Okay, so welcome everybody to the Tuesday session. I have to apologize that I wasn't

able to teach yesterday. The reason was that I was away. We just brought in yesterday a

completely different topic, perfusion imaging, something that is very important from a practical

point of view, that's interventional image processing at its best basically because this

is something that is really required while the patient is on the table. And today I jump

back in the story line a little bit and I apologize for this. We started out to look

into the problem, how can we generate 3D ultrasound images? And the idea was that we acquire the

ultrasound slices as usual with a 2D device and we track the ultrasound probe and compute

the camera motion and basically fill in the slices that we acquire into a volume. That

was the rough idea. Very easy to catch and very straightforward. And from an algorithmic

point of view so far we understood how we can use markers attached to the ultrasound

probe, cameras mounted to the seal and then we can compute the camera motion and even

the 3D structure of the markers out of these camera images. This technology is of course

not restricted to 3D ultrasound images. That's very generic and whole companies that work

on navigation systems, where we have stereo camera systems observing a scene, computing

how medical devices move in space, they basically use these algorithms to do their navigation

tracking and pose estimation. So I think basically you should be very well equipped now with

the algorithmic background that is required to know what BrainLab for instance is doing

in Munich. They are basically focusing on this type of technology. But there is one

problem left and this problem is easily explained. This is the ultrasound probe. We have here

our markers and here we generate our 2D ultrasound image with a horrible image quality as we

all know. And here we have of course an image coordinate system, X and Y. And this image

coordinate system basically allows us to access the various pixels in the image that we acquired.

And we all know that the major change in our life that was basically generated by this

lecture is that images are no longer just matrices, but images are 2D planes in 3D.

That's how we think about images. And that basically means that the vectors X and Y are

vectors in 3D spanning a two dimensional subspace. And if I access a point on this plane here,

I can express this point. Let's call this point P. Of course I can say P is alpha X

plus beta Y. And alpha and beta are the coordinates, the indices basically, the coordinates of

this point on the image plane. But on the other hand we follow our system, our ultrasound

probe by a visual tracker and this has a coordinate system of course that is attached to the markers.

And we have a 3D coordinate system right there. And what we need to know is basically the

transformation. You want to choose a color? Green, great idea. Here. And this transformation

is called, because it's unknown, and how do we call unknown transformations? X. Unknowns

are called X. We call it X. This is the so-called hand-eye mapping and we have to estimate this

mapping using a hand-eye calibration. And that's the topic for today and maybe next

Monday a little bit. Hand-eye calibration. We compute the transform between these two

coordinate systems. If you think medical engineering is not that exciting, you can also think about

a robot arm that keeps, let's say, a melting device in automobile industry. And you have

your coordinate system of your melting device or your laser and you have your coordinate

system of the markers of the robot and you want to compute the transformation between

the laser and the markers. That's also something that is used in practice. So this is a very

important problem and the major innovation is that this transformation is called X and

we have to compute X. Okay? So that's the picture we have to keep in mind. And I really

recommend that you always break things down to very simple scenarios. Break things down

to simple scenarios when you think about it and you work out the math. And I'm saying

this over and over again. Do not expect that you learn anything here in the lecture that

you can use one-to-one later on in industry because usually as an engineer you work on

new problems, on hard problems that nobody has solved this way and you have to be educated

in a way and you have to learn how to tackle these problems. And if you cannot solve the