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! 😄
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!!
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)
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"
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).
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!
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.
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
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.
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.
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.
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 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...
@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" ?
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 .
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
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'
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
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!
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 !!!!
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: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-3Ebvypj577E.html
