Good.

Anything else?

Yeah, so we are not switching the exercise.

The doodle poll seemed to be non-conclusive.

So the two best options would be either the option we already have or a different one.

And we are going to keep the option that we already have because some people have already scheduled their calendar according to this schedule.

So I thought the better option is to keep what we already have instead of switching.

Okay, good.

So if you decide to go for the early exercise, you might have very good supervision because only a few people are there.

It's Wednesday at 9?

Not 8?

It's in the universe. So you can look it up. It won't be changed.

Okay, good.

So today we are going to do a refresher course and we will talk about a tool that is called singular value decomposition.

And as you will see, this is a very powerful tool and we use it in image processing all the time because it allows us to solve linear problems quite efficiently and quite robustly.

And yeah, we will talk about that today.

Okay, so should we wait a bit?

Okay.

Okay, so let's start. So welcome back to medical image processing, interventional medical image processing.

Okay, there's still people coming in, but have a seat. Don't worry. This is a refresher course anyway.

So today we will talk about singular value decomposition and there is a certain rumor that is between the students every semester.

There is this rumor that if you're in the oral exam and you're not exactly aiming by asking very strange questions, then probably the answer is SVD because

probably there's something linear in the topic that we are discussing about and then SVD might be a good option to solve the entire problem.

Good. So what is SVD? SVD is a powerful tool. It's a matrix decomposition.

But first let's reconsider what matrices are and what we are doing, what we might use the SVD for.

So let's say we have a matrix. So we start from the very basics.

And of course we will then go into more detail. But let's consider what a matrix is.

Typically you have a matrix and you call the matrix probably A. You will realize that this is an advanced course.

So we are switching the symbols from time to time. Sometimes it's A, sometimes it's M, sometimes it's B.

But generally it's a matrix and because this is an advanced course we expect that you still can follow even if we switch the symbol for a specific matrix.

Don't get too much confused. If you get confused and I'm switching symbols too often, please just ask.

But let's call our matrix A. And our matrix A is composed of numbers and it has a specific dimension.

In particular it has M lines and it has N columns. This is how our matrix looks like.

And now you can also write down this matrix in components. So your components would be denoted as miniscule.

So we are not using capital letters. So generally if we use capital letters we are referring to matrices in this class.

And if we use bold notation, so matrices are capital letters and bold. And vectors are small letters and they are also bold.

So this is generally bold is some kind of entity that has more than one dimension.

And if we have scalar values they are just small letters and they are non-bold.

Because I'm having trouble writing bold letters onto the board I will mark vectors with a small arrow.

So this is generally the notation and most of the time we keep this notation in the class.

So if it's not kept then we need to fix it on the slide. Or I just forgot to put an arrow.

So then please remind me of putting an arrow somewhere.

So you have a matrix and it's composed of scalar values. And as I just said it has N columns.

So this needs to go to A1N. And here we have the rows and we have M rows.

So this goes to AM1 and then of course the last entry in our matrix is AMN.

So this is our matrix. Okay. So if I now multiply this matrix with some kind of vector.

Let's say I multiply my matrix with a vector X. Then I can essentially write this down.

So I could rewrite this matrix also as a set of column vectors.

And then I would have only vectors AI. And now I'm using the arrow to mark my vector.

And I have a couple of column vectors and this goes up to AN.