So we started last week to discuss image registration.

And the definition of image registration basically is you have two images, either 2D images or volumes,

and they have to be registered in a sense or in terms of mapping the two images into the same coordinate system.

That means if I pick out a certain pixel in the specced image and I pick the pixel in the CT image

at the same position in the coordinate system, associated coordinate system, it refers to the same point in space.

And there are many problems if you want to solve this issue to combine these two images,

to compute the transform between the two images.

And one problem that is important is do I have point correspondences or do I just have to rely on the intensity values?

Another problem is did the patient move in between the two acquisition procedures in a way that a rotation and a translation is not sufficient?

For instance, if you look how CT images are acquired of the human chest of the thorax,

people lay on the table like this with hands up.

And if they go to a PET or specced acquisition, they have to stay there for 15 minutes or 30 minutes.

And other people, they cannot sit like this on the table or lie on the table like this for half an hour.

So they lie on the table like this.

If you want to bring the two images together, the CT and the specced or PET image,

you have the problem that one image is with hands up and the other image is with hands down.

And now think about having a tumor here.

It moves up like this in the skin, on the skin.

And that causes, of course, a lot of problems.

And the question is how much deformation, for instance, is incorporated into the registration method that we are considering?

In summer semester, we look into the non-rigid case where we have deformations.

In the winter semester, diagnostic medical image processing, we focus on rigid registration.

That means we just allow rotations and translations in the 2D and 3D space.

So basically, we consider now the following problem.

Assume, just insert here one page.

So the problem we are currently considering is the following.

We have 2D points in image one.

So this is the X coordinate and the Y coordinate.

And then we have points here.

And then we have a second coordinate system or a second image.

And we have points here.

And we know that these two point sets are transformed into each other by a rotation and translation.

So we rotate and translate the points and get the right image.

And the question is how can I estimate the rotation and translation?

Estimation of rotation and translation.

How can this be done?

And last week, one of the students has already mentioned that we can just set it up as a least squares problem

as it was written here on the whiteboard there.

And I pointed out, and there is a good reason for doing so, that we can use complex numbers for solving this problem.

Let's just reconsider what we did basically.

We have seen that complex numbers, Z, are basically written in terms of a real part and an imaginary part.

And we can illustrate this here as a real part and an imaginary part.

And I can draw here a point where this is A and this is B.

So all 2D points can be expressed in terms of complex number by just taking the real part and the imaginary part.

So I just rename X and Y by real part and imaginary part.

And so far that's not very interesting what's going on there, but complex numbers now have a very nice property.

We can say this here is the angle that this vector AB has in real and imaginary space.

And I can rewrite the complex number now in terms of the Euler notation, saying this is the absolute length or the Euclidean length of this vector here,

times E to the power of I phi, where phi is this angle here.

And if I multiply now two complex numbers, let's say I multiply Z1 with Z2,