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Good morning everybody. Welcome back to Diagnostic Medical Image Processing. So today we will

talk about 3D reconstruction. So now we will go from 2D to 3D and look into these reconstruction

problems. And to get you back all onto the topic of reconstruction, we will just go through

fan beam reconstruction shortly. So yesterday I told you if you don't know this in the examination

you are doomed. But obviously if you only know this example you are also doomed. Okay?

So you need to know more than that. So tomography is basically solving for the slice that you

observed in your object. So you want to reconstruct the individual intensities or attenuation

coefficients in your slice. Today the slice will turn into a volume. So it's a set of

voxels, no longer pixels anymore. But the idea is still the same that we can observe

the ray sums. And with these ray sums we know something about our object and we can use

that to reconstruct. And we've seen that if we observe the object from multiple angles

and we apply a technique called filtered back projection we can do something to our projection

and then just smear it back into the slice or into the volume and this will give us the

reconstruction result. So this is a very efficient way of reconstructing and we don't have to

save like the complete system matrix which we've seen that it's far too big. So we can

solve that. So far we've seen that for parallel beam and fan beam geometry. So you remember

that in parallel beam we were taking one beam at a time so we had to translate and rotate.

And in the fan beam geometry we had the idea that we can acquire much faster if we have

a fan and acquire all the rays along the fan at the same time. So we have one source position

and a detector and using this geometry we can acquire a whole bunch of fans, a whole

bunch of rays. And this is much faster. So we've seen that this is built in into commercial

CT scanners and scanning became really fast using slip rings that you can have a continuous

rotation. Today we look into 3D reconstruction and we'll start in a similar way as we did

in 2D. So first we look into parallel geometries and in 3D we have a very similar

dimension. So in principle we can have two types of integral data. So we could on the

one hand collect line integrals but we could also use plane integrals for the reconstruction.

We'll see that in a bit how this could work. And then we will look into comb beam data.

So comb beam data is the data that we usually acquire and we have a total of three different

reconstruction algorithms that I will present in this lecture. Of course there's many more

reconstruction algorithms because you can tweak in two and many parts and in 3D you

can much more freely choose your acquisition geometry. Hey, welcome. Okay, so let's look

at parallel line integral data. What we're doing here is we are collecting the

lines. So we are using rays to which we cast through the volume and we observe the ray

sums. And the idea is now that we can acquire this parallel line integral data in different

geometries. So one thing we could do here on the left side, so we rotate around here

in this plane and we acquire these rays. So we would get a very similar geometry for every

slice and we would be able to reconstruct. So this would basically be multiple fan beams

or parallel beams. So we are in parallel here. So this would be just the acquisition of multiple

parallel slices using this kind of geometry. But we could also do something else because

we're in 3D now. We can acquire rays that are not in the same plane as the rotation

plane. So we still rotate around the same plane, but now we're acquiring rays that are

slanted with respect to the rotation plane. So we are not acquiring the rays which are

exactly in the same plane as we rotate, but we are slanted. So they have a certain angle

which they are rotated at. And we can even increase that so we can increase this angle

and still rotate around here. And now the question is, if we do that, how does that

affect our reconstruction? So do we get any problems? Can we still reconstruct in the

same way? How does that work? So there is, for this line integral data, there is also

a central slice theorem. And this theorem says that if you're acquiring, so now we are

not acquiring only a single sequence of detector positions, but we are acquiring like on a