So good morning everybody. New week, new topics. Before we start to look into new algorithms

for computed tomography, let's briefly summarize what we have considered so far. Good morning

sir. It's a beautiful Monday. We are all highly motivated. We like to learn. We have four

weeks, well we just have four weeks for Christmas and we have learned a lot so far. We talked

about acquisition, specific image enhancement. So we looked into the acquisition devices

that are used in radiology departments. We looked at the principles that are used for

building the sensors, for acquiring the images and we found out that certain artifacts are

a consequence of the mechanical, electronical setup of the device. And we have seen several

acquisition specific preprocessing algorithms. In terms of math, we have seen in this context

the concept of least square estimators. The least square estimator to minimize a parametric

function that was set up by using a least square estimator. In the second part now,

and that's where we are in currently, we look into the problem given multiple images,

how can we generate higher dimensional information out of the projections. And that's reconstruction.

We talk about reconstruction and in particular based on X-ray images. And what we have seen

in this context, first before we looked into the reconstruction methods themselves, we

learned about projection matrices and homogeneous coordinates. Just to remind

you that we have considered things like P times X is U, where P here is a three by four

matrix carrying the intrinsic, extrinsic camera parameters. And it's all you need to characterize

the straight lines, X-ray quanta are propagated through space before hitting the detector.

Also in this context, we have used the least square estimator to do calibration. Calibration

is the process to estimate P out of a set of known point correspondencies. And we also

have learned a crucial tool here in this context that allows us basically to solve any linear

algebra problem that we are usually facing in medical imaging. So that makes us also

happy. No special lecture on linear algebra, SVD does it all. If we know about the properties

of SVD without proving the properties, we can use it for rank computation, for rank

deficiency enforcement, for computing inverses. And remember, two weeks ago I was showing

to you a singular matrix and I said I'm going to invert it. Then one student said we cannot

invert it because it's a singular matrix and I said I can do it because I use the SVD and

if there is no inverse, it takes the matrix that is closest to the potential inverse,

for instance. So we have a very powerful tool here, tool box, SVD, least square estimator,

we know about projection matrices. And now we look into X-ray reconstruction and the

story is as follows. We have Bayer's law where we integrate the function that we want to

compute over a straight line and we have seen last week the concept of filtered back projection.

It's called FBP based on the so-called Fourier slice theorem. And what is the Fourier slice

theorem? So Bayer's law and then we have seen the Fourier slice theorem. You can write it

in terms of formulas, but for us it's more important to have the picture in mind. So

if you have to reconstruct a function like this and you project this object here by parallel

beams on this detector line here, we can say that the Fourier transform of this projection

is equal to the Fourier transform through the object that is parallel to the detector

line and centered here, for instance, in the object center. Or centered here where this

is the Fourier transform space, eta and xi. And this is the zero zero point where we have

the zero frequencies. So if you want to build up the Fourier transform of the object we

want to reconstruct, you just rotate around the object. You acquire detector line by detector

line the X-ray projections. You compute the Fourier transform. You put in the Fourier

transform into the frequency domain representation of the object we want to reconstruct. And

we basically get for free the Fourier transform of the object we want to reconstruct in polar

coordinates. So that's the Fourier transform. And that's also why CT systems look like they

look like with this donut structure where the detector and the X-ray tube rotate around

the patient. They rotate around the patient. They collect nothing else but the projection

images and the Fourier transforms and we sample the 2D Fourier transform of the object we