So, good morning everybody. Before we continue with the MR preprocessing algorithms we want

to consider, let's briefly summarize the topics that we have considered so far. The mind map

at the beginning. We started to look into the field of preprocessing in medical imaging.

And if you think about an imaging system how that is built today, a medical imaging system,

you have your sensor, you have your computer, you have your patient, and then you have your

monitor. And preprocessing is, in our context, is nothing else but the algorithms that are

performed on the sensor data in between the sensor and the monitor. So that's preprocessing.

In other words, before we see the image the first time on the monitor, algorithms are

applied and these are the preprocessing algorithms. And dependent on the principle we are using

for visualizing the inner of the human body, we get artifacts into the acquired images.

And the question is, how can we remove artifacts from the way of the signal from the sensor

to the monitor? And so far we have just considered one modality. That's X-ray. And we have seen

that there are three technologies out there to convert X-ray energy into images. And these

three techniques are films, analog technique that is no longer used in modern systems.

Then we have seen image intensifiers where a vacuum tube is used with an electron optics

to amplify the signal. It's also old technology but still available in the hospitals. So image

intensifiers, image intensifiers, and then we have flat panel detectors and the abbreviation

is FD, so flat panel detectors. This here is an analog technique, no algorithms applied.

So from our perspective, it's not that much interesting. The image intensifiers, they

imply now artifacts. Which types of artifacts did we consider? Artifacts. So here we have

seen that we get distortions into the images. The distortion is implied by the fact that

we have an electron optics for amplifying the signal. Electrons moving in the earth

magnetic field are deviated. These deviations cause distortions in images. And the order

of magnitude we observe is that distortion of 20 pixels at the boundary of the image

is something that is happening. So 20 pixels offset is something that can happen. And that

does not make us happy. And for image distortion, we have learned about compensation or undistortion

algorithms and in this context we talked about calibration. So we have seen calibration patterns

where by the manufacturing process we exactly know where the points have to be. Then we

acquire images. We see the points are distorted, the mutual distances of points are changed

and we use this type of information to estimate the distortion mapping. And the distortion

mapping is estimated by using, in our case it was a parametric function and polynomial

mapping and we estimated the whole thing by using a least square estimator. A very important

technique and we will also see today or maybe on Thursday another application of a least

square estimator. Very important technique. And then the second one was associated, the

second artifact we have considered was associated with FDs and FD detectors have the nice property

that they are not affected by magnetic fields. But the manufacturing process leads to defect

pixels and somehow we have to come up with sandpaper algorithms to smear the image information

in a way that we don't see that there are defects on the detector. So the second class

was defect pixel interpolation. And in this context we have seen two different approaches.

And let me specify this in more detail, the defect pixel interpolation. So we have seen

that there are two different methods. One is working in the spatial domain. Spatial

domain methods that use the image signal as it is acquired and then you do interpolation.

We talked about linear interpolation or bilinear interpolation if you consider this in 2D.

Remember just the picture where you have let's say two points and you assume that the two

points are linked by a linear function or an affine function and you want to have the

function value at this point. You just do interpolation, estimate the linear function

and then you just take the function value at this particular point here to get an estimate

for the unknown value. And the second method that I consider more powerful that can even

deal with noise are methods that work in the frequency domain. And we have seen two methods.

One method is using high pass filtering technique by just cutting out the high frequency, low