Welcome everybody to the Tuesday session.

We are currently in the chapter on 3D reconstruction.

So what are the topics in digital medical image,

diagnostic medical image processing? We talked about modalities.

Not in very much detail, but we have

seen an overview over different modalities. We talked about preprocessing.

We had also a few exercises on it. And currently we are in the chapter on 3D reconstruction.

So the idea is we capture images with a certain device. We do specific preprocessing methods

that take into consideration the setup or the measurement principle

and eliminates artifacts that are implied by the acquisition device and the setup.

And now we take multiple images of one and the same modality and we do a 3D reconstruction.

Now basically we are considering the problem, how can I compute from X-ray images

higher dimensional image information like CT datasets, computed tomography datasets.

And the fourth chapter after Christmas will be on fusion.

And that's the question, what do I do if I have multiple images, different dimensions from different modalities,

how can I transform these images into a joint coordinate system such that you can just switch back and forth

between the different modalities that have their intrinsic advantages and disadvantages.

So that's the storyline. Let's make it short. I'm pushing a little bit because we are a little delayed concerning the contents of the lecture.

And now we have started to think about 3D reconstruction and I also explained to you the integral equation.

This is something I want to repeat today as well in the next set of slides.

What we have considered yesterday and last week was basically the geometry that underlies the X-ray acquisition.

How are the X-rays propagated through the object?

How can I characterize the straight lines that more or less represent the projection lines from the X-ray focus to the detector?

And we have introduced in this context the homogeneous coordinates.

We have introduced different projection geometries.

For us it's most important to know the perspective projection.

Perspective.

Okay.

What was that?

The perspective projection.

Perspective.

You know what I mean. The perspective projection.

And then we also discussed yesterday a calibration procedure based on homogeneous coordinates.

How can we estimate the components of the projection matrix using a least square estimator?

It was a little difficult in a sense that we had to deal with homogeneous coordinates and we had not the identity but identities up to scaling.

And then we computed the ratio between the upper components and the last components and broke things down to the original spaces,

computed differences instead of looking at identities and then we started to look at the minimization process that is required to solve the

or to estimate the components of the projection matrix.

What's important to remember are two things.

If you take the definition right away of the projection matrix and the projection mapping,

you end up with a nonlinear estimator because you have a ratio where the P entries show up.

This can be solved by iterative methods. It's not a big deal.

But we also have seen that we can manipulate the equations in a way that we end up with a linear optimization problem

that can be solved either by eigenvalue and eigenvector problem or by SVD.

That's the main message of yesterday.

But I have to point out that the multiplication with a denominator changes the objective function in a way that the result will not be as robust

as the result you will get if you do the nonlinear optimization procedure.

Just as a side remark, this is something that I cannot explain here in more detail.

But what you should remember in the future, always try to optimize an objective function where the error that you want to minimize is in the space of observations.

Then you are usually on the safe side.