Good morning everybody.

Tuesday session, 90 minutes, so we have a few minutes to reconsider the storyline and

the major topics we have done so far.

So let me draw as usual at the beginning of the lecture the mind map.

I'm aware of that this has a high degree of redundancy and I'm repeating myself over and

over again.

But I also noticed that people appreciate that quite a bit to see the big picture if

there is some big picture and it will allow you to follow the storyline.

We talk about interventional medical image processing this semester and which topics

did we consider so far?

Sandra, you are my candidate.

Magnetic navigation, what was the idea?

So you have here the two magnets and here you have your catheter and then you can control

the direction of the catheter tip by a magnet.

And we built an interface for that, user interface that allows us to adjust the direction of

the magnetic field using two projections.

And what we did there was, well, we have considered two projection images, point and the two camera

centers and we have introduced the concept of epipolar geometry.

And this allows us to compute the essential matrix out of this equation here, that is

the epipolar constraint saying that two points, corresponding points have to fulfill this

epipolar constraint in terms of homogeneous coordinates, so these are 3D vectors that

represent homogeneous coordinates.

And the essential matrix has how many degrees of freedom?

How many degrees of freedom do we have with the essential matrix?

Sandra, it's a 3 by 3 matrix, so an upper bound is 9 and a lower bound, a sharp lower

bound is?

How is E defined?

How is E defined?

Come on.

Right, that's R, T, X, sometimes R transposed, doesn't matter.

So how many degrees of freedom do we have?

R is what?

R is a rotation matrix.

How many degrees of freedom does the rotation matrix have?

It's 2 by 2, so 4 degrees.

3, yeah, the three rotation angles around the X axis, around the Y axis, around the

Z axis.

And the translation vector, how many unknowns do we have there?

Three.

So E should have 6 degrees of freedom, but we can multiply here the equation with an

arbitrary lambda, so we reduce the translation vector to a unit length translation vector,

and we have 5 degrees of freedom.

We have 5 degrees of freedom.

How many degrees of freedom did we consider in the 8-point algorithm?

Sabine.

Right, we had 8.

So you see the point here.

We have the epipolar constraint, we get for each corresponding point pair, we get an equation

linearly in E, we can estimate E using, for instance, the 8-point algorithm with 8 degrees

of freedom, or using a completely different algorithm that is based on the 5 degrees of