Just two lectures left. Good morning everybody. Just two lectures left. For me it's a very

special situation. It's the last week of my life where I'm required to teach for the next

six years. So let's enjoy the upcoming two lectures. Currently we talk about diagnostic

medical image processing. To be honest, this is the most exciting topic at the whole university,

as you know. And we talked about many aspects. We talked about pre-processing of single images.

Then we combined the images and tried to compute higher dimensional information out of these

images. We called it reconstruction. And then currently we talk about registration fusion.

How can I merge images of different modalities, images that have been acquired at different

time points? How can I combine them and bring them into a joint and common coordinate system?

And we call this registration. And there are many different types of registration. Currently we are

considering the most obvious situation, or let's say the simplest configuration of registration

problems, where we say we have rigid motion or a rigid transformation. That means the objects are

just rotated and translated without deforming things. Think about the bend it pointer, the

broken point pointer. And we use point correspondencies. Basically we have points, and we have points in

2D, 3D, 3D, or 2D, and here also 3D or 2D. And we want to compute the transform of this point set

to this point set. And what we have learned now is, let me describe this here, point-based rigid

registration. We have seen three different types. Two we have discussed in detail in the context of

registration. One we have already learned. We just reconsidered the algorithm from one of the

previous lectures. So if we have 2D point-based registration, we come up with a linear solver for

this problem using complex numbers. So we have a linear solver of the problem using complex numbers.

That's the easiest way to come up with a linear solution. There are many others, but this is the

easiest one. So we transform our points into complex numbers, where the x-coordinate is the

real part, the y-component is the imaginary part, and then we make use of our knowledge that we know

by heart and forever. We knew that multiplication of complex numbers is nothing else but rotational

scaling. If I multiply one point, interpret it as a complex number with a rotation complex number,

then I get the rotation in the 2D plane. Whether I think about real and imaginary part or x-y

component, that doesn't matter at the end of the day. So we come up with a complex number-based

linear solver. Then we have considered the 3D image registration problem, and we have seen that

we can transfer the idea of complex numbers from 2D to 2D to 3D by using quaternions.

Again, we find that we end up with a linear solver, where we say the quaternion q prime is

nothing else but the rotation quaternion multiplied with the point multiplied with a

conjugate complex. That was the idea. So we have this quadratic form where we multiply from left

with a rotation quaternion and from right with its inverse. Last time on Thursday,

we have seen that we end up again with a linear solver. So we can estimate the rotation of point

sets given to point sets, knowing that these are linked by a rotation, 3D rotation. We can compute

the rotation using the quaternion setup. Then we looked at the situation 3D, 2D registration,

where we said the point in homogeneous coordinates can be transformed by a projection matrix P,

and we end up with this point again in homogeneous coordinates for all the points from 1 to N.

We can write this just to repeat it in terms of the row vectors of P. So that's the first row of P,

the second row of P, the third row of P. We write it in terms of row vectors. We multiply this with

Pi, and what we see then is we get P1 transpose Pi tilde. I should write it this way. P2T Pi tilde,

P3 transpose Pi tilde. So I get here the inner product. I get the components of this vector,

and then I can divide by the third component to get the first component of Q, divide by the third

component, the second one to get the second component, and then we multiply by the denominator

and end up again with a linear solver. See the chapter on camera calibration, where we discuss

the homogeneous coordinates and the projection models. There we did this in detail. So if we

have 2D, 2D, 3D, 3D, 3D, 2D registration using point correspondencies, we know it's rather simple

from an algebraic point of view to do the computations, numerical algebraic point of view.

We get linear solvers, so SVD is the tool to be used to do that. Okay. You look at me like

you're not so happy. You're happy. Stefan? Totally happy. Okay. Good. Good. Then over the years,