Can we start?

Good morning everybody.

We are currently in the chapter on image reconstruction.

In particular, we are considering computed tomography principles abbreviated by CT.

This is a very important technique that allows us to do 3D reconstructions from X-ray images.

In terms of the storyline of this lecture, so far we considered pre-processing algorithms.

What can we do with single images to enhance their appearance?

Then we take multiple images and try to do a reconstruction from multiple images.

That's where we are currently.

The core of the computed tomography reconstruction algorithms is Bayer's law.

Bayer's law is not associated with alcoholic beverages at all.

That's the name of the guy who did that.

Bayer's law tells us that the observed intensity at a certain pixel is the original intensity that we observe at the X-ray tube.

We have an exponential decay fxy and we integrate over a line dl.

This can be reformulated as logarithm I divided by I0 minus integral fxy dl.

That means for each pixel in the image we get a line integral through the function.

What type of picture do we have in mind?

If this is our xy space, if this is our object where each point here is an intensity value at position x and y.

If we consider the measurement, for instance our detector is at this place, then we measure here I.

We have here our x-ray tube I0 and then the energy, the original energy is reduced to energy I by this integral here.

That's the X-ray attenuation law for a single pixel.

Now let's continue.

Let's call this integral value here P.

P is the line integral over the two-dimensional function along a line.

Again we have P is integral fxy dl along a line.

On Thursday we have introduced Dirac's delta function that has the value 1 if the argument is 0 and 0 if any other value is used as an argument.

We can rewrite this as a double integral fxy delta.

Now we say that we consider only those points that sit on the line going through the origin.

Now we look at these projection lines here where this is the angle theta, this here is the distance 1, this here is length 1, this here is the sine theta, this is the cosine theta.

A point sits on the line that is defined by let's say this is, let me draw it again in a better way.

This is the projection line and this is the normal vector.

The normal vector we consider at the rotation angle theta.

You have the rotation angle theta and here you have your sine theta and your cosine theta here.

It's clear that a point sits on the straight line if x times cosine theta sine theta is 0.

It sits on this original line.

We can write here cosine theta plus y sine theta.

If the point xy sits on the straight line given by this normal vector going through the origin, then here we have a 0.

If this is 0, then we have the delta function at point 0.

The delta function at point 0 is 1, so we consider this function value.

For all the other points where this is not sitting on the straight line, this vector is 0.

That tells us if we integrate fx times the Dirac delta over all x and y values, we end up with a line integral.

That's the trick in this here.

We end up with a line integral.

If we look at P of S or at the projection point at detector element S, this extends to fxy delta x cosine theta plus y sine theta minus S.

Here we are.

This is dx dy.

That's the projection line.

That's the projection.

We always have to be clear about this.

This is the rotation angle of the detector element.