To help us better understand the field of back projection and how it works, we need to take a small detour and introduce something which is called the Fourier Slice Theorem. >> So the main theorem that makes tomography possible is called the Fourier Slice Theorem, I'll describe here what it is. So what we do is we've measured a projection. So I've got the projection here to the top left. So this projection was measured at a particular angle Phi. Now, we take the 1D Fourier transform of that and we keep it at this angle phi, and put it in 2D Fourier space. So we've taken a 1D Fourier transform of the projection, then we stick it in 2D Fourier space, but keep it at that same angle. Now, we do that for all of the angles phi we've measured at, so all of the 180 degrees. And as we do that, we're basically filling up the 2D Fourier space, as you've seen at the bottom here. So the nice thing about this is that what we're obtaining by doing that is actually the 2D Fourier transform [COUGH] of our original image. So it means that the 1D, one dimensional Fourier transform of our projection is actually part of the 2D Fourier transform restricted to this line. So in order to build the complete 2D Fourier transform, we need all of the 180 degrees and then apply this one dimensional Fourier transform, stick it in the 2D Fourier space and then we've got the full 2D Fourier transform. So this is quite nice because it gives us directly a simple reconstruction method - that's called the Fourier reconstruction method. So again, what we do is, we've measured our projections, so that's essentially acquiring the Radon transform data, so our projection might look like this. So what we do we, we 1D Fourier transform that, so we get something that looks like this. Then we piece that together in the 2D space as I talked about before. And then we can apply the inverse 2D Fourier transform, and that actually gives us back the original image. >> The Fourier slice theorem takes its name from the Fourier transform. And although we don't have to go into details with the mathematical background of the Fourier transform, it can be helpful to get a small primer on how it actually works. >> So the Fourier transform and I'm just going to be keeping it quite intuitive here, is really a frequency representation, you may already be familiar with it. So you have a signal and you can transform that into the Fourier domain which is essentially decomposing it into all the different frequency that are present in this signal. You can also transform it back. So you have frequency version, you can transform it back to your normal signal. This is also possible for functions of more than one variable [COUGH] so if you have a function of two variables, you can apply the two dimensional Fourier transform. And you'd get a two-dimensional frequency space representation of your function and then you can Fourier transform that back to get your function back. So what you have heard now, is that the Fourier Slice Theorem is sort of the foundation for the reconstruction method. You're also hearing that it is based on Fourier transforms, and this has a practical application which is called Fourier Reconstruction. We're not needing of you to understand the mathematical background of the Fourier transform. But we're just mentioning it so that you know that this is the background, the theoretical background for the methods that we're using. The practical application the Fourier reconstruction It has some limitations unfortunately that Jakob will now tell you about. >> In practice, the Fourier reconstruction method is rarely used. And that's due to a couple problems with the method in practice. So, I just described sort of the continuous version of this where you have an infinite number of projections and that's nice because then you fill up the Fourier space. In practice we only measure a finite number and that means we're only getting values on a polar grid. So these are the red points here that those will be all of the points in the 2D Fourier space we're getting. So in order to do the 2-dimensional inverse Fourier transform, it's easiest if we can get all of the points on a rectangular grid, on a Cartesian grid, like this, so that requires us to interpolate from the red point onto the black points. And that's a process that introduces errors, and it's been found to be introducing errors to an extent where it's not, doing the best job possible. >> So in practice, due to these challenges we are not using the Fourier reconstruction method normally. We're using a different method that mitigate these challenges somehow. And this is the method called filtered back protection which we will return to in a moment.