Good afternoon.

We are currently in the chapter on computer tomography and how we can do a reconstruction

from X-ray projections.

And just for you a brief reminder, so what is the problem we are considering?

We are getting a signal at each pixel that is more or less here an integral equation.

So we observe a line integral with each pixel in the detector.

And so we capture images from different directions.

We rotate for instance around the object and with each pixel we get a line integral.

And from all these line integral values we have to compute, basically we have to compute

this attenuation function.

It tells us basically what kind of material was exhibited to X-ray.

And we do that slice by slice in standard CT.

And what we have learned last week is the so-called Fourier slice theorem.

What is the Fourier slice theorem telling us if we have a function like the mean?

So this is X and Y and the values here they are denoted by me X, Y.

So we get density value.

And if we now use an X-ray system and look through the object like this here, we observe

here the projection values.

And the Fourier slice theorem is now giving us a relationship between the Fourier transform

of this 1D signal and the Fourier transform of this 2D signal.

If I consider the Fourier transform of this 2D signal, I know that the Fourier transform

along this line through the origin has exactly, oops, let me use now yellow, the values here

are exactly the values that we have in the projection.

That's the picture you have to keep in mind.

We can sample the 2D Fourier transform by rotating around the object, capturing the

signal, Fourier transforming the 1D signal and sort it into the 2D Fourier transform

of the function that we want to reconstruct.

And basically once we have computed this or sampled this with many X-ray projections,

then I can take this and compute an inverse 2D Fourier transform and I get the original

function.

The Fourier transform tells us the relationship between the 1D Fourier transforms and the

2D Fourier transforms of the function we are looking for and it's like a miracle that we

can use the X-ray projections to sample the higher dimension of Fourier transform.

Once I have sampled it, I do just the back transformation, the inverse Fourier transform

and I did the reconstruction.

That's a very, very exciting result.

I personally like it very much because this is something that is not obvious.

And the proof of this was done by us last week and it's a very basic proof.

There is no extended course on Fourier transform required to understand why this result holds.

It's very straightforward.

It's very straightforward and we have seen that the basic result that we are required

to use is saying it makes no difference whether we rotate the function and compute the Fourier

transform or whether we compute the Fourier transform and rotate the Fourier transform.

That was a very important component and then we also have shown the relationship between

the 1D Fourier transform and the 2D Fourier transform for a projection where we just have

a projection that is parallel to the X coordinate or to the Y coordinate.

I think the Y coordinate was the one that we set to zero.

And then we brought things together and finally ended up with the Fourier slice theorem.

So if I ask you in the oral exam about the CT reconstruction and the Fourier slice theorem,

what do I expect?