After the introduction of how the projections related to the object and the sinogram, we now need to get a little bit more into the practical problems related to doing the actual 3D reconstruction. Manuel will give us an introduction to this part of the problem. So, one of the basic concepts for computed tomography is back projection. So, as we mentioned before, and you see it in the upper part, the physical problem is forward projection, the photons are going through the object, adding up the information and you measure these shadows. The almost opposite operation would be to do a back projection. It's not exactly an inverse operation because the inverse operation is to reconstruct the tomogram, but it is an adjoint operation, it's almost the opposite. So, all you know when you look at this shadow over here, when you look at this point of the shadow, all that you know is that it came from somewhere along this line because you know what was the direction of these photons propagating. So then, the back projection is the most intuitive or the most natural thing to do in order to try to make sense of the data. Now, if we take all the data at different orientations and we start back projecting angle by angle, you can see the result over here in the animation that I have. So, basically, as we go through more and more angles, you can start to see some of the inner details of the sample. You can really see already some of the gross details like this black blobs over there and this white blob on top of this head phantom. The object that you saw formed here you might have recognized. It's called the Shepp-Logan phantom, and is a simulation of assemble, an object, that is commonly used in both teaching and also for testing algorithms for 3D reconstruction. But, it is actually somewhat blurry as you see it here. Let's hear more. If you remember what the object should look like, that's on the left of the screen over there, you see that the result of our back projection is actually blurry and the very small details are just lost. So, if I may point out, for example, on this area over here of the reconstruction, there should be this detail over here and also this detail here is lost. So, the image is blurry, the contrast is low, and details are lost, and the question is why? If I did this back projection, what am I missing in order to get a good reconstruction? We will shortly return to an explanation of why it comes out blurry. But to help us understand this, we will first take a look at a somewhat more simple problem, a really simple geometric shape, and see how it relates to the projections. So, first, I'll illustrate the process of back projection to you. So, at least to me the opposite operation of projection would be to back project. That intuitively, might actually do a pretty good job of reconstructing. So, if we look at this example where we have measured three projections in these three directions, back projection amounts to simply taking each of these projections and smearing them back across the image domain. So, when you do that, you get the image you have over here to the right. So basically, you assign the same value that you have measured back across along the line where it's measured. So, in this case, you're getting values here in the middle and then zeros to the side and similarly for the other cases. You see, you're getting something right. You're getting something that looks like this square back. But you're also getting some signal put back in the other parts of the image. So, it's not quite right. So, as you can see, the process of back projecting doesn't perfectly give us back our object. We need to add an extra step to achieve this. It turns out what we need to do before we do this back projection, we need to do a filtration. So, we need to do filtering by a so-called ramp filter. So, if we do that, our projections are changed into looking like what you see here on the right. So basically, it emphasizes the edges. When you back project that, you get something that looks like this. Initially, it might not look much better, but it turns out that if you keep adding more projections, you will get behavior like this. So, in the unfiltered case, you will keep adding projections and as you have more and more projections, you end up having this blurred image in the end. So, it's not recovering the original image. Whereas, if you do the filtered version, where you filter your projections before you back project them, you get in the end that all of these filtered projections, they end up cancelling each other out. So, in the end, you get the image back. So, you need quite a large number of projections to get there. But in the end, all of these filtered projections will mathematically cancel each other out and and filtered back projection recovers the original image. That was the first mentioning of the term filtered back projection. This is a term that I will ask you to try to remember because it's one of the fundamental methods that are widely applied to 3D reconstruction. In principle, as both Jacob and Manuel have shown, this technique allows you to recover your object perfectly. In practice sometimes, you will see in your data that you can have various kinds of artifacts which is an indication that something is not complete in terms of data acquisition, for example. It might be a case of missing projections. So, this is something to watch out for. There are other solutions to the 3D reconstruction problem that may address various kinds of artifacts. But for now, we will keep in the track of how the filtered back projection works.