Hi, it's time for a new chapter and we are going to spend the next few lectures to talk

about X-ray Tomography again.

We have talked about this in the beginning of this course, but I would like to spend

some more time so we can understand more deeply how these operators involved look like and

how the inverse problem is structured.

Let me start again by the setting of X-ray tomography.

Usually we think of the object being examined being the patient lying in an X-ray tube,

in an X-ray machine.

This is a section through the body of this patient, so there are some organs like this

where the denser than the rest of the patient.

Maybe this is a very dense object which I'm going to show like this.

Things like that.

I don't know how patients actually look like inside, how people look like inside.

This is the patient and we use X-ray penetrating the body in parallel lines and those X-rays

lose their intensity due to the fact that the part of this X-ray is being intercepted

by this different kinds of dense material.

Here we have very little attenuation that it's quite strongly attenuated here, goes

up again and it's very, very small here because there's this dense object here.

This is the measurement.

Measurement.

No one can read that.

We rotate this.

We don't rotate the patient.

Usually we rotate the X-ray emitter and the detector on the other side.

We already talked about the model that underlies this thing here.

We let F be a function of this domain.

This is the domain omega to R be a density.

Density not in the sense of probability density.

It's just a function.

We call this the metric density.

When the Radon transform of F in the variable S and omega was defined like this, integral

from minus infinity to infinity of F in S omega plus T orthogonal to omega dt.

We work with this definition but I would like to rewrite it slightly.

The mathematics won't change but the formulas will be a bit more verbose later.

We redefine the Radon transform.

Definition 5.1.

We set the vector theta to be cosine of 2 pi theta.

We don't have a small vector on top here.

Sine of 2 pi theta.

This is actually a function of theta.

It takes an angle theta in 0,1 and returns a vector in R2 with unit length.

EG, I'm going to write it like this, one half vector is then cosine of 2 pi one half, sine

of 2 pi one half, which is cosine of pi which is minus 1 and sine of pi which is 0.

We take an angle but this angle is now normalized to be between 0 and 1 where 0 means 0 degrees

and 1 means 2 pi so 360 degrees.

We can also visualize that.

For an angle 2 pi theta this object here is the vector theta vector.

It's a nice shorthand notation in order to define this unique unit length vector pointing

in the direction given by this angle.

Now we define, so that's not the definition, that's just notation, now we define the Radon