So, fan beam, once again the setup, it rotates three times a second just for your mechanical

inspiration here and here you have your curved detector and in modern CT scanners this detector

is not just a single line but it's multiple lines and that means that we have instead

of a fan beam a cone beam geometry and I told you that roughly 10 years ago industry started

to build in detectors with multiple lines so industry started with four lines, with

eight lines, with 16 lines, 256 lines and now they are no longer talking about lines.

So, that's the scanner we consider and now let's look at different acquisition geometries.

So, what you can do is you can first think of reconstructing an object, let's say, okay

and you can say I rotate around the object here like this and I rotate around the object

and I do a reconstruction of the central slice using fan beam, I move the table a little

bit, I reconstruct the next slice so I do a slice by slice reconstruction, a slice by

slice reconstruction and so I move the table for instance in

and I do a reconstruction slice by slice.

What we assume here is that the projection rays are always orthogonal to the rotation

axis.

So perpendicular to the rotation axis but what we also could do is for instance we could

tilt the CT scanner a little bit, we could tilt the CT scanner, we tilt it and so the

table and the CT scanner are no longer perpendicular to each other but will have a different angle

than 90 degrees.

That means that the line integrals are no longer perpendicular to the rotation axis.

And that's also a problem.

The question is how does it affect the reconstruction algorithm?

Now we always say we have a 2D function that we want to reconstruct and then we rotate

around the object and do a standard back projection reconstruction.

Now things are tilted and of course you can say we compute the tilted 2D sections or intersections,

the 2D planes that are not perpendicular to the rotation axis and reconstruct this step

by step and look at the reconstruction result.

And before we look into the general cone beam situation we look at the parallel projection

scheme here.

So think about the three dimensional object we want to reconstruct.

It's a tomato or something like that.

And we have parallel projection lines.

Parallel projection lines, that means the X-ray particles are projected on the detector

by using parallel X-ray beams.

And what we can do is we can then look at the Fourier transform of this 2D projection

plane.

And like we did it in the 1D, 2D case we can show that this plane here, this plane here

in the Fourier domain corresponds to this plane here in the 3D Fourier transform of

the function to be reconstructed.

So if I have parallel projections, I have a 2D image acquired with parallel projections.

I can compute the Fourier transform of that and take the plane and put it into 3D space

right parallel to the detector plane in 3D and I sample the 3D Fourier transform of the

function to be reconstructed.

That's a generalization of the Fourier slice theorem that we know from the 2D, 1D case.

If you look at the proof of the Fourier slice theorem, it really can be lifted easily to

the higher dimensional situation.

So we reconstruct the n-dimensional function from n minus 1-dimensional projections by

using the Fourier slice theorem.

It can be generalized.

And that means I have to walk around the object, acquire images, and I want to sample the 3-dimensional