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Okay, so welcome everybody. Let's start with today's lecture. I have three remarks in the beginning.

First of all, many of you, or most of you, have already scheduled their oral examinations.

If you have scheduled your oral examination on the 17th of February, 1-7, there won't be oral examinations on that day.

So you have to move your oral examination if you're on that day, 17th.

Go to the lab and reschedule your oral examination if you had the 17th as date for the oral examination.

The next thing is, I ordered five of Larry's books, and I will put them in the lab,

and you can go there and borrow them in order to prepare for the examination.

The next thing is, it will take some time until they arrive, so I guess they will be available from the end of January.

They won't be available right away. But you can go to the lab, ask the secretary for Hannes Hofmann's office,

and you can go there and borrow them. So I will put five copies of the book there.

Nonetheless, you can also get your own copy.

Another thing is, over Christmas I got an email from Larry saying, and he told me that there are a couple of errors in his book,

so he sent me a list of errata, about one page, with very small, minor errors, about small things which are wrong in the book.

So I will also put that on the website, that you can see the list of errors.

That's quite a convenient service that the author himself sends out the errors of his book.

So he was quite happy about that. So he even encouraged me more to make use of his book.

Okay, so these were the remarks in the beginning.

Today we will have the last lecture on image reconstruction, and today's topic will be iterative image reconstruction.

So let's have a look at this puzzle again.

So the idea, what we want to do, we have some kind of volume or a slice of a volume,

and we want to reconstruct the information contained inside this volume.

And we can observe something, in this case x-rays that traverse the volume, they are being absorbed.

And what we can see is the remaining energy that was not observed within the volume,

and we can use that information to, these raysums, we can use this information to compute the inner structure of the volume.

So we've seen that we could put this in a matrix notation, but if we have a bigger problem,

we cannot easily solve that with a matrix inverse, because our matrix would be really huge.

So we won't do this with a matrix inverse.

Another thing that we discussed were different geometries.

We've seen that we can find the central slice theorem and parallel beam geometry,

that we can derive a relation between the 2D Fourier transform of the image and the 1D Fourier transform of the projected image.

And this led us to a reconstruction algorithm for parallel beam geometry, filtered back projection.

And using filtered back projection, we could reconstruct the image.

Then we've seen that we cannot directly apply this algorithm for fan beam geometry,

so if we want to acquire faster with a whole fan of rays, we couldn't easily use the parallel beam reconstruction algorithm,

but we found several ways how we could reformulate the problem and we would be still able to reconstruct the image.

So one thing would be just reorder the rays and put them into a parallel geometry.

Another thing was the change of variables where we could derive a reconstruction algorithm for fan beam geometry.

Then we've seen we can also reconstruct using a comb beam geometry,

where we acquire not only a fan of data but a whole projection image, and we use that for reconstruction.

And we've seen that we don't have complete data if we only rotate on a circular trajectory,

but we would get comb beam artifact in parts of the volume,

only the central slice of the volume is reconstructed exactly.

And we've seen that we can create a helical trajectory by moving the patient through a scanner,

and this helical trajectory was really nice because it had complete data.

Okay, so much about that.

Today we talk about iterative reconstruction, and we will go back to linear equations

and see if we can maybe still derive something to solve this problem without inverting the matrix.

So let's look at these linear equations again.

Well, we've seen in our simple example that we just see the sums of the different voxels or pixels in our volume