Looking through Objects - How Tomography Works!

During my studies, I became really fascinated by the math and visual illustrations in biomedical imaging. I hope that I can share my passion about this topic with you in this video. The idea of 3b1b's "Summer of Math Exposition" was quite motivating to make this project happen.
Resources used for this video:
* Lecture notes from DESY: Milan Zvolský www.desy.de/~garutti/LECTURES...
* Lecture notes from University Göttingen: Prof. Tim Salditt
* Digital Image Processing: Rafael C. Gonzalez
* Rose CT data by microphotonics, • 5 Minute Micro-CT Scan...
* Animation Software: Manim, www.manim.community
Special thanks to Veerangna Kohli for aesthetic advice, mental support and help with the audio recording.
I really appreciate the people who took time for reviewing this video and giving me technical feedback:
Rishabh Jha,
Martin Krzywinski (mkweb.bcgsc.ca/),
Oliver Schön, and
Matthias Schröter.
Lastly, I would also like to thank the great people in the Manim Community and the Institute for X-Ray Physics in Göttingen.
Small typo at 13:40:
p̃(s,ϕ), not p̃(ω,s).
p̃(s,π*m/M), not p̃(π*m/M,s)
imgur.com/a/qsZwJdI (corrected proof as an image)

[@parhamzolfaghari7394]

Mind blowing!!! I wish most professors and lecturers stop teaching and just play videos from people like you. Thank you for your fantastic explanation.

[@benjaminhezrony5761]

This is SUCH a great video on the topic. Fantastic animations! I do research in CT super resolution, so this provided a lot of good background and visualizations on the source of the images I super resolve

[@kphk3428]

I always wanted to learn about this! The animations are amazing. You put a lot of effort into it. It saddens me that it isn't getting more views...

[@jubieralonsojimenezcamargo5732]

Amazing; Thanks for taking the time to doing this

[@phonix6494]

Thank you for this video, it really remindet me how connected everything is, so many concepts that I already had a glimce at shining in a new light. Aesome video music and Visuals

[@MariaJose-ub9rt]

Thank you so much! i study Nuclear Medicine and this part of tomographic images is so hard to understand! but the video really helped me. Greetings from Argentina

[@NoNTr1v1aL]

Absolutely amazing video! Subscribed.

[@shailesh4673]

Thank you very much for this video!! my friends and I were struggling on a past year problem for our medical imaging class and this video cleared everything up

[@rohitkumarsaini221]

I have been facing problems with removing the tomography effect from OPG dental image. Anybody can help me please

[@ahmadayoubi1830]

YOUR EXPLANATION IS PERFECT WOW! you made a complicated idea sound so easy thank you so much

[@error200http]

Excellent animations!

[@AForce295]

Amazing video, well explained! Helped me in studying for my medical imaging exam :D

[@alirezasheikhsofla2764]

The video was so great and informative for the basic concepts of tomography. Thanks!

[@SirTravelMuffin]

So good!

[@camilogomez775]

Hi. This has been very usefull for me. You did a sublime work. You are a great teacher. Thank you so much. 😄

[@thebees955]

Very cool video!

[@Caspar__]

Wow, this is great. I am really looking forward to when I learn fourrier Transforms next semester, so I can appreciate the beauty at play here. I have been thinking about going into medical physics and I appreciate, that even there many mathematical concepts are at play.

[@zign2044]

Merci infiniment pour cette vidéo!

[@lavimuia7612]

Excellent lecture.

[@vishalnimesh3846]

Amazing

[@BrianAmedee]

The visualisations were really satisfying to watch, and thanks big time. I have never 'understood' how CT scanners actually work. This makes so much sense now!

[@Marie-yx7qs]

Danke Bruder

[@matthewjames7513]

Thanks so much for this video. I learned a lot. At 13:41 you make a discrete approximation for the inverse radon transform. I'm interested in coming up with a matrix representation for the Radon Transform [R] and Inverse Radon transform [R]^-1 such that: [R]f = p and [R]^-1p = f Do you know what expressions I can use for [R] and [R]^-1 can be? As context: I'm running an optimization algorithm where I need to calculate [R][R]^T f. I have no idea how to find [R][R]^T though.

[@MooImABunny]

I can't find a formula for the Shepp-Logan filter online it looks from what you graphed and the fact that it should start like |w|, that its 2|sin(w/2)| for -pi

[@shervinnia7456]

Hello! This animation is extremely useful, and I'd like to publish some parts of it to open-source resources for understanding tomography in the context of structural biology. Would I be able to post parts of this video with credit given and a link back to the channel? I'd also love to have a look at the source code if you're open to that. Thank you!

[@Kolibril]

Thanks for your feedback! That sounds like a really nice project, and I would like to make my animations available to it. Further details we can discuss via direct message.

[@corentink3887]

you genius

[@user-vl6bt6st4x]

Wait. When we have circle (2:50) and rectangle (3:50) as objects in a sinogram we get some shit picture. But when we have Pi letter object we get an image of Pi (4:10 ). Did I miss something? And what is the difference between what we do here (4:10) and there (2:40) ?

[@matthewjames7513]

Thanks for the video! I was following the proof very carefully. I believe you've made a small typo at 13:16 on the second last tline. I believe p_tilde is a function of phi and s. Whereas you have written p_tilde(w,s). This is because once you evaluate the inverse FFT the w should get removed. The next line, you just integrate out the phi and leave the s untouched. But s is a function of phi as well so I think the final summation should look different too, right?

[@Kolibril]

good spot! it sould be p̃(s,ϕ), not p̃(ω,s). p̃(s,π*m/M), not p̃(π*m/M,s) imgur.com/a/qsZwJdI (corrected proof as an image)

[@matthewjames7513]

@@Kolibril Thanks for your quick reply! Although I still think the very last line is wrong. Since s = s(phi) = xcos(phi) + ysin(phi), this needs to also be in the discrete sum. Right?

[@someonetoogoodforyou]

Thanks for the video. I see you made it using manim! May I access the code somehow?

[@MyDum]

Can you share code of this video, please?

[@atlasxatlas]

Why no english subtitles? Why yt's auto generated subtitles in German?! I'm so confused

[@Kolibril]

I just changed the video settings, now there should be English subtitles as well.

[@71sephiroth]

nice! but... why do we use integration? (6:15 and onward)

[@Kolibril]

Imagine a snail race. And now imagine the "pi" symbol would be made out of salad. On their paths, each snail would eat all the salad that it comes across. When the snails reach the finish line, you see how much weight they gained, but you can not see anymore, where along their lines the salad was. Similar, the integration "collects" all the intensities along the u-axis (as shown at 6:59). Note, that the function p(s,ϕ) is not u dependent. Because we integrate over "du", all points along the u-axis are projected.

[@71sephiroth]

@@Kolibril Thank you!

[@byG027]

@@Kolibril Shouldn't the integration showed at 6:37 be over ds instead?

[@moesterer]

Sequel: diffraction tomography?

[@Kolibril]

I already have the script for another sequel "Artifacts in Tomography" : nosy image, moving sample, misaligned rotation axis and missing wedge. But making these animations takes really a lot of time. Maybe sometime in the future.

[@fuckyoutubengoogle2]

4:20 frequencies of what?

[@fuckyoutubengoogle2]

Explanation is out of order.

[@aymaneelfahsi8048]

can i contact you. Needa help for my research

[@simonmaverick9201]

I have got a headache...

[@alexkart9239]

4:30 poor explanation. there is no any info about this "frequency" and wtf is these -150~150 values. i cant see anything negative on the picture, i see only positive brightness from 0.0 (black) to 1.0 (white)