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

Okay, welcome everybody. Welcome back to Diagnostic Medical Image Processing.

Today we will talk again about reconstruction and we will talk on a special problem that occurs or can occur, which is truncation.

But let's first have a look at what we did in fan beam reconstruction and just get back to the topic of image reconstruction.

So let's look at that. I bring this nice example.

By the way, if you're in the oral exam and you can't show this example, you're doomed.

If you can't show how to solve this example, you're really doomed. Remember that.

So basically image reconstruction is solving for the unknowns in the slice we want to reconstruct.

And these unknowns, I denote them here as four variables, X1 to X4.

And what we observe on our X-ray detector is the sum of the respective rays passing through these pixels of your slice.

And we can do that from multiple directions. Then we get basically a system of equations.

And this system of equations we can solve and then reconstruct the slice.

We've seen that this direct solving, putting this into a matrix is not feasible because the matrix is far too big to solve this easily with a matrix inward.

So we came up with a different way of solving this. And we are using filtered back projection.

And the idea is that we can take the projections, the point that we observe on the different detectors, and we do the filtering.

And then we can back project and get exactly the image.

So you remember that if we would not filter, we get something that is wrong, that is like this peak shaped shape that I showed previously.

So we need filtering and then we get the correct reconstruction.

And the filtering that we are applying requires a convolution and that implicitly requires our object to be finite.

So we need the object to end within our detector in order to do proper filtering.

And we've seen that we can do that in a parallel beam geometry.

In the parallel beam geometry, we have this very nice Fourier slice theorem where we can really show that the Fourier transform of our projection equals to one of the lines that goes through the origin in our Fourier space.

And then we can apply that to derive the filtered back projection algorithm and we get a beautiful reconstruction.

Then we've seen that this is kind of infeasible for a real scanner.

So if we put that into a real scanner, we have to move the source and the detector every time upwards.

And we also have to rotate them. So we have this principle of shift and translate.

And this will take in a real scanner five minutes for, in this case by the way, it's already a multi-row detector.

So this has two rows, so two detector elements that are located on your parallel beam.

Anyway, it takes five minutes to acquire this thing and while we can reconstruct this much faster now, acquisition time was one of the major limiting factors here.

Hey, welcome.

Okay, so we can't do parallel beam geometry in this scenario because our acquisition time takes too long.

So we wanted to do something more clever and we came up with the fan beam geometry.

There we have a single detector that acquires a whole fan of rays because we have a single source but multiple detectors.

So you get a fan and they are no longer parallel.

And we've seen that the Fourier slice theorem does not directly represent this geometry,

but we have to do some modifications to the reconstruction algorithm in order to make them applicable for this kind of geometry.

But we've seen that even if we have a fan beam geometry, we can re-bin this or we can transform it into a parallel geometry problem.

We have slightly different requirements for a full data set,

but we can transform it from a fan beam into a parallel beam problem and vice versa.

We can take a parallel beam reconstruction algorithm and transform it into a fan beam reconstruction algorithm.

So we can still solve that thing just using the fan beam geometry and we can acquire much faster.

And you've seen that in modern CT scanners, they are approaching a comb beam, so with multiple detector rows,

they're approaching this kind of geometry, but fan beam reconstruction for the center slice will work perfectly.

We will look into 3D reconstruction tomorrow a bit further, but can already acquire pretty fast with that.

And the first fan beam scanners became available in 1975, so this is already a quite old principle,

but it's feasible for reconstruction. So we can do that.

The algorithm that emerges is again a filtered back projection type of algorithm.

So we do filtering and projection domain and then back project and we get our reconstruction.

Today we will look into truncation, the truncation problem and what the implications are for our scanning geometry.

So truncation is a very specific problem, but it commonly occurs.