So welcome to the Monday session. We are still in the chapter on magnetic resonance imaging

and we will conclude the chapter today or the section today by considering probabilistic

correction methods for bias fields. We all know what bias fields are. These are the low

frequencies inaccuracies we have or distortions, intensity distortions we have in the MR images

due to heterogeneities of the magnetic fields. And these artifacts can be reduced by various

methods. For us it's important to remember frequency domain filters, high pass filters

basically, homomorphic filtering, homomorphic unsharp masking and polynomial surface fitting

and then we also considered entropic methods and KL divergence based methods to eliminate

the bias field. The KL divergence is something that we will see for many, many other applications

within this lecture and the upcoming lectures on medical image processing. The clustering

approach was introduced around about five to seven years ago and is still heavily used

in practice. The clustering method for bias field estimation is basically combining two

things and that's important to remember. It's combining segmentation and the intensity correction

simultaneously. That means that we know something, we have prior knowledge about the tissue classes

that are present in the image and based on this knowledge we can set up a clustering

method that allows us for correcting the bias field. We have seen one basic clustering method,

decay means clustering, where we do clustering pixel-wise without considering the local neighborhood

and then we have discussed the fuzzy C means clustering where we basically regularized

the fuzzy C means objective function by an additional term requesting similar classes

in a local neighborhood. And the formulas we have seen last Tuesday, they were incredibly

large, they looked so complicated but basically it's nothing else but a gradient computation

of the objective function and we compute the zero crossings and get these closed form solutions.

For you it's important to know that we have no global method to solve for the minimization

or maximization of this objective function. We have used coordinate descent where we said

okay let's look at one dimension and then the other dimension and then the other dimension

and we iterate over one dimensional optimization problems for all the dimensions. That's also

an important concept. Please do not stick to this particular application whenever you

have to deal with the clustering method. Sit down and think about local neighborhood relationships

and whether you can add additional constraints to the objective function and then think about

the optimization problem. I mean due to the fact that most of the students in the audience

are computer science students, please have also a look at the complexity of these methods,

for instance the k-means clustering method or the optimization of the objective function

that has to be considered in k-means clustering. The optimization problem is rather hard, it's

an NP-hard problem. So the theory tells us it's NP-hard. In practice we have pretty fast

convergence. That's the two things that fight each other. Another application where we observe

a similar situation is the solution of linear programming problems. Maybe you know the simplex

method for solving linear programming methods. It's also known to be an NP problem and in

practice it can be shown that this can be solved rather efficiently. It converges pretty

well and works out pretty well in many practical examples. At least that was the saying 10

years ago. Nowadays we have way better algorithms than the simplex method for solving linear

programming problems. Just for you to set up the context properly. And now we will talk

about a probabilistic correction method and be sure you will love it and it's very exciting

and most of the most successful methods in pattern recognition are based on a probabilistic

setup and that's why we do a lot of probability theory in our classes and lectures. And it

also emphasizes that it is important what you have learned in the basic lectures. So

what do we intend to do? Like in bias field correction using the c-means clustering, we

now again look into a hybrid approach. And in this hybrid approach we do simultaneously

bias field correction and image segmentation. So we combine these two things. And image

segmentation in this particular situation is nothing else but a proper labeling of the

pixel values. So there are different methods for image segmentation. For instance you can