Good morning. Before we continue in the text, let's briefly look where we are. This semester,

we focus on diagnostic medical image processing. Diagnostic medical image processing means

that the system is used for diagnostic purposes. We have no tough time constraints and images

just have to be as nice as possible for a diagnostic procedure. And after having considered

different modalities as a motivation, we are currently in the sub-chapter on pre-processing.

Once again, pre-processing is all that we do in between the detector, the sensor, and

the monitor. The images on their way to the monitor are processed. And in particular,

we are looking at algorithms that take into consideration the physics, the principles

that underlie the detector technology that is used, and the implied artifacts by this

technology. For instance, we looked at two imaging modalities so far, X-ray and MRI,

magnetic resonance imaging. And in X-ray, we have seen that we can use image intensifiers

for converting the energy of X-ray particles into intensity images. And we have seen that

in this system, we have an electron optics that basically has electrons being accelerated

within a vacuum tube. And if we have moving charges within a magnetic field, we know that

there are forces on the electrons and these forces cause deviations of the electrons and

these deviations finally end up in terms of a distortion in the image that has to be corrected.

And we have introduced the concept of calibration. We calibrate the distortion mapping and use

this calibrated mapping for image correction. In terms of mathematics, that's also what

we have to keep in mind. Image processing is a lot of mathematics. We have seen SVD

that can be used to solve any kind of linear algebra problem, hand waving type of argument,

any type, mostly any type, I should say, singular value decomposition. We also have seen how

to set up least square estimators. In particular, the calibration problem was formalized as a

least square estimation problem. And then we have looked at flat panel detectors abbreviated

by FD. And we have seen these are very difficult to manufacture and we get some defects into

these detectors and business guys force us to sell these defect systems or these defect

force to a certain extent. And so algorithm experts are required to come up with sandpaper

algorithms that smear the defect pixels in a way that our perceptive system, our eye

is not seeing any corrupted image. So here we have discussed two different ways to do

the sandpapering, the interpolation of the defect pixels. One was working in spatial

domain, basically interpolation of intensities in spatial domain. That's actually also the

method that is applied in today's systems. No Fourier method is applied in today's systems

due to the fact that they are very computationally expensive. And we also have seen frequency

domain methods. And we have seen impressive examples where we are also able to some kind

of interpolate the noise that appeared in the image. So these are very nice algorithms,

very nice algorithms. And the idea we had or the picture we have in mind is in this

case, I just do it now, okay? You can't do that in your notes, but I can do it. Hi. The

idea that we have had in mind is there is the ideal image, there is the defect mask

for instance, here the defects and then we have a pixel-wise multiplication and we end

up with the observed image, observed image. We end up with the observed image including

the defects and the rest of the ideal image. That's the picture we have in mind and by

these methods we can estimate the values here where we have zeros in the observation. So

we can divide by zero basically. Good. And the next preprocessing subfield that we have

considered here is MRI. And that's what we have started to look into in the past lecture.

The idea is we have an ideal image. We have a bias image. That's a low frequency intensity

gradient. These are added or multiplied pixel-wise. We have seen the two models with the gain

and the bias field and then we end up with the observed image. So that's the bias field

and that's the observed image. And what we have to do later on is we don't know this,

we don't know this, compared to this situation where the mask image is known. We can measure

the defects of a detector by just capturing an image and look where are defects. In this

case of MRI we cannot calibrate for the bias field. So what we have to do basically is