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Okay, so good morning. For the last lecture this semester, I'm very amused about that.

As I said, I received the course evaluation and I am not going to go through all the details

because this time it was not so good. We were not well above the average as usual.

And I basically have two outliers in the evaluation where we are way below the average of the technical faculty

and that's the degree of complexity. So maybe you consider the lecture to be too trivial.

So maybe this is the problem or it's too difficult. Either way, it was producing some disappointment from your side.

We also got some requests for courses, refresher courses on linear algebra for MAOT guys.

And two of my PhD students volunteered to give a refresher course next semester for students that are not that familiar with linear algebra and all that stuff.

And the second outlier was regarding the amount of work that is required to understand the lecture.

Maybe it was not so much that you had to do or it was way too much. Either way, it caused some dissatisfaction.

Overall, I hope that you are, in average, quite happy that you decided for this lecture.

I also hope that you learned something. And believe me, if you want to work in this field later on,

you are well prepared for many tasks that are currently done in industry.

So let us conclude today the lecture with a few considerations of the image registration task using neutral information.

And after that, I am going to summarize the big picture that you get, the roadmap again of this lecture.

And you can just reflect your experience with the overall storyline.

And let us see whether you are well prepared for the oral exam next week.

I am also going to explain you the answer for the first question.

And I am convinced that 50% of the students next week will not be able to answer this question properly.

That is my personal experience.

Yesterday we looked at the problem, how can I estimate the histograms?

Once again, we have to compute the Kalbeck-Lioula divergence, the neutral information that is basically requiring to compute the joint histogram

and the single histograms for the two intensities.

And if you decide to compute a histogram, you have to address a few problems that will appear in practice.

First of all, it is the number of bins that you are considering,

basically the number of intensities that you are considering in the image.

The second problem you have to consider is what is the bin size?

Are all the bins the same size or widths or do you adapt the widths of the bins automatically?

And the other problem is how do you deal with histograms that are very sparse in a sense that many entries have zeros?

And yesterday I told you that in pattern recognition and image processing,

people are applying the Parson window estimation, which is basically nothing else but a convolution of the histogram with a Gaussian filter.

Convolution means it is just a filtering operation where you smear the bins over the other bins,

such that all the bins are a little bit higher than zero.

That's the core idea that is basically applied.

And I told you here that you also can compute the number of bins automatically by an optimization problem.

I also told you that the variance of the Gaussian kernel for the Parson estimator can be estimated using cross-validation.

Leave one out, cross-validation was the algorithm that we have used.

And now look at this.

I told you an issue for many practical implementations is the runtime.

If we have a volume of 512 by 512 by 512 to compute the statistics, it takes a while.

And if you want to optimize the neutral information, you have to compute the histograms and all the stuff for each point of the function that you want to optimize.

And I mean, think what happens if you have to do hundreds of thousands of function evaluations.

You have for each function evaluation, you have to compute a histogram.

And this here is a plot of the histogram where we merge or where we register a PET and a CT image.

And this is the distribution of intensities, the joint distribution.

So you see here the intensities of the PET, you see here the intensities of the CT image, and here you basically see the relative frequencies.

And now you can estimate the function using all the voxels in the volume or you take just each tenth voxel, for instance.

Instead of taking all of them, you take each tenth voxel.

Here we have used 10% of the voxels and got this intensity here, two-dimensional intensity.