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Okay, so welcome. Maybe we will make a few minutes longer.

Okay, I have to apologize for this. This was something I haven't had under control.

So we are going to talk about defect pixel interpolation today.

And you might assume this is a problem that nobody faces in the daily business.

This is actually wrong. If you order a detector today, you have high chances that this detector will have defect pixels.

And to a certain degree, these defects, pixels providing actually no image signal,

they can be interpolated using the surrounding information.

Or think more illustrative. You take sandpaper and somehow smear it out and hope that nobody notices that there is something going wrong.

And what we are going to do now and in the following is we will discuss algorithms that allow us to compute and estimate the intensity values of defect pixels.

So the situation is as follows and this is well described here on this figure.

We have the image that is not corrupted. That's our ideal image that we actually cannot measure.

We know that there is a defect mask telling us which pixels are corrupted or dead pixels and which pixels are alive.

Basically, this is nothing else but a zero one image where we have ones where the pixel values are valid and zeros black values, black intensities where the pixels are dead.

And then we multiply the two images and what we get is what you exactly expected.

We get a new X-ray image where those areas are cancelled out where our waiting mask or our mask image shows zeros and defects.

That's a very simple idea. And what do we want to do? We want to compute this image while knowing this one and this one.

And the simplest idea is to say, I compute now the ideal image Fij is equal to Gij divided by Wij.

This is a very good idea and it works except for those points where Wij is zero.

But exactly for those points we want to have the information. So we are very sad about this.

Okay, this is something we can't do. And our problem is now how can we divide by zero to say it in simple terms?

And in the following lectures I will show you how you can divide by zero. What was this?

I'll show you, really. This is medical imaging image processing at its best, pattern recognition at its best.

You have a very sound mathematical model. And then you try to solve a practical problem and in pattern recognition and image processing the golden rule is it will not work.

So whenever a student shows up in my lab, you have explained to me this algorithm in the lecture and I tried it and it doesn't work.

My comment usually is, welcome to the club. Image processing is very difficult and most of the things we intend to do usually do not work.

But you can tell your dad this afternoon it's worth paying 500 euros tuition because I learned something new and it's dividing by zero.

Without breaking the computer. Who cares? Just for the naming and the notation and the following we have three signals.

And instead of the 2D signals, image signals, we consider 1D signals. Why am I doing this? There are several reasons.

The first reason is that the papers where I copied out these ideas have used this notation and why should I transfer all the results into the higher dimensional spaces with or by incorporating additional errors.

So I just copied it as it is. And that's what I say here. And it's straightforward to extend it to two dimensional areas or volumes.

And it's completely sufficient to look at the one dimensional signals while explaining the structure of these methods.

And we have three signals. We have the ideal signal, that's F. Images are usually called F in our community. That's the ideal image.

Then we have a binary mask. The binary mask is usually denoted by W. W stands for weight.

Waiting mask. And then we have the observed signal and because the O notation is already used in computer science, we have used this little hook down there and call it G.

That's a nice explanation, but it's not true. We call it G. For what reason ever.

That was just a spontaneous explanation and you can explain it to me the same way in the oral exam and I will enjoy your exam. So write it down.

That's exactly the point in the oral exam where my PhD students are ashamed of their boss because he's telling stupid things in the lecture.

So we need to talk about Fourier transform now. What is the Fourier transform? Do you have a good intuition?

Thomas?

What is the Fourier transform of a signal or a function?

Please compute the Fourier transform F zeta of F of let's say X. How do you compute this?

First of all, we usually have a pre-factor independent on the professor and the book. The professor whose lectures you attend and on the book you are looking into these pre-factors differ from each other.

Just have to make sure that the Fourier transform and the inverse Fourier transform lead to the identity, but the rest is not so exciting.

So I call it usually C because I don't care. Later on I will just skip the C.

So if your father is an electrical engineer or a mathematician and he will see my notes, he will look at it and say, this is wrong, there is a constant missing.

And I say, I'm an engineer, I don't care. It's a constant. Doesn't matter. This is an advanced lecture so you are in charge of constants.

I'm not in charge of the constants. What I'm telling, come on the formula.

Should I tell another story to make you thinking?

There is the huge symbol of integration.