# How to get the scale bar of an FFT for a 2d image

I've been analyzing some images from a transmission electron microscope, including their FFTs, and I'm not sure how to apply a scale bar to the FFT images. I have calibrations for the real space image, and I was thinking it might just be the inverse of that scale, but I really have a hard time visualizing reciprocal space and would like to be sure.

This seems like it would have an easy answer, but I haven't been able to find it and I would appreciate your help.

• Would Signal Processing be a better home for this question? – Qmechanic Jan 7 '17 at 11:39

## 1 Answer

If you are having trouble visualizing the reciprocal space, perhaps an analogy to 1D temporal FFTs will help (personally I'm more familar with these, so an analogy helps me anyway)

In a 1D temporal FFT, the input is N samples, at a rate of Fs samples/second, for a total time of N/Fs seconds. The output is N samples, distributed between 0 and Fs (or between -Fs/2 and +Fs/2 depending on your point of view) (units 1/second). So each output sample has width Fs/N (again units 1/second).

In a 1D spatial FFT, all we do is replace seconds by meters and samples by pixels. The input is N pixels, and a rate of Fx pixels per meter, for a total width of N/Fx meters. The output is N pixels, distributed between 0 and Fx (or between -Fx/2 to Fx/2) (units of 1 / meter). So each pixel has width Fx/N (agains units 1/meter).

So an output scale bar of M pixels has a width of M*Fx/N in units of 1/meters.

To go from 1D spatial FFT to 2D spatial FFT, just need a sampling rate in each direction. Presumably you have the same sample rate in each directin.