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So welcome to the Monday session. I'm surprised that we have so many attendees today.

I was suggesting last week that this is one day and then another day off,

and the motivation to show up here should be rather small,

but I'm surprised and I'm happy that you are here and that I have to teach now for 90 minutes.

Okay, so what we are going to discuss today is we will continue on detecting points and features

in the image, and we will discuss one algorithm that will allow us to reduce artifacts in CT images

to make them look really nice for interventional procedures.

First of all, at the beginning of the lecture we had a refresher course on

singular value decomposition that's especially important for those of you who did not attend the lecture

in winter semester on diagnostic medical processing where we have introduced these concepts as well.

And we had a second refresher course on homogeneous coordinates.

Also for those of you who homogenous coordinates,

that was meant for those of you who had no idea what perspective projection is

and how perspective projection can be incorporated into a higher dimensional or embedded into a higher dimensional space

such that you can characterize also perspective projections by linear mappings using matrix calculus.

The important thing is if you get a vector x, y, w for instance and you know this is a homogenous coordinate,

then you can start to or you can begin to become nervous and think, oh, horrible, I don't know what to do.

Homogenous coordinates always sit down, relax, that just means take the last component

and divide the above components by the last component to get the vector in the dimension one reduced space.

So you just divide by this.

And then Philip is saying, well, that's nice, but what happens if w is zero?

Well, I can tell you it's well understood what's happening if w is zero,

but for all the practical problems we have to solve, w will never ever be zero.

Of course, you can discuss it and you can think about it and there is whole theory

what that means in terms of point infinity and blah, blah, blah.

But for all the practical problems we have to deal with, this is not something we need to think about.

I give you a four dimensional homogenous vector.

How do you transform it into a useful vector?

I divide the first three coordinates by four coordinates.

What do you do if the last component is zero?

This will never happen.

This will never happen? Okay.

Interact this. Okay.

What can go wrong goes wrong, but for all the problems we consider here, it will never happen.

Okay. Good.

We looked into a horrible chapter.

One of the students approached me after the last lecture and was telling me,

look, I did image processing.

I learned a lot about edge detection and corner detection,

but my impression is that what you teach is the most complicated thing I have ever heard.

Why don't we just look into the simple strategies?

Well, I mean, edge detectors were built in the 60s and 70s with very simple approaches,

but with modern approaches like structured tensors and covariance matrices of gradients,

you can do way more and that's why I explain these things.

So what we discussed was gradient computation.

I mean, we have to do real time image processing in an interventional setting.

That means the doctor is there with the bloody fingers and wants to do image processing,

image visualization to treat the patient in a proper way.

So we have to compute gradients. What is a gradient?