So let's start. Good morning. Good morning everybody. The number of students already

reduced unfortunately. I'm very sorry about that. Well, this semester we talk about diagnostic

medical image processing. And so far we have seen an introduction that was more or less

a motivation of all the exciting topics that we are going to consider within the lecture.

And last week we started to look into the algorithmic part of the lecture. First we

started with the singular value decomposition that is basically nothing else but a standard

decomposition of matrices where the factors allow for a very nice geometric interpretation

and SVD and the decomposition of a matrix is very supportive to solve many, many problems

that can be formulated in terms of linear algebra. So if we know SVD, we are on the

safe side with respect to our linear algebra knowledge.

So then Marco introduced last week the first chapter on image preprocessing. That's the

first part of the lecture where we consider preprocessing algorithms to improve the image

quality of those images that are used by the physician, by the radiologists for coming

up with a diagnosis. And we also know that the imaging tools that we are using in the

radiology department nowadays always relies on effects that we know from physics. And

based on these effects and the physics modeling we are using, sometimes we know which artifacts

show up in the image and we can think about how to remove these artifacts by knowing a

lot about the image acquisition procedure. And we started to look at one very specific

case last week where we use image intensifiers in X-ray imaging. We know about the principle

how X-ray images are generated, so we have our X-ray tube, then the photons of a certain

energy are generated by the X-ray tube, the photons are propagated on straight lines through

the object, through the patient, and at the end we just look at the balance sheet of the

energy of the photon. So while a photon is traveling along a straight line through the

human body, hopefully along a straight line, is traveling through the human body, dependent

on the tissue classes the photon has to walk through, it loses energy. And at the end we

can measure how much energy was kept by the X-ray particle or the photon. And this energy

is visualized in terms of an intensity value in our image. And the problem the image intensifier

solves for us is the transformation of the photon energy into an intensity image. That's

what the image intensifier does. And the image intensifier is making use of a rather old

technology, this is known since the mid-50s that this can be used, it's making use of

a vacuum tube like the old TV systems, the big ones with the vacuum tubes. And what is

done is the photons, they hit the surface of the image intensifier, Marco explained

this to you, and then the photons, they just release some electrons. And these electrons

are accelerated within the vacuum tube. The photons are accelerated using an electronic

optics and this is done to amplify the original signal that is generated by the photons. So

if we accelerate electrons in a vacuum tube, then we know that we have particles that carry

an electricity, they are moving within the earth's magnetic field and due to our basic

physics knowledge, we know that these particles are deviated by the magnetic field. And this

leads to a distortion in the image. That means the electrons in the vacuum tube, they are

deviated and this type of deviation should be corrected within our algorithmic setup.

And that's what we have learned last week, how this can be done. So we have two images.

We have two images. The corrected image, that is the undistorted image and the distorted

image. And the distorted image. And here we know that the image points that we observe

are due to a deviation of the electron beam. That means the position of the intensity value

in this image is somewhat distorted. So if we use a regular grid for instance and map

a regular grid by our optical system, this regular grid will somehow be distorted. So

if I have the situation like this, I have here a grid with these points. This grid will

be somehow distorted by the image acquisition device. The question is how can we for instance

align the points that we have here, parallel lines as in the original one. And what we

did is we said we are computing a mapping from the corrected to the distorted image