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

Good morning everybody. We will talk about fan beam reconstruction today, but first we

will start looking at what we have been talking about so far. So I just bring this example

again that you remember what we are talking about. We want to solve this puzzle. We consider

the slice that we want to reconstruct as a matrix of pixels and we want to know what

their actual values are. And what you observe are basically line integrals that are the

summation of these variables and I can write this as an equation. And we observe the sum

of these pixels and later we want to solve for the actual value of the pixels. We also

discussed that we could put that into a system of linear equations and we could basically

just solve this maybe with a matrix inverse, but we were considering the problem size and

we have seen that maybe this is not a good way to solve this just with a matrix inverse

if we need something like 65 terabytes to store this matrix. So we came up with another

idea how we could solve that and this involved looking at projections and back projection.

So you remember in a parallel beam geometry if I take a detector, these are detectors

and I rotate it around the object, I get observations of the object at different points and then

I could take the idea and just smear this projection back into the volume using the

back projection and if I do that from a certain amount of locations I would see a maximum

where the original point was. And we also thought about that that if we really do that

with an infinite amount of projections we would end up having something like this. So

this isn't a single point anymore but it smeared across the whole reconstruction domain and

we derived the filtered back projection algorithm where we applied this filtering which introduces

this kind of negative wings so we are removing the false effect of our back projection and

then we get the correct reconstruction. So that was parallel beam filtered back projection.

So in the parallel beam geometry we remember that was the earliest acquisition geometry

and it had the principle rotate and translate. So I would pick a small point source and I

would pick a small detector and then I move it all along here and detector all along here

and we get a projection and then you rotate the whole system. So again we would have to

move the source and the detector and we get one ray at a time in this geometry. And we

remember that if we look at this, so this is our image and if we put this into a sinogram

we can see that if this is y, the y coordinate in the image and this is the x coordinate

and we project and let's see we start with a projection across here. So let's say this

is the angle theta equals zero and then we put this single observation in here and I

rotate around that I would see something like this that the objects are moving across the

detector and the shape forms a kind of sine wave. So this thing here is called sinogram

and here you would see your detector index s and along this axis you see the rotation

angle theta. And what you also observe is that if the objects are behind each other

you get a higher intensity like in this case or here they move behind each other so the

intensity is increasing because we are observing the sum of all the mass that's in the beam.

And we also see that the farther the object is away from the center the higher the amplitude

of your sine wave. So you can see that for example this object is pretty far away so

it has a higher amplitude in the sine wave this guy here right corresponds to the object

here and the object that we have here in the center it has no amplitude at all it's stationary

throughout the rotations because it's always in the center of the detector so the amplitude

of this object here is zero. And then we have some other object that is slightly behind

it and you see that it's closer to the center than the other object so it has a slightly

lower sine wave. So looking at the sinogram I can somehow figure out what's going on and

in fact this is enough to compute the reconstruction if I know the geometry. In this case it's

parallel beam so I can just do the reconstruction from this sinogram.

Okay everybody understood what a sinogram is? And yeah okay so you also see that these

blobs they are uniform here but in the sinogram they are not uniform there is a difference