So good morning everybody, happy new year. I hope you had some time to enjoy life and

forget about the university and all the hard work you have to do. I wish you all a successful

2015 and an exciting 2015 and maybe one or the other will be even very successful in

terms of examinations and thesis projects. So good luck.

Today we want to continue in the text on Diagnostical Medical Image Processing and I have to reduce

the number of chapters a little bit because I'm not as fast as Ani Maia who did the lecture

in the past two years. So I will skip Katziewicz algorithm and I also will skip the MR reconstruction

and I will now continue with the chapter on algebraic reconstruction and then I will close

the reconstruction chapter because we need the upcoming lectures to talk about image

fusion that's as important as reconstruction and you have to have some basic knowledge

and you have to be equipped with the basic techniques with respect to image registration

and I want to make sure that you get this. So we talked so far about modalities. Let

me just check whether I can switch on the light here. Yeah. We talked about pre-processing

and currently we talk about reconstruction and in particular we are considering reconstruction

from X-ray projections and if we talk about X-ray projections it's important to remember

Bayer's law that tells us the observed energy, the observed intensity in the image at a certain

pixel is the intensity we get if nothing is in between the X-ray source and the detector

times and now we have an exponential decay minus f X Y integrated over a line L. That

means we integrate the function we want to reconstruct. In this case it's a 2D function

where we observe 1D projections. We want to reconstruct the function F given the projections

and equivalently we said okay this is equal to minus integral f X Y D L L and for each

pixel we get this type of equation and we have to solve the system of integral equations

if we want to compute f of X Y given an X-ray image. That's something we have considered

very, very, very detailed and we started to look into reconstruction methods that make

use of this analytical characterization and we found out that there is the Fourier slice

theorem. What does the Fourier slice theorem tell us? Well once again that's the figure

you have to keep in mind. We have here our projection rays, we have here our detector,

the 2D function has to be reconstructed, we have a 1D detector and the Fourier slice theorem

tells us that the Fourier transform of the 1D signal can be found in the Fourier transform

of the 2D signal that we are looking for if we look at the elements in the 2D Fourier

transform that sit on the parallel line to the detector line. That means if we rotate

around an object and capture different X-ray projections we basically generate here the

Fourier transform of the 2D function that we need to reconstruct and then we apply the

inverse Fourier transform. Then we have seen this is given in polar coordinates so we have

to do a coordinate transform and then at the end of the day we came up with a very powerful

algorithm, the filtered back projection algorithm that decomposes the reconstruction problem

given this set of integral equations basically into a two-step algorithm where we do a convolution

of the observed signals with a high-pass filter, a REM filter that is basically nothing else

but the absolute value of the Jacobian that we get out of the coordinate transform from

polar coordinates to Cartesian coordinates and after the filtering we have to do a back

projection which means smear the projection through the slice that we want to reconstruct.

And then we looked at different geometries, we said this is not the way X-ray systems

work nowadays, usually the geometry looks like this, we have the fan beam geometry or

if we have 2D detectors we have the cone beam geometry and we did a lot of geometric analysis

and it turns out at the end of the day that all the reconstruction methods that we can

use in practice even for these modified geometrical setups are ending up with a filtered back

projection type of algorithm.

For us it's important to remember filtered back projection algorithm is the working horse

of current computer tomography systems.

So if you go downtown in our digital radiography or radiology department of our university