/10 as wide as the other, it represents 10 times as much smearing in the spatial domain.

If you choose the right cutoff magnitude, then the cutoff frequency, in cycles per pixel, or cycles per meter, or cycles per whatever, is the reciprocal of the resolution in pixels or meters or whatevers.

I'm not sure what you'll want to choose as a cutoff magnitude. I expect that you'll want to go with 3dB or 6dB below peak magnitude -- that would imply that the difference in intensity between a dot center and the space between two dots, at the minimum resolution distance, is 1/2 the feature intensity (3dB) or 1/4 the feature intensity. The image processing community has standard numbers they like to use, but I don't do that work often enough to remember -- you may wish to look it up.

  1. Can the effective resolution be directly read out by distance between 2D FFT function cutoffs?

You need to find the reciprocal -- remember that the FFT output is in the frequency domain, where the input is in the spatial domain.

  1. What do vertical and horizontal axis of FFT result represent? Do they represent vertical and horizontal resolution?

I don't know, because it looks like your tool is massaging the results somehow. Specifically, I see a rectangular ROI selected, and a square FFT output. This means that your tool is "helping" you, or is choosing to do power-of-2 FFTs. If you can't find this spelled out in the tool's documentation, and if it's at all possible, shove an image of a grid of known size into that, or horizontal or vertical stripes, or whatever, and get the FFT of that. Use the known, easy relationship between the frequency of the stripes or grid to the expected FFT output to calibrate your tool.

  1. How am I supposed to acquire vertical and horizontal resolution from my images?
  1. Take the FFT and use the relationship I explained between the cutoff frequency and resolution -- after checking the literature to see what cutoff magnitude you should use.
  2. Just eyeball the thing -- it looks to me like the resolution is somewhere between the spacing of the little dots and about half that much. If it's not enough to just eyeball it, then at least use that eyeball calibration to make sure you're in the right ballpark with your calculations from your FFT results.
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  • $\begingroup$ Thank you very much! I also did find other materials and books and with your explanation together, I am no longer stucked. $\endgroup$ Mar 10 at 14:50

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