Good morning again. As I said yesterday we had a guest speaker and I was rather ashamed

that only four students attended. That was something that is totally unacceptable. I

should invite or if you want me to invite medical people then at least I expect that

500 people attend or something like that. That's not good. Good. Okay, so we are in the

final for this semester and the final topic we are looking into or one of the final topics

is non-rigid image registration in various applications. And yesterday we have seen why

image registration is very, very important and yesterday we also have seen that especially

the registration of multi-model images is important and Professor Kuvert has also pointed

out why non-rigid registration is so crucial and where the weak parts of the existing algorithms

actually are. So what we are going to discuss today is we will continue with a mathematical

formulation and please reconsider what we have done so far in terms of image registration.

Let me just add here another page to come up with the mandatory mind map for the Tuesday

session. What is this? I want to make this a little thinner. So in interventional image

processing we are currently discussing non-rigid image registration. And what is meant by non-rigid

image registration? Well non-rigid image registration is nothing else but the mapping of two or

more images into a joint coordinate system, into a common coordinate system. And while

we map things into a joint or common coordinate system we can deform it. That's the idea of

non-rigid image registration. And what we have seen last time is that basically this

turns out to be an optimization problem that requires to minimize a functional equation.

That requires to minimize a functional equation. And we have seen that this functional equation

dependent on the displacement vector field is basically some similarity measure that

depends on the two images. Hold on, we call it source and target image. Source and target

image plus some regularizer that only depends on you. And this regularizer only depends

on the displacement vector field. Only depends on the displacement vector field and not on

the observations, not on the actual images that we want to register. So in terms of pattern

recognition this is also called some kind of prior knowledge that we are applying here.

For the pattern recognition guys in here, just a weak pointer, if you optimize the,

just for the pattern recognition guys, if you optimize the a posteriori probability

and you do the logarithm of the a posteriori probability that is the same as optimizing

log p of y plus log p of x given y. And this here is also not dependent on the observation

and is called the prior. So there is some similarity between these two things here and

basically also basing classifiers can be reduced to a problem that is stated as above. Just

for your information how to look at these things. So let me just remove that because

some of you might be confused by that. Good. And how do we solve such a functional equation?

How do we maximize that? Well a sufficient and necessary condition is that the Euler-Lagrosian

differential equation is valid. And what is the Euler-Lagrosian differential equation

telling us? Well basically it allows us to optimize a functional that says it's u prime

dx from x1 to x2. You remember that. And if I want to minimize this, a necessary and sufficient

condition is that the Euler-Lagrosian differential equation is fulfilled and the Euler-Lagrosian

differential equation looks like what? Sabin? The function derived by variable times that

variable derived again. So it's f of u minus d dx f u prime is 0, right? That's the Euler-Lagrosian

differential equation. We had one whole chapter on variational calculus and we have also shown

by the first variation that this is a necessary and sufficient condition. Okay. So we can

deal with these things and here the s and r actually can be exactly written in this

integral form so that we can apply the Euler-Lagrosian differential equation. And today we will compute

the Euler-Lagrosian differential equation for the SSD similarity measure and the diffusion

and the curvature regularizer. That's something we are going to discuss and we will also look

at the derivative, the first variation of the mutual information for the multimodal

case. But we will see the details later on. So non-rigid registration is part of a huge

chapter that we are currently discussing. Then we have talked about the tracking of