We will continue the discussion of bias field correction and I will consider this as the last subsection on this topic

and we will skip the probabilistic correction methods of bias fields.

So we will leave the probabilistic approach to some more advanced lectures or whatever.

So today we want to talk about clustering and how to use clustering methods for bias correction.

And if we talk about clustering, the idea, or it's important to understand what is meant by clustering.

Let me just go here, okay, white and white, clustering.

The idea of clustering is the following, you have a feature space that's for instance a two dimensional space and you have features.

Let's say the blue ones here.

Maybe they are black in the projection.

And let's use the yellow ones.

Do they look somehow yellowish?

And greenish?

And now some reddish.

And now we see, if we look at these point sets and point clouds, you might say even without the colors, you might say there are three clusters, right?

There are three groups of points or three point sets that seem to belong to each other and that are well separated from each other.

And now think about the following problem.

I give to you the 2D coordinates of these points without telling you to which cluster things belong.

And you have to find out how many clusters there are and where the clusters are.

So what we want to compute is nothing else but, you know, the set of these points that belong to cluster one,

the set of these points that belong to cluster two, and the set of these points that belong to cluster three.

We call this process clustering while assigning the points to different clusters.

So this process is called clustering.

And now the question is, how can we do this algorithmically?

And why might this be important for MR bias correction?

Well, if we think about the images that we are considering, there are different tissue classes, for instance, soft tissue, bone, water and other tissue classes.

And basically, we want to assign the intensities to tissue classes.

And this is also some kind of clustering process that we say certain intensities are mapped to certain class numbers that are associated with a tissue class.

That's the idea.

And there are whole books on clustering.

So we could have a whole lecture on different clustering methods, but that's not the purpose of this lecture here.

So what I want to do is I want to give you a brief idea how clustering might work in principle.

And one method that is well established in pattern recognition and many applications in multimedia or image processing where clustering is required is the so-called K-means algorithm.

Is there anybody in the audience who has heard about this algorithm?

Yes.

Where did you hear about this?

Pattern recognition.

Pattern recognition, Alice's lecture.

OK, that's where it belongs to.

So how does it work?

How does it work? Any ideas?

OK, we have to say what K is.

So look at the data.

K is three, maybe.

OK, so let's fix K to three.

That's, you know, that kind of magic that is involved in clustering.

By the way, that's also one of the hardest parts in clustering algorithms to find the number of clusters reliably.

I mean, just to say that you minimize a certain error is not sufficient because then you might say the best thing is to use for each point a single cluster.

So you have to penalize also, you know, an increasing number of clusters.

So that's a non-trivial problem and it's a discrete search problem.

Why is it a discrete search problem?