Welcome.

Good morning, everybody.

I'm sorry that I was out of office yesterday, but there was something that was a little

more important.

So I had a short trip to Bonn yesterday, and Alex was replacing me, and I think he did

a pretty well job because it was on his master thesis project, so we knew pretty well what

he was talking about.

Well, today, what's on the programme?

Today we will talk a little bit about a basic concept that we need a lot in image processing.

It's variational calculus.

For those of you who attend the diagnostic medical image processing, they know basically

what this all is about, all the others, they see it the first time maybe, but it's a very

important concept, and we will need that for the final set of topics that we are going

to discuss in this lecture.

The final lectures will all be on image registration, image fusion, and in winter semester, we have

considered image fusion, we have already considered image fusion, but only the rigid case.

So do I have to press something?

Here we go.

So we will talk now about image fusion or image registration, and the problem of image

registration basically is, and I will explain that next week in much more detail, that we

have two images and that we want to find the corresponding pixels in both images.

We want to compute these correspondences.

And in winter semester, we considered rigid registration methods, rigid registration methods

where we basically assumed that this coordinate system can be transformed in this coordinate

system by a rotation and a transformation, a translation.

So we have considered just rotations and translations.

So we had no deformations.

And now in this semester, we consider in addition to that the non-rigid case where we allow

for deformations.

Why is that important?

Think about the following situation.

You have a patient, gets a CT, then the patient drinks some water, he goes to the PET scanner,

and the bladder is filling up with water, yeah, and so it's a little deformed.

And to find a proper matching, we need to have a mapping that does not only map the

coordinate systems by rotations and translations, but also allows for deformations.

Or if we think about reconstruction of the beating heart, we need to find out how has

the heart moved within two projections.

So we capture two images, and then we compute the displacement vector field telling us which

pixel is mapped to which pixel in the other image.

And if you just look at the abstract problem of image registration, we have already considered

one mapping in winter semester where non-rigid image mapping was basically considered, and

that was the image undistortion procedure that we have considered in the context of

image intensifiers where this mapping here was basically characterized by a bivariate

polynomial in X and Y.

So for the rest of the semester, we will talk about non-rigid image registration in various

dimensions and we will consider basically the current state of the art.

So there happened a lot since 2000 in research, and basically we try to catch up with the

major ideas that were generated within the last nine years already.

For instance, the problem is how can we formalize this in terms of an optimization routine?

How can we optimize the functional equation that we basically set up?