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

So welcome to the Monday afternoon session.

Before we go into one very important problem in computer vision in general,

and in particular in many medical image processing applications,

hand-eye calibration is a very, very important problem which is required to be solved.

And that's basically computing the transformation between the coordinate system of your eye

and the hand's coordinate system.

So if you have a robot arm and a camera and you want to compute the transformation

of the camera coordinate system into the coordinate system of the robot arm and vice versa,

then you need to do hand-eye calibration.

And that's, for instance, also required if you have an ultrasound probe or an endoscope

with markers on it and a camera.

You need a transformation between the camera coordinate system and your markers,

or you need a transformation between the markers and your image coordinate system

that you are moving around like the endoscope device or the ultrasound probe.

And how to compute and estimate this transformation is something that we want to look into today.

But before we do so, let's briefly summarize what we have considered so far.

One second. Next page.

Do never try to understand Microsoft here.

So in the summer semester, we will talk about how imaging devices are used during intervention.

So we have an endoscopy system, we have an ultrasound probe,

we have x-ray imaging during the treatment procedure for the patient.

And what we have considered this semester in more detail was we talked a little bit about linear algebra

and projection models, and then we looked into features that we have computed like the structure tensor

that tells us something about the behavior of the gradient in a local neighborhood.

And instead of the gradient for each pixel, you can also think about higher dimensional features

that are computed for each pixel, and then you combine these by looking at the covariance matrix

and basically the eigenvalues and eigenvectors.

What is this here?

Eigenvalues and eigenvectors.

And I tell it now for the hundredth time, if we have similar eigenvalues far away from zero,

we have more or less a cyclic structure of the gradient, so with a high probability we have a corner.

If we have one eigenvalue high, the other one close to zero, then we have an edge.

And if both eigenvalues are zero, then we have a flat region.

What tells us, and this is something you should think about,

what tells us that all the eigenvectors are real numbers?

I'm always saying, okay, we compute the covariance matrix of the gradients in the local neighborhood,

then we compute the eigenvalues, and I'm saying eigenvalue larger than zero, equal to zero.

What tells us that the eigenvalues are real valued?

Because the matrix is symmetric.

If the matrix is symmetric, all the eigenvalues are real numbers.

Right?

Good. So think about that.

If you don't remember this, think about that and try to repeat the properties of symmetric matrices,

because this is something you will need also later on when we will look at other situations

where we have to deal with quadratic matrices.

Then we looked a little bit into the problem how to detect lines, detection of lines using the Haft transform.

And I also extended this to the detection of circles or ellipses.

Whenever you have a parametric representation, you can apply the idea of Haft transform.

And then we started to look into one special topic, and that was magnetic navigation.