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So we should start. Welcome for the Tuesday session.

We started to discuss yesterday basics of variational calculus

and I'm very much aware of the fact that many of you have seen this now for the second time,

but on the other hand, I just wanted to make sure that you all know what we talk about here.

And so I gave a little refresher course on this.

Let me check. It's not saying no signal.

It should say the signal here.

Ah, okay.

So, once again.

And we have seen yesterday basically how to deal with functionals.

And functionals are something that we will consider in the following quite often.

And this type of problem characterization in medical imaging

is something that came up over the past 15 years or so.

Over the past 15 years or so.

So this is not something that we do forever,

but we bring in more and more application of variational calculus to image processing and pattern recognition, for instance.

If you remember Eli's lecture on pattern recognition,

when we discussed, for those of you who have heard this lecture, cross-linked to other lectures, five points,

we have discussed a problem, compute a linear mapping that maximizes the spread and the projection of the feature points.

You might remember this.

What happens if I don't say this is a linear mapping, but that this is any mapping?

Then I'm looking for a function and I can do a variational PCA.

When was variational PCA published? Three years ago.

So, simple idea.

Instead of saying I'm looking for a linear mapping, I'm looking for any mapping,

and I do a functional maximization or minimization using Euler-Lagrange partial differential equations.

Or other examples, if we talk about object tracking,

so you have a car driving by and you capture an image sequence to track the car,

you can basically consider the function that describes the motion of the car,

the change of the image function over time, and you can formalize this as well as a functional optimization problem,

and you can use variational calculus.

So there's a whole field.

My candidate.

Hello!

So there's a whole field in image processing, and we call it PDE-based image processing or something like that.

There are books on this and whole lectures on that,

and I think the lecture of Michael Fried is also in the direction of using partial differential equations for solving image processing problems.

So that's a very modern approach, and many groups in the world work on this.

Or if you think about segmentation, if you think about segmentation problem, for instance, you have the following.

You have a picture, medical image of the human heart, and you want to find,

so this is the human heart in a CT data set, and you want to find, for instance, this contour line.

What you can do is you can compute, for instance, the gradient, and compute those points where the gradient is showing up a large value,

or you can compute the structure tensor to detect the boundary here.

But there are very modern methods that are no longer looking at local operators and differentiations.

They just say the boundary line is a function, and then they write down an objective function and say,

we look for a boundary line where the inner part is showing the density of the heart muscle,

and the outer part is showing, I don't know, the density of water, for instance.

And then you compute the function, the boundary function in the registration process.

So we will see a lot of applications of this technology, and as I said, it's one of the core technologies we have available nowadays.