So we are approaching now the final few lectures on medical engineering one. As

you know summer semester will be on biomaterials. This semester is more on the

imaging component and device component. What did we discuss so far? During the

lecture usually you have the feeling it's not so much but if you summarize

things it turns out that we learned quite a bit and quite a lot. So what did we do

so far? We discussed about 1D signals.

We discussed about 1D signals, EMGs, ECGs and so on and in the lecture on

anatomy you should also have heard something about these measurements. We

talked a little bit about system theory and signal theory. Remember we talked

about the Fourier series where many of you complained about the fact that I did

that right away in the second lecture and many were asking what is it good for.

This is a tool that you will use forever now and at that time you haven't heard

about or most of you haven't heard about complex numbers but the way from the

Fourier series to the Fourier transform that we have seen also last week in the

context of MR imaging is not so far because we use basically the relationship

e to the power of i phi is cosine of phi plus i sine of phi and that means you

can rewrite using the relationship i to the power of minus phi is equal to

cosine phi minus i sine phi. You can rewrite the cosine function this is a

equation where cosine phi and sine phi can be considered as unknowns so you can

express now cosine phi and sine phi in terms of the e to the power of i phi

function and that means that the Fourier series can be rewritten or we call it k

larger or equal to 0 a 0 or a k cosine kx plus bk sine kx. That's the Fourier

series as we have defined it yeah and now you can take these two equations

write cosine in terms of e to the power of i phi e to the power of minus i phi

and then you can replace the cosine function and the sine function here

with this Eulerian expressions and then you get a complex representation of the

Fourier series and this is right away the Fourier transform if you consider k to

be continuous. So this is not something I want to have reproduced from your

side but this is just something that you should keep in mind if now the journey

through medical engineering is going on and if you learn more and more about

Fourier series. So if you are highly motivated sit down right away at 945

write cosine and sine in terms of e to the power of blah blah replace it and

look at the expression you get and then suddenly you have a complex or you have

a function with complex terms. Looks very complicated in fact it's not

complicated it's right away produced out of the Fourier transform. I would have

done that in much more detail if you would have been equipped with complex

numbers at the beginning of the lecture for us it's not very crucial now so keep

that in mind this is important and the transition to complex number and to the

complex domain is not a miracle and you should not collapse when you see things

like that it's very controllable right. So we talked about Fourier series and

this is good for many things. For instance modern computer tomography

reconstruction methods they do not work with a catch-march approach with the

iterative scheme that we have discussed they work using the Fourier theorem and

properties of the Fourier transform. They work with the Fourier transform and use

the Fourier transform for reconstruction. So you will learn about this later. So the message

again periodic signals can be written as linear combinations of sine and cosine

functions and this is highly correlated with the Fourier transform. Okay good

what else did we discuss we talked about OCT optical coherence tomography

interference and these things and have learned about a imaging method that is

able for instance to show the nerve layers of the retina and in anatomy last