Good.

So welcome everybody.

Okay, so everybody feeling well today?

Yes, yes, feeling well today?

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 will 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 function now.

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.

So this is what we can put in.

And now we don't want to think of the discrete images as we traditionally use them in this class.

Now we think of the image as a continuous function.

And the nice thing is we basically, if we stay in a continuous domain, we don't really have to describe how this image is exactly modeled.

So we could model our image with pixels.

Then we have a discrete version, a version that we know.

But we could also model the entire image as a mixture of Gaussians.

And so it could be any other type of function or our image could be a polynomial.

But in the following, we won't discuss how we actually model the image.

And we will do the entire math without even looking into the model that we use for the image in high detail.

Of course, in the end, when you go to an application, you probably have to choose a model.