The following content has been provided by the University of Erlangen-Nürnberg.

What did we do last week? We looked at the image registration problem.

We have the source image, we have the target image, we have displacement vectors or a displacement function.

And we look for a function that warps the image properly such that source and template image are basically the same.

And instead of computing parameters, we compute here a function, a warping function.

And that means we have to look at the functional that has to be optimized and we have to set up the Euler-Lagrangian differential equation.

We have to discretize the derivatives and look what type of mathematical problem we end up with.

That's a very brute force definition or explanation.

As I said, there are whole lectures on how to solve partial differential equations.

I cannot wrap up all the algorithms that are out there.

First of all, I don't have time.

Second of all, I do not know these methods either.

I have just a vague understanding and I usually find a good way through the type of problems I have to solve using numerical recipes and other libraries in the web.

It works pretty fine in most cases.

So we looked at the functional.

You remember how the functional looked like.

We have the sum of square differences.

Here is the functional that we want to minimize.

SST stands for sum of square differences.

The alpha is what?

Engineering constant.

Engineering constant and R is the regularizer.

That does not depend on the measured data.

So it's some kind of prior knowledge.

And what did we do as a regularizer?

First of all, we looked at the gradient and said we want to minimize the sum of square differences for the warped image and the original image.

And we penalize high gradients, so high changes in the deformation field.

We want to have a smooth and nice deformation field.

And then we wrote things down into this form and brought it into this form.

And this is the typical situation that we have for applying the Euler-Lagroshian differential equation.

We had this refresher lecture on Euler-Lagroshian differential equation where basically this type of optimization problem was considered.

So what do we have to do?

We have to compute the Euler-Lagroshian differential equation.

And we all remember what the Euler-Lagroshian differential equation looks like.

It's fu minus dx fu prime.

It has to be zero.

That's a necessary and I think also sufficient condition for the extreme that we are looking for.

And then we compute fu, fu prime, and the derivative.

And basically we end up with this PDE.

S of x minus T of u of x times dt du.

So we derive the deformed template image with respect to u.

And this has to be alpha times the Laplacian, the second derivative, the Laplacian.

And I derived that last time and it took me some time to fix that.

Here we go.

Where is it?

Somewhere I wrote it here.

We did the derivation here, where do we get in the second derivative.

And this is the type of equation we have to solve.

And everybody here in the audience should be able to write down this in terms of a system of equations

by discretizing the derivatives that we have here.