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

We start with the topics of last lecture.

Let's consider the storyline of this lecture series.

This semester we talk about interventional medical image processing techniques

that can be used by a doctor during the therapeutic procedures while the patient is on the table.

So that's the big picture.

And so far we have considered two mathematical concepts in more detail that are important

for us.

That's the singular value decomposition, which is a kind of norm representation, normal representation

of matrices.

And we talked about homogeneous coordinates.

Homogeneous coordinates that are important for representing perspective projections in

the language of linear algebra.

By going up one dimension we were able to characterize perspective projection mappings

by linear mappings.

And then we looked into one chapter that was dealing with preprocessing.

And here in particular we looked at gradients and gradient information and we have introduced

the structured tensor which is built out of the gradients, tensor, sorry, tension, structure

tensor that is built out of the gradient, that is built out of the gradient and that

allows us to find points.

And points are important for doing all the geometric stuff we have considered within

the chapter on magnetic navigation.

There we have considered the problem of using point correspondences into images and characterizing

first of all the overall geometric properties of the scenario.

We have introduced the central matrix, we have introduced the fundamental matrix and

we also have introduced one possible algorithm once the projections, the projection mappings

are known how you can compute out of two corresponding or out of a pair of corresponding points how

you can compute the corresponding 3D point that belongs to the two projections.

That's something we have quite under control in this context.

We also discussed problems like building robust algorithms to estimate the essential matrix

using point correspondences.

We also considered one aspect that is very important for many practical application and

that is the scaling of the input data that was also considered.

So we learned quite a lot on magnetic navigation.

And why was it important to consider in the context of magnetic navigation the epipolar

constraint and all the geometry underlying the epipolar constraint?

Well, the reason for that was that we basically were required to build a user interface that

allows us to tell the system from where to where or what the orientation of the magnetic

field has to be.

So you click a point in the two images, you click a second point and the difference vector

is basically the orientation of the magnetic field that we want to apply.

And this is important for stereotaxis procedures when you want to guide the catheter through

the human body by using external forces.

And the system is also built and shipped to hospitals and it's in practical use.

But we also have to admit at this point that the expectations that have been associated

with magnetic navigations have not been met at all.

So doctors have expected way more impact on their daily work than it actually had at the

end of the day.

So for the standard routine work, these systems are not necessarily required.

And last time we looked into a device that is used during surgeries a lot, that is used