Good.

So welcome everybody.

So everybody feeling well today?

Yes?

Yes?

Feeling well today?

Yes.

Yes, good.

That's good.

That's good.

Everybody should feel well, at least before this lecture.

Okay, so today we will do something really, really cool.

So we will do a refresher course.

Who knows variational calculus?

Yes?

One?

Who else?

Okay, today we'll do something really cool.

So we will talk about variational calculus and why variational calculus is a really nice

and useful tool.

So variational calculus is essentially an extension of the analytic algebra that you

already know today.

And it's really, really useful because you can not just work with functions of variables,

but you can, we will introduce something which we call a functional.

And this functional is a function of functions.

And then we can operate on the functional in a quite similar way as we do with functions.

And we can look at extreme situations and things like that.

And the nice way, the nice thing about it is that I don't have to do many constraints

on that functions that we are dealing with.

So I can solve a functional for a function.

And in the end we will see a quite trivial example.

But this is a quite nice way of looking at things.

So I'll try to keep it slow.

Ask all your questions on the way.

If so, if there's only one guy in the audience who's already familiar with this, then we

probably take the kind of slower approach.

But this is, so if you get the idea, you will immediately agree that this is pretty cool.

Okay?

Good.

What do we want to do?

Why is it a useful tool?

Well, let's think of a problem as we have it in image processing all the time.

Let's say we want to compute a smooth image.

And we want to compute it according to some selected optimality criterion.

And the model, the assumptions that we put in should be that we have a filtered image

G and G, the result of our filtering, so the smoothing, the result of our smoothing, G

should be as similar as possible to the original image F.

And then, so they need to be similar, so probably the difference or the distance between the

two functions should be small.

And of course, G should be smooth.