Let us now hear how the filtered back projection is applied in practice and how the various elements that have been introduced play a part in this reconstruction method. So Jacob is helping us piecing these elements together and showing you in sort of a symbolic way how the math operates to create the reconstruction through the use of fully transforms and filters applied, for example, in the frequency domain. >> We basically manipulate the mathematics until we get a more convenient expression. So that's going to be our filtered back projection. So it's doing the same, in some sense, but we're just moving the symbols around. So what we do is we start out by basically writing this identity here that our function is basically the inverse 2D Fourier transform of the 2D Fourier transform of itself, so that's quite intuitive. And then, and I'm skipping many of the details because it's quite straight forward. It's basically application of polar coordinates and the Fourier slice theorem. If we do do that, then we end up with an expression like this line here where we've got an integral, and we've got some one dimensional Fourier transform, and the Radon transform is suddenly showing up. So that's coming in from the Fourier slice theorem. So I just want to highlight what the different components are here. And the R of f, that's actually the projections we've measured. So I'm sort of just substituting in the projections here. And then if you look at the next thing, we've got the one dimensional Fourier transform of the projections multiplied on to something here. And this is actually coming from the coordinate transform, so that's the Jacobian. And then we've got another one-dimensional Fourier transform, the inverse actually. So that's actually a filtering operation with this ramp filter that I mentioned earlier. So this is basically written as a filter. And then, the remaining integral turns out to be the mathematical way to write back projection, actually. So in the end, we have the operations to our projections. We apply filtering with the ramp filter and then we apply the back-projection. And that's how this method gets it's name, I suppose. We take projections. We filter them. We back-project them. And that's our reconstruction. >> So an important addition here is this ramp filter. Which has this function of suppressing the higher frequencies and making sure that we get rid of this problem of the blurring of the final reconstructed object. We'll go a bit more in detail with the function of this filter and modification to it. >> And the filtering we need to do is by applying a so-called ramp filter, which is basically an absolute value function. You apply that in the frequency domain, so that's what's shown here. >> So the addition of this filter that Jacob is talking about here is the very same filter, as you'll remember, was mentioned earlier when it was discussed, this problem of the blurred reconstruction of the object that we have to somehow deal with. And this is exactly what this, so called, ramp filter is doing. It is suppressing the higher frequencies in order to remove this blurring effect. Jacob has some more details on how this work in practice. >> In practice, we need to do additional filtering because then the data is typically noisy and the noise tends to be high-frequency. And the ramp filter is a high-pass filter, meaning that a high-frequency components, such as noise, will be emphasized. And in practice, that means that the reconstructions will be quite noisy. So what we do in practice is we take some low pass filters like some of the ones I've shown at the top here. We multiply them in the Fourier domain on to our ramp filter. So the the ramp filter, which is basically absolute value function, is then bent off a little bit so that we de-emphasize the high frequency components towards the high frequency components. Now, you can do that by the different standard filters, the Ram-Lak, and the cosine, and Shepp-Logan and Hann filters that apply different amount of smoothing and are found to be useful for different purposes. So yeah, we have the ramp filter, which is needed for the mathematics to work out. It's part of the inversion formula. And then on top of that, we apply another filter to make sure that the high frequency noise is damped and then you do the back projection. >> What you should understand by now is that the ramp filter is in fact a fundamental premise for the back projection method to work. The filter back projection method relies on the application of the ramp filter. The other effect of this is that, for example, high frequency noise will be amplified in the reconstruction. And that is why, for actual data, it is necessary to introduce, typically, some kind of modifications for the filter that can, again, dampen this high frequency noise in the reconstruction. And the form of this filter can be a little bit different depending on the type of sample and how the noise is distributed. Let's try to hear an explanation of this one more time, but this time in the words of Manuel. >> And that we can use the back projection operator to get a reconstruction but we have to apply a suitable filter, and this is the main method that is used for computer tomography, and the most popular one is called filtered back projections. So it's basically applying this filter and doing back projection. So the way it works is have your measured projection, your measured shadow, your computer fully transform. You apply this filter that emphasizes the high frequencies and de-emphasizes the low frequencies. Because you're going to repeat them over and over as you rotate so you have to de-emphasize them linearly. Then you compute the inverse for your transform, and then you can back project this function. So what you can see in the animation is that, okay, at the beginning if I just have one back projections, of course, it looks edge enhanced. But if I then start adding up all the different angles, and you can start to see how this start to come together and start to form a reasonable object. So you can see the edge is forming and you can see the different details. And in this case, we don't have the case of this blurriness. And when we finish going fully around the object, then you have a nice complete reconstruction. >> So as you can see, now we recover faithfully the layout of our phantom, this Shepp Logan phantom. And this is due to the filtering that make sure that we don't get this blurring effect that we mentioned before. This concludes then our presentation of the filtered back projection method, which is the most widespread used algorithm for reconstructing tomography data. There are alternative methods which are applied in other cases where, for example, data is missing or where one wants to apply some special knowledge about the sample, and these are called algebraic methods. We will touch briefly upon them in the latest video.