Okay, so let's get started. Welcome to the last session of computational MRI of this

year. We made a lot of progress. We are now at the stage where we leave the linear realm.

So we're turning to nonlinear optimization methods, nonlinear reconstruction. And as

I already said last week, this is always a favorite for the students I find both in the

lecture and in the exercise. And the reason for that is that so far what we have done

usually is in line with the theory that you have learned during your undergraduates, Nyquist,

linear systems of equations, matrix inversion, inverse problems a little bit. And we just

use these concepts in the context of imaging and image reconstruction. But we are not doing

anything completely unusual. Everything is very well behaving the way you would expect

it. But what we're going to do today and then also in the next lectures is essentially we

are going to violate all these basic assumptions. And we still will see that we actually get

a solution. So in particular today with compress sensing, we're going to say that the 100 year

old, not quite 100 year old, 70 year old Nyquist theorem actually doesn't hold for us. And

we're still going to get a nice solution of a signal that we sampled at a much too low

frequency rate. Just curious, anybody has already heard about compress sensing? You?

A little bit? Okay. All right. So it's going to be just one person. So for you maybe not

as revolutionary, but the rest of you, you're probably in for an exciting session. So let's

just give me a very, very intuitive picture for you. So what we are going to do is we

are going to find some sparsity in our solutions, in our images, and we're going to exploit

that. And to give you an example, if you take a look at these two images, this project recently

well, so this is a Shep Logan phantom with multiple different gray values in the image.

This here is its gradient. So you just do a finite differences derivative and we can

see that everywhere where the contrast or the signal intensity changes, you get a signal

value here everywhere else that the relative of this function is zero because it's essentially

just a combination of piecewise constant functions. So if you take a look at these two images

and now I'm asking you from everything you have learned so far, if we are treating this

as an object that we want to image with MRI and we are doing a Fourier acquisition, we

use our gradients to acquire the images, but what this particular image, which image requires

more Fourier coefficients or more phasing coding lines? Or does one require more than the other

or the same? Any thoughts? Do you have any ideas where this might lead? Who thinks we

require the same amount of Fourier coefficients when we want to acquire this object or this

object? They have the same matrix size and the same resolution, same pixel size. The

same. So we have one line of thought. Who agrees with this statement? One other person.

So everybody else thinks we need fewer or more. You think fewer? So maybe I think we

need a bit fewer because the gradient we have mostly high frequency information. So maybe

so in the Fourier space we might have less low frequency information. Yes. And that leads

us to an interesting line of thought. You're absolutely right. This is essentially the

representation of the high Fourier frequencies. The problem that we have is we usually don't

know this when we image the object. Right? So before I resolve with any other thoughts

about more or less Fourier coefficients for this one or for this one, you said you think

it's different. I want to hear why. I don't know less or more. Why not? Less information.

That's an interesting observation. Anybody? Any other thoughts? I think we have covered

the range of opinions. And the answer and the reason why this is controversial is that

the Nyquist theorem way of thinking conflicts with your common sense. If you go strictly

by Nyquist, as you have said, you have the same field of view, you have the same matrix

size, you have the same resolution, you need the same number of samples. There's absolutely

nothing that goes against this. We need to define our k min, our k max. We need to have

the correct distance between our case-based sampling points. Otherwise, our field of view

is too short. So it is exactly the same. On the other hand, just looking at the images,

if you think about it in terms of information content, this image here has an extremely