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 University Göttingen: Prof. Tim Salditt
* Digital Image Processing: Rafael C. Gonzalez

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,
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)

[benjaminhezrony]

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

[parhamzolfaghari]

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

[kphk]

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...

[jubieralonsojimenezcamargo]

Amazing; Thanks for taking the time to doing this

[HanGyooKang]

Excellent video about sinogram and FBP with comprehensive explanation!!

[phonix]

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

[ahmadayoubi]

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

[MariaJose-ubrt]

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

[shailesh]

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

[alirezasheikhsofla]

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

[camilogomez]

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

[AForce]

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

[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.

[stoyantodorov]

Brilli
ant explanation! Thanks a lot!

[matthewjames]

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.

[ДаркШнайдер]

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) ?

[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!

[matthewjames]

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?

[shervinnia]

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!

[sephiroth]

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