Fourier transforms in cryo-EM | Convolution, Gaussian filters, Fourier Shell Correlation, FFT
Here is my lecture on Use of Fourier Transforms in cryo-EM I was supposed to give in person at the course in Sao Paolo (“CryoEM São Paulo School 2023” on July 11, 2023). Unfortunately, I could not attend the event in person, so I recorded this video talk.
CryoEM São Paulo School 2023
sites.usp.br/cryoemschool2023
Timestamps:
00:00 Introduction
01:28 Use of Fourier transforms in cryo-EM in past and today
02:55 What is a periodic function and how to sum them?
07:07 Fourier decomposition - inverse operation of summing functions
08:41 Mathematical definition of Fourier transforms and its important property - Invertibility
11:03 Example of decomposition of a periodic function and Nyquist theorem
14:29 Two domains: spatial (real space) and frequency (Fourier space)
15:54 Convergence and sampling
18:23 Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT)
20:52 What are images (photos or micrographs)?
24:52 What is 1D, 2D, 3D?
26:24 Representation of images as linear combinations of base vectors in real space
27:22 Examples of 2D Fourier transforms
29:46 Fourier decomposition in 2D
30:55 Amplitude and phase spectra in 2D
32:24 Scaling images in real space
34:06 Scaling images in Fourier space (Fourier cropping)
35:55 Filtering in Fourier space (Gaussian low-pass, band-pass and high-pass filter)
39:06 Why should we avoid using sharp masks?
41:40 Generalisation in 3D
43:18 What is convolution?
48:48 What is correlation?
51:51 What is Fourier Shell Correlation (FSC)
56:57 Fourier transforms in an electron microscope
57:46 Take-home messages
59:11 Useful references for deeper understanding
Пікірлер: 9
Important correction for slides 31 and 33 (38:20 and 40:55): regarding the masks for band-pass filters (in Fourier space), I need to clarify that they should be inverted. A band-pass filter suppresses both very low and very high frequencies, resulting in the central part (circle in the middle) of the power spectrum appearing black, as well as all four corners of the square. I made a mistake during the slide preparation as I used shapes in Powerpoint for didactic purposes. I want to emphasize that the "Barbara" test image with the artefacts in slide 33 is indeed real and was generated as described in the video. My apologies for the oversight and any confusion caused. Thank you for your understanding.
Really clear and well-structured explanations !
@afanasyevp
6 ай бұрын
Thank you!
Thank you for the exceptionally clear explanation! :D
thank you, professor!
Excellent content. Thanks for sharing it!
Thanks, professor ! ❤
27:17 there is a mistake on a slide: second row, first and second cells (counting from left) must be switched, because 1 x 0 != 255 and 0,56 x 255 != 255
@afanasyevp
4 ай бұрын
Yes, indeed! Thank you very much, well spotted!