So folks, we are currently discussing the projection mappings.

We are characterizing how are the 3D points mapped onto the image plane or into the image plane.

That's an important question. Why is it an important question?

Because we are in Chapter 3, where we are going to talk about 3D reconstruction.

And last week I pointed out that 3D reconstruction using X-ray images

is basically nothing else but solving a system of integral equations.

And the integrals are line integrals.

And the lines that are considered within these integrals are the projection lines.

So these are the lines which are used by the X-ray particles

to be propagated through the object and to hit the detector.

So we have to find a mathematical characterization.

How are 3D points mapped into the image plane?

And we have considered a few projection models,

parallel projection or orthographic projection,

scaled orthographic projection,

paraperspective projection and perspective projection.

And within the chapter on perspective projection,

we have seen that this is unfortunately a nonlinear mapping.

And unfortunately this is exactly the model that we have to use for our X-ray setup.

So we are in a bad situation in terms of engineering issues

that we are going to be involved.

But we have introduced homogeneous coordinates

by lifting the problem into a space that has one more dimension.

And a point in 2D or 3D is mapped to a line in 3D or 4D.

And within the space of homogeneous coordinates,

we can formalize the perspective projection also as a linear mapping.

That was the key message of the session on Tuesday.

And now we are considering the following problem.

Assume this is the calibration pattern.

Like a calibration pattern we had for image undistortion at the beginning of the lecture.

This is a cylinder.

And within the outer shell of the cylinder,

we have embedded steel or I don't know which material is it.

I think it's steel or some steel-like or some metal type metal spheres are embedded there.

So we talked to our mechanical engineering friend and told him I need a cylinder with metal spheres.

These metal spheres should have different diameters

because then I can generate a pattern within the helix that I have here on the surface of the cylinder.

And this will simplify later on the matching between the point that is observed in the X-ray image

and the point that is actually in the calibration pattern.

We have to find a one-to-one correspondence between the 3D calibration pattern

and the 2D points that we observe.

We put the calibration pattern into the X-ray system and capture a 2D image.

So just for illustration purposes, here in more detail.

So basically we have here our 3D points and this is in the 3D coordinate system.

And here we have a 2D image plane and we map these points into the image plane.

And what we need is a one-to-one correspondence between the 3D points and the 2D points.

And these 3D points in the world coordinates are given basically by the mechanical engineer who built it

with a precision of a tenth of a millimeter for instance.

And the problem we have now to consider is the following.

And that's basically the calibration problem.