The following content has been provided by the University of Erlangen-Nürnberg.

So, welcome to the Tuesday session. We are currently in the chapter on MR image pre-processing.

In particular, we are going to discuss the problem how to remove inhomogeneities in MR images,

which is basically the same problem in standard CCD color image processing.

If you have inhomogeneous illumination in a picture, can we, for instance, make sure that the darker areas are brightened up

and the bright areas are brought on the same level like all the others?

So, a very, very important problem, and it's a problem that always appears if you deal with MR images.

So, it's a fundamental issue.

Before we start to look into the details, I want to give you the big picture again.

There is the cloud of knowledge, unsharp knowledge on diagnostic medical image processing.

And as I said, this lecture is built up on four basic columns.

One is basically the introductory part on modalities.

That means we talk about imaging devices.

And then we talk about acquisition specific pre-processing.

That's the full name.

Where we basically study the acquisition devices and the artifacts that come in with the acquisition devices

and try to find algorithmic methods to remove these artifacts or to reduce these artifacts.

The third chapter will be on reconstruction, 3D reconstruction.

So, we will look into the problem if I have multiple images from a patient, how can I do a three-dimensional reconstruction of that?

If you look into the web and you look for 3D reconstruction, today there are many nice applications in general in computer vision.

For instance, some researchers deal with the problem to get all the pictures of, let's say, the computer science building in Flickr.

And then they try to reconstruct the three-dimensional shape of the computer science building out of these pictures

without knowing which cameras were used or what was the focal distance of the camera and so on.

So, having multiple images, you can gain additional information on the 3D structure of an object and how can you compute that

is an issue that we will consider in the third part.

And the last part is on combining images from different sources and it's on image fusion.

And we are basically still in this chapter.

And to discuss all the algorithms here, we were required to extend our island of knowledge here in terms,

you shouldn't remember these things, I do these things very spontaneously.

The island of knowledge is growing by some mathematical tools that you should be aware of.

And usually I have several refresher lectures where I basically provide information on numerical tools

that you should be aware of and that should be known for you.

And we had one fundamental method and it's the so-called singular value decomposition that's a normal form for matrices.

Quite often we can reduce our problem, our image processing problem to a matrix problem

or a problem that can be solved in linear algebra terms and dealing with matrices is heavily supported

by knowing the SVD and the properties of the SVD.

That's why we had one whole lecture on the singular value decomposition and its properties,

which goes basically hand in hand with many optimization problems like least-quare estimators or something like that,

if we have to estimate parameters.

And parameters had to be estimated for getting the undistortion map that is required for X-ray images

if you use image intensifier technology for acquiring the digital X-ray image.

And we have seen that Fourier methods are required to do the defect pixel interpolation,

which basically work in the frequency domain.

And of course it's important to know how this undistortion mapping is built up,

how we can efficiently use these models during runtime of the system, how to estimate the parameters.

We had a lot of sidetracks where we explained what does it mean to have a fair parametrization,

what does it mean if I have data with an arbitrary scaling,

do I have to do a data balancing before I go into the numerical methods, we considered this in very much detail.

And I also told you if you have some regular structures in the problem you are considering,

usually you can use these regularities to gain performance in implementing things.