How MRI Works - Part 3 - Fourier Transform and K-Space

How MRI works, Part 3 - The Fourier Transform and k-Space
Part 1 - NMR Basics: • How MRI Works - Part 1...
Part 2 - Spin Echo: • How MRI Works - Part 2...
Part 4 - The Gradient Echo Sequence : • How MRI Works - Part 4...
FFT code: github.com/thePIRL/fft-code-f...
Powerpoint slides from this lecture can be purchased here:
www.patreon.com/thePIRL769/sh...
0:00 - Intro
1:00 - The Sinusoid and phasors
5:48 - Fourier Theory
9:05 - The Fourier Transform and Inverse Fourier Transform
11:10 - Adding phase to our plots
13:42 - Fourier transform of sin(w0t)
17:40 - Hermitian Fourier transforms
19:26 - The Dirac Delta Function
21:00 - Fourier Transform Examples:
21:48 - Decaying Exponential/Lorentzian
29:09 - Square Pulse/Sinc Function
31:08 - Gaussian/Gaussian and Fourier Shift
32:39 - Discrete Signals, Fourier Transforms, and Nyquist
36:12 - The Fast Fourier Transform
40:26 - kSpace
40:48 - t/w and x/k convention
41:55 - Intro to kSpace
48:05 - Hermitian kSpace, half Fourier, and spatial filtering
51:29 - kSpace frequency units
55:08 - FFT organization of kSpace
56:56 - Outro and GRE Teaser
Drinking game: Take a drink every time I say 'notice'
Do I talk too fast in this one?
#some2

[@10monthspregnant]

This channel could be called "Applied 3Blue1Brown"

[@chimetimepaprika]

Jajajaja such a brilliant observation

[@jeunjetta]

It's just what we needed !

[@yoyobom1130]

I just discovered this series couples of days ago , am so lucky 😂

[@winterforlife]

After a long wait, the third episode finnaly came. And it was well worth the wait. Cant wait for the next episode in a year.

[@lexinwonderland5741]

I can obviously tell you're a busy guy, but you teased the 4th video too much for me not bug you about it. WE WANT ANOTHER LECTURE ON MRIS, DAMMIT!! your teaching is just too good!!

[@vma011]

AWW YEAHHHHHH Part 3 is here fellas

[@apollonyo5776]

What a coincidence! 2days before my medical imaging systems exam😂 thank you so much. I appreciate the godlike visualisations.

[@yazar8]

Best series of videos on MRI. Thank you.

[@luc7478]

The wait was worth it *-*

[@BIBECHYOUTUification]

The best KZbin channel for academic. Thank you so much.

[@h2o1066]

We need more videos like this, on quantum mechanics and Lagrangian Mechanics. Awesome video!!

[@felipemonteiro5877]

Yes! Awesome, thank you for posting this!

[@tinynudges]

Thanks! I have been waiting for this!!

[@bengoodey]

I don`t think I`ve ever been this exited to see a video. Just finished part 2 and really hoped you had made the one about the Fourier Transform. You are really helping me conseptualise MRI and its inner workings! So glad you followed up on your plans, hope the fourth video also gets made and uploaded. Thank you! 😄

[@akekalukardjnaphai464]

Finally, part 3 comes. Thank you for very good lecture of MRI series.

[@rodas4yt137]

Thank you. Really hope the next chapter can come out as soon as possible. These animations are stupidly awesome, for lack of a stronger adjective.

[@user-tc8tg4ry7p]

I’m a medical student who study physics of medical instruments in high school but never finished it! Now I’m studying it for entertainment in my free time. Thank you for make it all easier for me and many people to learn!

[@finnheinemann3682]

I thought I would have to wait for another year. This is awesome thank you so much

[@paintTitTblack]

Part 4? 😇 the videos are amazing, thanks!

[@Bozlee22]

i’m watching all these before the ARRT exam for extra preparedness

[@francismanno3822]

Part 4? Absolutely amazing content

[@hugodavalos4423]

This is insanely good! Thank you. I just started my section on MRI and this is a life saver

[@DaviidReiis]

Amazing video series. compressed sensing is pretty interesting too

[@MrDscrow]

Excited for the next one! These are great

[@gtau3144]

Nice explanation of fourier transforms

[@rexfung6376]

Thanks for the amazing animations and education! Can't wait for the next episode!!!

[@lauravillafane631]

I want moreeeeee. This videos are amazing!

[@mattn8775]

THANK YOU! been waiting for this

[@giuseppefusco1189]

Can't wait to see the next video, hopefully in a couple months ❤

[@Gigaswing]

love it, tnx for another great part in your mri series ❤❤❤❤❤

[@xiaoleixu5447]

Fantastic video! Thanks!

[@electronicbeats2010]

This was very helpful! I am doing a talk about MRI scanners and I wanted to get an introductory understanding for it, thank you!

[@devrim-oguz]

You just summarized the entire electronics engineering degree in one video 😂

[@cocoabutter5888]

This is absolutely amazing

[@laurentiurudeanu4102]

This a remarkably thorough and clear-cut presentation. Very well done, congratulations and many thanks for the huge effort that surely went in !

[@generaltheory]

I woke up while watching the first minute of this video how in luck I feel

[@h2o1066]

So nicely explained. Awaiting the next part eagerly!!

[@kaushalgagan6723]

It's was amazing, Thank you for creating this video. Hoping to see more videos in near future

[@futurekelley1898]

Great video!

[@andreaboscutti1194]

Craving for part 4 now

[@TheSpyFishMan]

Can’t wait for the 4th episode!!

[@andrejpauliny3581]

Wau! Very nice. Can't wait for next episode. Thank you. :-)

[@camicanu]

Thank youuu!!! Your videos are great

[@atrijasingh8868]

Your explanations are so helpful. I am doing a course on MRI Sequence Programming in my university FAU Erlangen, and now I feel more confident on the topics. Really looking forward to the next video on gradients. It would be great if you can talk about compressed sensing methods.

[@SgtNicholasAngle]

This is incredible, I know the DFT and FFT functions from an electrical engineers perspective and your explanations are so good it's clear you've worked very hard to present the topic in this way. Amazing, thank you .

[@lukec7525]

I am a new MRI tech and your video series have been invaluable to understanding the NMR phenomenon. It's kind of mind blowing. I think if I watch your videos a few hundred more times I might finally understand it 😜. I can't wait for the next video!

[@ahmadaljazeeri9287]

Is this level of understanding required for MRI techs or is it an engineering level of understanding?

[@lukec7525]

@@ahmadaljazeeri9287 It is not at all required... but it is certainly helpful in educating individuals such as resident M.D's as to why they should not order MRI hip scans to rule out osteomyelitis on patients that have undergone total hip arthroplasties...

[@scottdow5171]

Yessss. Been waiting!

[@holulu777]

Thank you. You are great.

[@Nxck2440]

I needed to understand k-space for my solid state physics class, a pretty much unrelated context, but your explanation of k-space was so intuitive that I was able to 'translate' to what I needed! Thank you!! To anyone in the same situation needing a bit more help: at 43:00, we have the physical setup on the right, where the surface represents the complex wavefunction of a given state, and on the left is its corresponding encoding in k-space. For crystals, we have three spatial dimensions instead of two but of course the idea is identical. So, when we look at a point in k-space, we are looking at a wavefunction that oscillates within the unit cell. Due to quantisation, these oscillations must fit an integer number of times within the crystal unit cell size, so our k-space is discretised: k_n = (2 * pi * n) / a.

[@yablaker]

Worth mentioning, that for the FFT to work best, the range should be a power of two. So 32 pixels would have been better than 30. 🤓

[@user-cu9vy9qp4e]

I'm very happy to see that great imagination and animation for one of the most difficult physics topics, despite all equations which I'm not interested due to my designation as MRI technologist but I passed through to next step in my To do list

[@eeemagic7729]

thank you for your great tutorial. excellent !!

[@allanvi2011]

more thoughts like this pleaseee

[@anmolsingh009]

This was amazing!! Now will have to wait for another year :/

[@tanchienhao]

awesome!!

[@hmshahparan3088]

Amazing intuition of fourier transform. Could you please make some videos on control engineering? Why we use step and impulse response? The real meaning and reason behind using of laplace transform, state-space model etc. Thank you

[@sajiruddinsk8129]

Very good

[@scramah195]

At 53:42, for anyone confused about why delta k_x=1/FOV_x, note that each horizontal coordinate (for some fixed vertical coordinate k_y) in k-space is expressed as some integer number of cycles per FOV_x. Therefore, between two consecutiv discrete points on the given horizontal line in the discrete k-space grid, we have that k_n=n/FOV_x and k_(n+1)=(n+1)/FOV_x. Subtracting the bigger from the lesser gives: (n+1)/FOV_x - n/FOV_x = 1 /FOV_x. Similarly, to understand the "pixel dimension_x", which I will refer to as "delta w", relationship to the k_(x,Nyquist), consider the following: There are 3 relevant formulas: 1) sampling rate_x = (# of pixels_x)/ FOV_x 2) (sampling rate_x)/2 = k_(x,Nyquist) 3) (# of pixels_x) * (delta w)=FOV_x. Putting all of this together gives us the following sequence of algebra: (sampling rate_x)/2 = k_(x,Nyquist) = (# of pixels_x) / (2* FOV_x) =(# of pixels_x) / (2*# of pixels_x * delta w)= 1/(2 delta w). This gives us: k_(x,Nyquist)=1/(2 delta w)...multiplying by 2 and then inverting gives us: delta w = 1/ (2* k_(x,Nyquist)

[@nas5432]

PLease PLEASE bring out the next one

[@gabcuba]

very very interessting

[@susantpanigrahi1149]

38:15 Note: The Nyquist frequency is as stated in 52:45 is half of the Sampling frequency \omega_s. Thus the total frequency space will vary between (- \omega_s/2) to (+ \omega_s/2). Where \omega_s = Sampling Frequency.

[@JSieberer77]

Hi, this serious is so great! Is there any way to support you in continuing this work?

[@SergioMiletto8]

Bro keep up with KZbin as you are extremely good at explaining (and i have watched many videos of the same genre like 3b1b).

[@wesslesyt3304]

Yessss

[@mihaimoldovan7291]

you're an amazing lecturer! can't wait for part 4 even though I don't understand 90% of the video. May I ask what software did you use to animate the graphs? also thank you!

[@achimbuchweisel2736]

your work is amazing! How long did it take you to make this!?!?!

[@impossiblephysix2633]

How did you animate this and how much work was it? Its so fantastic

[@dmacleanlaser9059]

Really excellent !!!! best course ever I watched on youtube ! You have a talent, really !!! Can't wait to see your next "chef d'oeuvre" :) :) Please, hurry ! We are all addicted now !!!!

[@ahmadjr199]

OMG it's a great Video thank you so much but I need the sources 😭

[@Kris_M]

Very nice video. One very common mistake though @34:38, only frequencies *below* the Nyquist frequency can be sampled. => "make sure your sample rate is *more than twice* as high as your max frequency"

[@matutejohn6994]

Where can I donate? I need keep these kind of videos coming! Thank you!!!

[@ajitmaurya3226]

Please make a video on 2D NMR of biomolecules . Please 🙏

[@sanawarhussain]

@50:30 if the complex conjugate phasors have the same magnitude wouldn't it be possible to just collect half k-space and get the original magnitude with: ∣A∣^2=AA∗ to reconstruct the "prefect" real image instead of a "reasonable reconstruction" ?

[@shoopinc]

Hello sir, will you cover Compressed sensing at some point?

[@JanPBtest]

An interesting bit: the FFT was in fact invented by... Gauss! It was found in his notes after he died, he apparently thought the result was cute but not important enough to warrant publication (he was right: without computers it's practically useless and with nothing terribly illuminating about it from the POV of pure mathematics either).

[@fuminguo2564]

would you be able to upload the slides for these videos? I believe it will help a loooot.

[@abeldeincognito]

50:43 why do you still lose information despite de asymmetry?

[@TheTauntaunCardtrick]

What are the main physical differences in MRI and MRSI?

[@jeunjetta]

Wow

[@kaishihchi]

plz! part4 now

[@efeerenceyani8457]

Which tool do you use for the visuals?

[@thepirl903]

These are all made in Powerpoint. The animations are gifs I made in either R or Blender. Maybe if people are interested I'll show my process in a future video.

[@efeerenceyani8457]

@@thepirl903 Thanks for the reply. I honestly myself would love to see a rundown.

[@TheOneAndOnlyOuuo]

@@thepirl903 Just be sure to make part 4 first! You left us with such a cliffhanger there ;)

[@JeffMTX]

If you’re a student, try to get the matlab student license. Matlab is the ULTIMATE platform for doing scientific visualizations. World renowned…

[@omridrori3286]

Is it necessary to watch the previous videos in order to understand this one?

[@headpox5817]

Watch them to find out.

[@HaraldSjellose]

im sad that this episode was released after my exam

[@scramah195]

Thank you for the video. At 43:34, you say the following: "...the 2D function we're describing, f(x,y), will increase in phase, and thus sinusoidal wiggles, according to the sum of phases k_x*x and k_y*y". What do you mean by this? Why are you describing the terms k_x*x and k_y*y as 'phases'? Aren't these terms analogous to omega*t...rather than phi? Why would 'increases in phases' have anything to do with 'sinusoidal wiggles'

[@thepirl903]

It's perhaps not the best phrasing, but the point is that k_x *x + k_y*y is indeed the phase of the complex exponential (units of k are rad/m or cycles/m) which is actually the spatial analog of phi = omega*t (units of omega are rad/s or cycles/s). Hope that helps? Cheers

[@scramah195]

@@thepirl903 cheers~

[@ahmadaljazeeri9287]

Is this level of understanding required for MRI techs or is it an engineering level of understanding?

[@michael_nel]

Definitely well beyond the level needed for MRI techs (the math especially). Knowing weighting, spatial localisation and sequences is crucial though.

[@longluo1788]

Thank you very much! Could you open source the code of the animation?

[@thepirl903]

Yes, I think I will do this and show my process in a future video.

[@longluo1788]

@@thepirl903 Thank you 😀

[@giovannispeziali1491]

@thePIRL Good morning, I am a US-based physician and researcher, is there a way to send you a DM?

[@thepirl903]

@@giovannispeziali1491 Yes, I can be best reached at [e-mail]

[@josephdays07]

For to do the wavelet transform or Function Wave i am not requiere the complex number or Fourier series. Just I need the new methodoly I discovered.I left this video to compare: kzbin.info/www/bejne/aXbFp6ymn5pqbac

[@sanawarhussain]

profound........

[@tommasoagrumi501]

Jumpscare at 40:26

[@Jacob-ye7gu]

your diagrams only serve to confuse the concepts