Visualising 2D k-space and Fourier synthesis (1D & 2D, helps for image reconstruction and analysis)

k-space, 2D Fourier transforms (analysis) and Fourier synthesis (inverse 2D Fourier transform)
Visualises radial trajectories in k-space, useful for subsequent understanding of the central section theorem, filtered backprojection, PET, CT and MRI image reconstruction

[@055shubhamsonkar6]

Dear professor ,
I was never this much clear about the concepts, Thank you for this much effort in each videos .
please share the slides for the videos , if possible.

[@florentb8578]

Excellent visualisation

[@AndrewJReader]

Many thanks!

[@anatolicvs]

Dear Professor,
I greatly admire your TH-cam channel and have learned a lot from your valuable knowledge and techniques. Thank you for sharing your expertise with us.
I'm particularly interested in the marvellous visualizations you use in your videos. Would it be possible to get access to the codes for these visuals?

[@AndrewJReader]

Thank you so much for your feedback, means a lot. The code has taken me a lot of time to write from scratch, and at present I am not yet planning to distribute the code. But I do share code for some of my examples (e.g. th-cam.com/video/FPzi8cUhNNY/w-d-xo.html), so I hope that has been useful. Thanks again for your support!

[@jaewonlee8147]

Hi Professor Reader, thanks for another great video! If you don't mind, I have a question about the sampling method you used for the Einstein picture reconstruction.
Around 26:00, the sampling in k-space was done in a radial direction, and it very much reminded me of the CT image acquisition, where each projection is a slice in a 2D Fourier domain. In the case of FBP in CT image reconstruction, the argument is often that the lower frequency region is oversampled compared to the high frequency region, and we need to somehow normalize the sampling density by applying the ramp filter.
So from the radial sampling of the Einstein picture, I expected to see something similar; as the sampling continues, we obtain a low-frequency enhanced image. However, it does not seem like the Einstein picture suffer from the overrepresentation of the low frequency samples. I am quite surprised and confused!
Is there a reason why in this case the reconstruction is good without the ramp filter? Would we get better result with the ramp filter? Thanks.

[@AndrewJReader]

Thanks for the feedback and great question. You are right about the sampling in the radial direction and the Fourier slice theorem, which indeed needs a ramp filter to compensate for sampling density. In this video I did not re-include k-space samples that had already been include in previous radial lines, hence I had no need to compensate for sampling density. I actually clarify this explicitly in my video on FBP, please see time point 32:40 in this video: th-cam.com/video/dmBCxUFHk44/w-d-xo.html. Thanks again for the comments.

[@jaewonlee8147]

@@AndrewJReaderThanks for the answer professor, it makes totally sense that by skipping the k-space samples that had been included already, the reconstruction does not suffer from the low frequency overrepresentation. I appreciate you taking time to give a thorough answer to my question. I also just discovered that you'll be teaching a short course at the next NSS/MIC conference in Vancouver. I might sign up for it, if I get my advisor's blessing. Looking forward to the course and learning more about your work at the conference.

[@AndrewJReader]

@@jaewonlee8147 Many thanks for your support and interest in my courses! Hopefully see you in Vancouver!