Okay, so please apologize.

So we start.

Good morning for the Tuesday session.

Today, 60 minutes on image reconstruction.

And yesterday, I was at the beginning a little bit annoyed about the fact that the slides

were corrupted and I couldn't read it.

But I think we got a very good intuition what is actually going on in CT reconstruction

or to say it in other words, what are the basic problems that are required to be solved

to do a CT reconstruction.

So we have to know about the line integral, the projection ray.

We know how to characterize the 3D projection ray given a point source.

Projection matrices is basically what we need for that.

We have discussed these things in detail.

We have seen that we basically have to solve a system of integral equations or in other

words, we measure a set of line integrals of a function and the problem is to compute

the original function.

That's CT reconstruction.

And then we discussed the Fourier slice theorem.

We did not prove it.

We discussed it.

And we have referenced, let me say we have referenced a very nice result saying that

the 2D Fourier transform of the slice we want to reconstruct has a relationship to the projections

and their Fourier transform.

And basically the picture we have to keep in mind is this one here.

We have our slice.

We want to reconstruct.

We have our, let me draw it this way, our detector and then we have our projection rays.

And if I consider the Fourier transform of this two dimensional slice I want to reconstruct,

I know by the Fourier slice theorem that basically the Fourier transform of this 1D signal, what

we measure is identical to the Fourier transform, the 2D Fourier transform sampled along the

line that is parallel to the detector.

And what we started out to discuss yesterday at the end of the lecture, we looked at the

very specific case that the y coordinate is zero of the slice we want to reconstruct or

let me say it this way, the rotation angle of our detector is zero.

And we were able to see that the Fourier transform of this 1D signal is identical to this one.

The final argument is still missing.

We can have a look at it closely in a few minutes.

And the second stage of the proof of the Fourier slice theorem is basically the fact that it

makes no difference whether I rotate the function and compute the Fourier transform of it or

whether I compute the Fourier transform of a function and rotate it afterwards.

That means if I have a 2D function and I rotate the function and then I compute the Fourier

transform, the result is the same as if I compute the Fourier transform of the original

function and then rotate the Fourier transform.

So this commutes.

And what we basically can do then is if we know this relationship here between this theta

angle is zero and this line here, we basically rotate back the Fourier transform and consider

always this straight line here and successively sample the Fourier space by rotating the Fourier

transform or rotating the detector around it.

That's the argument that we apply.

So this is the picture.