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Good morning everybody. We are currently considering...

We are currently talking about non-rigid image registration.

We have a pretty good understanding what image registration is, Matthias?

We have two images, one is deformed and one is transformation.

We want knowledge about how to give from one image to the other.

We learned about the general principle that is applied to compute the transformation of these images into a joint coordinate system.

There are four things we need to solve.

We have to find a way to characterize the transformation.

Is it parametric, non-parametric, parametrization?

Do we have to have a variational approach or a parameter estimation thing?

If we decide for parameter estimation, remember image distortion was an example where we did some kind of image warping.

The second is how to define an objective function that basically judges the quality of the mapping that we are considering.

Then we search over all possible mappings in the worst case and find the one that fulfills the objective function the best.

It can be either a minimization or maximization problem.

From a certain distance you can say image registration is nothing else but solving a huge optimization problem, a high dimensional optimization problem.

The third thing is once you have set up the objective function, you have to find an optimization algorithm that is efficient, robust, and guarantees to find the global maximum or minimum, depending on what you set up.

The fourth ingredient is the clinical evaluation.

It is required that the algorithm that you design is not working for just one image pair for a proof of concept, but you have to do a very, very extended evaluation to make sure that it works for clinical data of the given framework that you are considering.

Once again, what is the picture we have to keep in mind?

All the chapters we are currently considering in this lecture are driven by some key figures.

The key figure for non-rigid image registration is basically you have here the mapping of one pixel X to the pixel U of X.

As similarity measure, we have seen the SSD yesterday.

What does SSD stand for?

What does SSD stand for, Norman?

Sum of squared differences.

Sum of squared differences.

So we have our source image, we have our target image.

The source image minus the target image at position UX.

That is basically the warped coordinate over omega DX has to be minimized.

That's the data term that we are considering.

Now look what we are doing.

This is the way we are considering image registration.

We have a point X.

We move the point X in the image plane and look at the new point, the moved point, the deformed point, and read out the intensity value.

That's the standard way how to formalize image registration methods.

Now Cecilia is saying, well, that's fine, so we work on this.

Can you think about an alternative version?

There is a way to define this alternatively.

Because instead, and this is something that is not discussed in the literature, there are some disadvantages, but just to make you think of different alternatives.

If you read an approach like this, you should really be a fresh thinker and say, why should I move around the X coordinate?

Why don't I just add an intensity value to the current position to achieve by adding to this intensity value another intensity value, the intensity value that is here at the position X.

I mean, instead of moving around the point, like the belly is moving like this and this, I just say it's not moving in space, but we are deforming our intensity values.

You see the different idea?

I haven't seen this somewhere in the literature so far, but you could also say alternatively, be a fresh thinker.

We have S of X plus T of X plus AU of X, which is actually an intensity value.

Just correct this here. Oops. Take the difference here.

That's an alternative way of looking at it. This U of X is not a displacement, but this is an intensity offset.

How do I have to change the T of X by plus U of X that can be positive or negative to be as close as possible to this one here?

What would you say according to the dimension of the problem?