Okay, good morning everybody. Please apologize that I'm a few minutes late. I had a phone

call and you know how these things are. Let's start. We just have this week and next week

and then we are, no two more weeks we have, right? No, just next week? Oh my god. Okay.

I'm telling too many stories I guess. But anyways, as you see I always show the slides

right away from our webpage. So there was one comment in the evaluation that the slides

I show differ from the slides that are in the web and I don't know what to do with it

because I think I always use the most current version that is available online. So maybe

someone can tell me in more detail what happened and what went wrong. Good. So let's talk about

image registration. We are currently in our story with the modalities, with pre-processing,

with reconstruction and now we combine things in terms of we use images of one and the same

modality and transform them into the same coordinate system or we also have images of

different modalities and fuse them. So we talk about fusion and registration. That's

our current chapter. And from 10 or 30,000 feet looking down to the scene we see what

we did. We capture images, we pre-process single images, we combine multiple images

to gain higher information in the reconstruction chapter and now we take different images acquired

at different time points using different modalities and we merge the information in a way that

the radiologist can easily analyze the patient and come up with a proper diagnosis as fast

as possible. That's the story of diagnostic medical image processing. And image fusion

is basically defined as or registration, we look for a transformation such that the source

image is mapped to the target image. How do we call it? I and this mapping guarantees

that images that we are consider are aligned or registered. And what we did so far is we

looked at the simple case, we have corresponding point features. Corresponding point features

and we looked at the situation of 2D, 2D points and last week, no on Monday, we started to

look into the situation where we have corresponding 3D points. So we have here 2D, 2D and here

we have 3D registration. Okay. So the first part, and that's just repetition of what we

did on Monday, we can also rewrite the 2D points in terms of complex numbers where the

X coordinate is the real part, the Y coordinate is the imaginary part and then we use the

observation or we use our knowledge about complex numbers where we know the fact that

the multiplication of two complex numbers is nothing else geometrically than a rotation

and a scaling. If I multiply one complex number with another one, I rotate and scale the vector.

And having this in mind, we can write the optimization problem in terms of complex numbers

and then we just do a comparison of the coefficients, the real part of the left and right side has

to be the same, the imaginary part has to be the same and we end up with linear equations

that have to be solved. We did that on Monday. And now we try to do the same for 3D, 3D registration.

Our goal is to come up with a linear method that allows us to compute the rotation and

translation in 3D using just a system of linear or solving just a system of linear equations

or in other words, we want to find an SVD-based rotation translation estimator. I just want

to call the function SVD in MATLAB and that should solve the problem of point-point registration.

And to come up with an algorithm like that, we started last week, we started on Monday

to look into different representations of rotation matrices and we have seen two representations.

One is the Euler angle representation where we say, okay, this matrix can be decomposed

into, I think I wrote it this way, phi x, phi y, phi z. This can be decomposed into

three rotation product, a composition of three rotation matrices around the x, the y and

the z-axis. And now you immediately notice that I changed the ordering of the rotations.

That of course affects the final rotation matrix because the product of matrices is

not commuting. So it makes a difference whether I first rotate around the x-axis and then

around the y-axis or the other way around. For the computer graphics community, they

might know that you can assume here that the matrices commute if these angles are very

small. If the angles, the rotation angles are very small between different views, then

we say in terms of a numerical approximation, it doesn't matter in which direction or in