We should start, welcome to the Tuesday session.

Unfortunately we have to have a shorter session today

because I have to leave at 12.30 for a talk,

a very important talk.

And that means that I don't want to summarize today

where we are, but that should not encourage you

to answer the question in the evaluation

is there a storyline or not in the wrong manner.

We have a storyline, okay?

So yesterday we started to look into the problem

of image registration and in this context we said,

okay, let's look at the following problem.

It's very easy to characterize.

We have points and we have points in the second image

and these points are rotated and translated in plane.

So we have a 2D rigid transformation,

no deformation at all.

And the question was how can we compute the rotation

and translation out of these point correspondences?

This is a trivial problem, by the way.

This is not something where anybody is very excited

if you present a solution.

What we found out yesterday is that our knowledge

about complex numbers helps a little bit to describe

and to formalize the problem in a way that we end up

with a system of linear equations.

There are also other ways to do that.

This is not the only way.

But I personally find this pretty nice

and we do the following.

We say these points, the 2D coordinates

are considered as complex numbers.

The X value is the real value,

the Y value is the imaginary value.

And then we look what happens, I mean,

if I rotate and translate a point.

Basically, I can say complex number

representing the point PK1, PK2

is generated out of the point QK1, QK2

by a multiplication with a rotation complex number

and by the intercept of a translation complex number.

And the reason why we can do this here is

we looked at the algebraic operation of multiplication in C.

Not C, yeah?

And it turns out that we have a very nice geometric

interpretation of multiplications of complex numbers.

Multiplication of two complex numbers

is basically nothing else but a rotation and scaling.

And if the complex number has the norm one,

we know this is just a rotation and no scaling.