Hi, the last part of this week's lecture will consist of understanding filtered back projection

a bit better by seeing it in action. Here you can see again the formula for filtered

back projection. So given data acquired from X-ray tomography, I'm sorry my nose is a bit

congested today, I hope you can understand me a little less. So given the random transform

of some intensity field F, which is what X-ray tomography outputs, so this is the raw data

from a CT scan for example, how do you get back the image hidden in here? And the formula

we have derived as follows, what you have to do is apply this pipeline and we look at

this pipeline a bit more closely. And what you have to do is you take RF, the X-ray data,

you put this into Fourier space by applying the one dimensional Fourier transform, then

you apply a high pass filter, then you go back to the actual space via an inverse Fourier

transform and then you do back projection in the way we have understood so far. And

let's now see how this works. I've shown you this before, so this is the image we're

looking at, this is just an L-shaped object inside this X-ray machine, left marked this

edge for orientation purposes. And the forward round-trips form looks at this image via slices

for a given angle and it accumulates intensity along this line. So for example if we take

angle zero then you take vertical slices through this image and you can see that there's a

lot of intensity attenuation here and a little bit here and nothing at the boundary of course.

And rotating this gives you all those slices in the sinogram. So for example for angle

34 degrees which corresponds to this rotated image here, you get this shape here and this

is this black slice inside the sinogram. And I always mark the position of this edge here

in red. So again, different angle looks like this, different angle looks like this, you

can see that we're exploring the whole sinogram that way. So we're looking at this two-dimensional

data set, a sinogram, angle slice by angle slice. And we talked about how this could

in principle be inverted naively with back projection. So you take such a slice and you

project this mass here upwards and you distribute it along this direction and this is what you

can see here. For example for angle 56 you take this sinogram and you smear it out like

this along the image. And if you do this for all the angles and you add up all those smeared

back projected contributions you get this blurry image of L which is okay but not perfect

and you can't make out fine details anymore. Now filter back projection, what do we do?

First we have to go into Fourier space, apply a filter and then go back. So what you do

is you take the sinogram, this is the sinogram here and first you have to take one dimensional

Fourier transforms. This is what you can see here. So this is an image of only the real

part of the Fourier transform of the sinogram. So this is taking the sinogram and taking

the one dimensional Fourier transform in every slice. Now you apply this high pass filter,

this is what's happening here, and this turns this image into this image here. So as you

can see there is structure hidden in here which you can't see but if you amplify with

this high pass filter this just gets visible. And when you go back to the actual space by

one dimensional inverse Fourier transform you get this image here. And as you can see

I hope the quality is alright, you can now compare this, let's maybe put these side by

side, let me do this for a second. Okay so that's it, you can see the original sinogram

here, sorry here, and the filtered version we have just acquired is this one. So as you

can see this kind of only looks at the edges inside the sinogram. So what we've done is

we've taken this, applied a high pass filter and we get this sinogram here. This is not

the original sinogram, this is not the data that you get from a CT scan but it's now a

filtered CT scan, a filtered sinogram. If you now take, so if you take the original sinogram

and you do back projection then you get this reconstruction here which is bad because it's

too blurry. But if you take the filtered sinogram and you do back projection then you get this

here. So you take all the slices through the one dimensional, sorry the slice through the

filtered sinogram, these look like that, so this is a slice through the filtered sinogram,

you back project like this and you accumulate that for all angles that you can have then