Hi everyone, just a small correction I need to point out: at 17:40 on slide 12 when I describe the W matrix, the rows actually correspond to different values of dk (row 1 is dk = 0, row 2 is dk = 1, ... row N is dk = N-1) while the columns correspond to different values of n (column 1 is n = 0, column 2 is n = 1, ... column N is n = N-1). Mathematically, W is equal to its transpose, so switching dk with n has no impact on the final result, but I still wanted to point this out to be consistent with the summation to calculate the DFT value for a given dk shown on the prior slide (slide 11).

Another important reason that FFTs and DPs are not equivalent is that the image used in the FFT is a 2D projection of a 3D object (the sample). Great presentation. Slide 16 is an excellent visualization.

@NicholasRudawski

3 ай бұрын

Yes, thank you very much for pointing out the 2D versus 3D difference. This is something that will be returned to again when I get to part 2.

That was a fantastic lecture that answered so many of my questions lingering for a while. I do not want to sound pushy, but when is 2nd part coming? 😅

Another high quality content video. Many thanks!

@NicholasRudawski

3 ай бұрын

You're welcome.

Awesome lecture! Helps a lot!!!! I have a quick question, if the fine structure of the TEM images are composed of round dots but with different radius size, say one image is composed of small white dots and the other image is composed of larger white dots with both random orientations, what differences should I expect to see when I do FFT for both images?

@NicholasRudawski

8 күн бұрын

Hey, thank you for your support and I'm glad my video helps you! About your question, it sounds like you are asking about the appearance of an FFT from two images with no long-range order. I would definitely expect to see some differences between the FFTs due to the size difference of the dots, because this affects the short-range ordering and average nearest neighbor distance (assuming the dots are packed together like a solid and not spread out like in a gaseous state). Presumably, there would be some kind of diffuse ring in the FFTs for each image, but the radius of the ring should be larger for the image with the smaller dots because the nearest-neighbor distance should be less. If you have a chance to try this "experiment" please let me know what you find!

Great tutorial! Clear and easy to follow. Slide 16, when we talk about the amplitude of 2D FFT, I just wordering how does it relate to the amplitude in real space? According to the Fourier theorem or conservation of energy, the two should be equal, which I don't understand clearly in the video.

@NicholasRudawski

2 ай бұрын

@user-wy4sy7dz3m thank you for watching my video. The best way to explain FFT calculated in terms of amplitude is as follows: consider a sine (or cosine) wave with spatial frequency = dk and amplitude = A. If I compute the FFT of the wave in terms of amplitude, I will see peaks at +/- dk and the height of the peaks will be equal to A. In other words, I can use the amplitude version of the FFT to determine the actual amplitude of each "wave" (spatial frequency) present in the image. I hope this helps and thank you for your support.

@user-wy4sy7dz3m

2 ай бұрын

​@@NicholasRudawski Thank you very much for your reply! Forgive me for asking this question for the first time in the comments. When I used the program to verify the formula about amplitude on Slide 18, the formula of 1D was fine, but the formula of 2D seemed to have different results. My python code is as follows: import numpy as np complex_matrix = np.array([[2.48 - 0.46j, 1.02 + 1.28j, -0.760 + 1.70j], [- 0.230 + 7.51j, 0.400 + 2.68j, -0.760 + 1.70j], [0.180 - 0.10j, -1.22 + 0.51j, -1.66 + 0.86j]]) print(np.sum(np.abs(complex_matrix))) print(np.sum(2 * np.abs(np.fft.fft2(complex_matrix)) / 3 / 3)) The two outputs are completely different results. I made more attempts and found that, in 2D: when N is an even number, the formula is revised as |S(Δk)| / N when N is an odd number, the formula is revised as 2|S(Δk)| / (N+1) which can give a reasonable result.

@user-wy4sy7dz3m

2 ай бұрын

I'm sorry, I just found out that the formula I gave is wrong, and I'm even more confused. Is there really no connection between the sum of the module length of real space and the sum of the module length of its FFT? I thought the amplitude you mentioned was a bridge between them.

@user-wy4sy7dz3m

2 ай бұрын

@@NicholasRudawski Ohh, I see! The following is reasonable complex_matrix = np.array([[2.48 - 0.46j, 1.02 + 1.28j, -0.760 + 1.70j], [- 0.230 + 7.51j, 0.400 + 2.68j, -0.760 + 1.70j], [0.180 - 0.10j, -1.22 + 0.51j, -1.66 + 0.86j]]) print(np.sum(np.square(np.abs(complex_matrix)))) print(np.sum(np.square(np.abs(np.fft.fft2(complex_matrix)) / 3))) The two output results are extremely approximate. I should calculate the modulus square. Thanks again for your video.

Thank you very much for this useful lecture! I've learned a lot of interesting things for myself. I hope your microscope will be repaired soon.

@NicholasRudawski

3 ай бұрын

I'm glad you found this video helpful; thank you for your support. Hopefully, the other S/TEM is up again soon.