Okay, welcome. Monday session, 45 minutes, and we are currently in the chapter on image

reconstruction. Tomorrow I will give again an overview over the storyline. Today we will

continue in looking into the algorithms. For reconstruction, or if we talk about reconstruction

in this lecture, we basically mean how can we compute higher dimensional volumes from

X-ray projections. We have seen one fundamental mathematical result. It's the three slice

theorem and then we have incorporated the coordinate transform from polar coordinates

to Cartesian and we ended up with a very nice algorithm that combines filtering with a normal

numerical integration, that's filtered back projection. The nice thing is that we have

to do filtering in terms of a kind of pre-processing step and this can be easily combined with

other filtering operations like smoothing or something like that so we can easily change

the reconstruction method and adjust it to certain noise sources, for instance, to enhance

the reconstruction result. And as I told you, there are many different filters out in the

world and if you end up to operate a CT system you will find out that there are different

filters that you can select optionally and a few of them have technical names, others

have more intuitive names like bone reconstruction or soft tissue reconstruction or things like

that. Good. As I also told you in my lab we have current research projects where we try

to accelerate the reconstruction algorithms in terms of hardware support and we also have

a huge number, huge number, we have a few research projects where we try to do a reconstruction

of a dynamic object that is moving where the motion is unknown. For instance cardiac reconstruction

is a problem that we consider in detail in our research activities and the Fourier slice

theorem is basically one of the basic algorithms where all our research is focused. And today

I will look into a different type of reconstruction algorithms. These algorithms are called algebraic

reconstruction techniques and the abbreviation is ART. So whenever you hear something we

use art for reconstruction that basically means that algebraic problem is solved and

the reconstruction problem is approached from an algebraic point of view. These techniques

and that for me was kind of surprising, they were originally used. So if you look at MacBormat's

first CT scanner they have implemented the algebraic reconstruction technique, not the

Fourier slice theorem or some derived algorithm. So the algebraic reconstruction technique

is rather old. The disadvantage is that the algebraic or the system of linear equations

that have to be solved for modern CT datasets, the size of these systems is incredibly large.

I mean think about you want to reconstruct a 512 by 512 by 512 volume so you have 2 to

the power of 9 times 2 to the power of 9 times 2 to the power of 9 where 2 to the power of

27 unknown. I mean you all remember the first year here at university. What was the typical

size of linear systems you had to solve in your first year here? Okay, four unknowns.

But that was very challenging. Usually you have two by two equations, two by two matrices

and if the math professor is really challenging you he asks for three by three and if he doesn't

like you he's asking for four by four. But we have here 2 to the power of 27 unknowns.

Not only four unknowns but 2 to the power of 27. So quite a huge number and you can

imagine that the right answer to solve this system of linear equations might be to use

SVD and I just would answer good luck. Usually you do not even have enough memory to keep

all the equations in your memory. So that's a complete new dimension of problem and if

you read the literature and I mean scientific literature is one thing, the marketing brochures

is another thing and if you read the marketing brochures of General Electric for instance

they claim to have an algebraic reconstruction method now implemented in their modern CT

systems and if you go to exhibitions there will be these tie-wearing guys who will tell

you we are now doing algebraic reconstruction. And now you might think why is there a good

argument? Because it's usually known in the literature that the algebraic reconstruction

techniques tend to produce better reconstruction results. You remember the streak artifacts

I have shown to you if you don't have sufficient projections. We also know that the filter

back projection is basically requiring a 180 degree rotation and you have to sample the