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As usual, I mean in this lecture we need a lot of math, right?

And again I have the feeling that we need a brief refresher course on variational calculus

and the mathematicians in the back should correct me if I'm doing something wrong.

This is the engineering point of view as usual, so I'm always hand waving

and come with some intuition, but I can tell you if you want to build something

that you can sell for money without getting lost in some proofs,

that's more or less sufficient to have this hand waving approach.

That's my personal experience, but do not quote me.

My valued colleagues in the mathematics department I think will have a lot of comments

on the way I teach these things.

Today we are going to talk about variational calculus.

I think I was asking the same question last week already.

Who has never heard about variational calculus before?

Okay, and all the others, they heard it in algorithms of continuous systems I guess, right?

I read it in the book.

You read it in the book? Oh, look at this guy.

Good. Me too. I also read it in a book.

I never heard about this during my studies.

This is not considered to be that important, but for pattern recognition

and for medical imaging in particular it's very important

because most modern registration methods that combine images from different modalities,

they require to use variational calculus and methods from variational calculus.

So we will spend another 90 minutes roundabout on variational calculus

and the core ideas of that.

I will motivate this by a problem that the guys who attend pattern analysis

have seen already in the lecture on pattern analysis.

But don't worry, it's very straightforward.

We will talk about variational image smoothing.

And now switch off your ears if you're not attending pattern analysis.

That's basically a Markov random field based objective function

that we want to optimize and we have seen it several weeks ago already.

So what we want to do is the following.

So let's assume we capture an image.

So we press the foot switch.

Our X-ray tube is generating X-ray particles.

They are pushed and propagated through the object and we measure the attenuation

and we get an X-ray image.

And of course we want to apply not too much dose,

so we want to keep the dose on a very reduced level.

And there is always the trade-off in medical imaging between dose, acquisition time, and image quality.

And if we reduce the dose in X-ray imaging, we will get very noisy images.

And the radiologists, they look like this, they squeeze their eyes a little bit

and then they try to interpolate and as long as they see what they want to see,

they are happy with the dose.

They want to keep the dose on the lowest possible level.

And we as engineers are required to implement algorithms

that might increase the image quality based on measurements

and maybe some domain knowledge.

We have seen the bilateral filter, for instance, that is doing this job perfectly.