Hi, we'll dive right in and the first topic we're going to look at is the topic of X-ray

Tomography.

We are going to start this course with some examples in order to motivate inverse problems

and X-ray Tomography is one of the most paradigmatic examples for inverse problems so we will spend

quite some time on it, especially because its applications are obviously quite interesting

and relevant.

The setting of X-ray Tomography is not too difficult so we consider let's say a patient

maybe and it could be any other object, I'm just saying as a patient but it could also

be some type of material that you want to study, it's in a decomposition or things like

that so there are applications for X-ray Tomography in industrial applications as well but for

simplicity and because it's closer to our heart we are thinking of this as being a patient.

So this is a patient in cross section, so there is a cross section of the human body, someone

is lying on a bench in X-ray Tomography machine and there are regions inside this body which

are more denser than others so this is not too dense, maybe mostly water but there are

some organs here, some bones and obviously this is not a good description of the human

body but it serves as an example.

And now what happens is that from some direction, let's say from the top there is X-ray radiation

coming in and traversing the body and we assume that the intensity of this radiation is known

and we measure the remaining intensity at the other side after the radiation has passed

through the body.

So what's the physical issue here, so the denser the matter in the path of this, well

let's think of these as being photons going through the body and the more matter there

is, so the denser this object is, the more this ray will become attenuated.

So if we measure the intensity of these rays after having traversed the body it will look

something like this, so this doesn't hit much of the body so there will be a lot of intensity

remaining, then there is some denser object, maybe an organ or something and this will

drop down a bit and it will go up slightly and then drop down again because there is

something very dense here, go up again until it encounters another object here, so something

like that.

Those rays who travel through a dense object will become attenuated very heavily and of

course if we now look at the same thing but from another angle things will look different,

so for example let's have obviously the same object again but the x-rays will travel into

the other direction, they will travel from the left and what happens now, well there

will be some slightly different intensity plot on the other side, strong attenuation

here where this hard object is, even more when they overlap and well this is probably

slightly less dense, something like that, so this is maybe what we get on the other

side, so this is again intensity of x-ray radiation on the other side of the body.

The next way of tomography works by having this machine rotate around the body and measuring

the remaining intensity of x-ray radiation on the other side, so that's the basic idea

here and I have a small simulation for that, I'll just switch to this image here, so this

is of course not a patient but it serves as an example anyway, so we put something in

the shape of an L inside this machine, black means it's just air and white means there

is matter there, so it's just either one or zero, so one is white and zero is black and

as you can see this leads to something called the sinogram, we'll derive this rather and

transform later but that's just notation, just think of this picture here as the result

of looking at this object from various angles, so let's think of that, so if we pick projection

angle zero that means that we're looking at this object from, well let's say above or

something like that, so the x-rays come from the top and they travel through this L and

they are then measured at the bottom here and as you can see for position from zero

to 40, so this is this range there's no attenuation at all, so there's a lot of radiation here