Okay, as most of you can see, Professor Honegger is not here today, so I will give the lecture.

My name is Alexander Broest.

I'm in the medical imaging group at Professor Honegger's chair.

And yeah, so I have to give the lecture today.

What we did the last week was, I guess, about ellipse fitting in two dimensions, right?

Okay.

And we will continue today with quadric surfaces, which is a little bit the extension to three

dimensions.

So you have ellipsoids or parabolas which are rotated.

And this is just by extending the equation that you have with just two components for

the ellipse, extending by another coordinate set.

And then you can put this either in this equation or you can reformulate this as a matrix equation

where you have a world point as you had it before with a 2D point and homogeneous coordinate,

also with a world point.

And this Q matrix has a similar shape as you had it for this C matrix, I guess it's called

for the ellipse equation.

With this quadric surfaces, you have a little bit more possibilities what it can be in 3D

space.

So you have ellipsoids, paraboloids, cones, also intersecting planes or coincidence planes.

There's also a distinction between if it's a real object or if it's an imaginary object.

And to distinguish all these objects, you need the rank of the Q matrix, of the 4 by

4 Q matrix, and also of this Q1 which is the 3 by 3 left upper sub matrix.

And when you consider the ranks of these two matrices, you can classify these surfaces

in 3D space.

What we need for the ellipse reconstruction, or the model generation of the lasathy catheter,

is the cone, and there we have a rank of three for both matrices, and we have an intersecting

plane, what we later need forward on reconstruction.

So now I will come to the ellipse reconstruction.

First of all, what we know, we have a 3D object, which is something like a circle or an ellipse

shape which is just a degenerated.

It's not really a 3D object because you can't come up with a real

mathematic formulation that gives you an complete ellipse in 3D

because it's just in the section of a 3D cone and a plane.

But we know that in both projection images we have an ellipse

and we can compute this ellipse, that's what he did last week,

and put up these two equations for both images.

So for plane A and plane B we have this equation with the points.

And these points PA and PB

are in fact the projection of this W points in the 2D images.

So we can write here instead of this small PA and small PB,

just the projection of the world point from left and right.

And then we can compute from these two matrices

a cone, a 3D cone, as a quadratic surface.

So these two matrices here describe this 3D cone,

which is from the optical centre of the camera

going through the imaging plane to this ellipse

and give you a 3D cone in 3D space.

What you can do a little bit as a homework, I guess it's obvious to see,

just make sure or figure out why this matrix Q is ranked 3

when you have an ellipse in the image.