Okay, so please apologize that I'm a little late today.

I was in the dean's office and it took a little longer.

So we will skip today the mind map with a big picture and just continue right away with

the refresher course on the mathematical characterization of projection mappings.

So that is basically the image that you should keep in mind while we talk about projections

and projection models.

And our plan for the lectures until Christmas is how can we reconstruct a 3D volume, a 3D

cube out of 2D X-ray projections.

So the task will be how can I compute the density values, the attenuation values in

space using line integrals over the X-ray or given by the X-ray projection.

And for doing so it is very important to understand in detail the propagation path of the X-ray

particles from the source to the detector element.

We have to characterize these projection lines from the X-ray focus, from the optical center

through the object to the grid point on the detector that we use for measuring the X-ray

image.

And it's clear if we look at the geometry of projection, if I project a 3D point that

sits here on the detector I get the same projection like a point that sits here.

So all the points sitting on this projection line here they end up with the same point

in the image plane.

Or if I think this is the image plane, this is the projection ray, if I project this point

here on the projection ray, this point to the image plane I will end up at the same

point like the geometric projection of this point, this point.

Let me do it this way.

If I project this point to the 2D plane I get the same result as I project this point

to the 2D image plane.

And that means that we have here a relationship of 2D points and 3D lines.

A 2D point using a selected projection geometry can be associated with a projection line.

All the points that sit on the projection line they are in one equivalence class that

is defined by the projected points.

All the points, the 3D points that end up to be the same 2D point in the projection

they sit on a straight line.

And on Thursday last week we have introduced the concept of homogeneous coordinates.

And the idea was if I have a 2D point, let's say U and V, we associate with this a 3D point

W times U, W times V, W.

And that means we consider the point U, V, 1 and we scale it by an arbitrary scaling

factor and this is the homogeneous coordinate of the 2D point.

And if we want to come back to the 2D point what we have to do is we just divide by the

third component and get the first and second component of the point in 2D.

That's the idea.

So if you hear something about homogeneous coordinates just remember add an additional

component and if you want to switch back to the original vector you just have to divide

by the third component if you have a 2D, 3D homogeneous representation.

Is this additional component always the same for all pixels or does it change for every

pixel?

No, it can be any value, any scaling factor.

So if I ask you write the point 1, 2 and the point 2, 3 in homogeneous coordinates you

can write, okay, this is 25, 50, 25 and this here is 10, 15 and 5.

These are homogeneous coordinates of the point with different scaling factors.

If I tell you this is a homogeneous vector associated with a 2D point you start right

away to divide by 5, the first, the second component and you end up with this period.