So good morning everybody. Let's continue with the most exciting topic at this wonderful

university, medical image processing. We are currently in the chapter that students actually

appreciate the most in the past. We always ask which topic did you like the most and the feedback

was always very positive with respect to the chapter on reconstruction. And that is the reason

why we extended the chapter on reconstruction a lot and put way more mathematics in it and let's

see how things develop. Before we talked about what we can do with respect to single images,

so we are familiar with basic preprocessing methods that allow us to do image correction.

So this is somehow deviating from standard image processing books where you read filters

like mean filtering, Gaussian filtering and all these standard technologies. Here we looked into

very specific new specific and rather new methods to do a proper preprocessing that decides at the

end of the day on the final image quality that you have in a radiology department. And we know

a little bit about modalities, but I have to emphasize that we are not considering these

modalities in very detail and we also didn't talk about physics due to the lack of knowledge of the

professor in this field. And the medical engineering students, they had a whole semester in their first

year on different modalities and maybe have a better understanding of it. But just for you,

if you heard it the first time, it's totally enough what you have learned to write algorithms

for preprocessing, reconstruction and image fusion that is going to come in a few lectures

from now. And if we talk about reconstruction, we focus here in this lecture on X-ray computed

tomography. You all know that also in MR, magnetic resonance imaging, reconstruction is required and

these methods that we actually discuss here can be used as a basis for the MR reconstruction

technique. So we focus on X-ray CT, X-ray computed tomography. Good morning. X-ray computed tomography

and the basis for reconstruction is that we have projections for a given angle at a given point

on the detector that are basically nothing else but a line integral through the function that we

are going to reconstruct. So we integrate over a line and we have different lines for different

angles and we integrate over these lines and get here for each element on the detector transformed

element on the detector. We have to apply this logarithm transform and integral of the function.

So the task of CT reconstruction is nothing else but compute a function in the most or in the

simplest way, compute a two-dimensional function using different line integrals. So we have

different line integrals and integrate along the line the 2D function we want to reconstruct and

based on these line integrals we compute the original function. The question is how many line

integrals do we need to come up with a good solution and there is a mathematical proof that

we need an infinite number of line integrals and then you have to prove that there is you need an

infinite number of line integrals and then you can say as a mathematician I don't go for the

system because I need an infinite number of line integrals and I will never get that so I can't

build the system. Engineers say we ignore the mathematical result and we just build it and say

it's good enough and it's actually good enough. That's also what engineers have to learn it's

good enough. I know that there is no solution but I have one and it's good enough. That's

engineering. So how many line integrals do we need from which directions do we need the line

integrals? So there are a lot of theoretical questions that have to be answered and it's

also important for engineers to understand the theoretical constraints of the whole system but

at the end of the day for coming up with a practical solution we have this in the back of

our mind and we always double check things but when we build it sometimes we have to ignore

theoretical constraints and just do it. Okay and the method we carved out was the so-called filtered

back projection. Back projection it's really a pleasure to write with this. It's the filtered

back projection and the filtered back projection is based on the Fourier slice theorem. What is

the Fourier slice theorem telling us? That's the picture you should keep in the outer shell of

your brain. That's the function we want to reconstruct. That's the X-ray projection using

parallel projection rays. We compute the Fourier transform and we find this Fourier transform in

the Fourier transform of the function we want to reconstruct. And this is parallel to the detector

line. So we look at the 2D Fourier transform along a line that is parallel to the detector line. Yes