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Okay, great. So today's projection models and some camera calibration.

So we will talk a bit about the projection models and we will introduce a concept called homogeneous coordinates.

And you've already seen this in the previous lecture.

I didn't detail it too much, but the concept is fairly easy to follow and it absolutely makes sense

and it will allow us to calibrate our projective cameras much better.

So let's talk a bit about the motivation.

So we've seen that we require quite some detailed knowledge about the projection and given an X-ray image, for example,

you want to be able to figure out that any point that you observe in your scene or also in a normal camera image,

you want to figure out the ray that connects a point of interest and a pixel in the scene.

So this is the ray that is acquired by the camera.

And it doesn't matter too much whether it's an X-ray image or whether it is a typical camera image.

You typically have a bundle of rays that is emerging from a focal spot or from a camera center

and then you are acquiring those rays and the respective information about those rays.

So the thing that we will deal with today is how can we mathematically describe all of those rays

and we will actually consider different projection geometries.

First and foremost because some of these projection geometries can be modeled much simpler in a mathematical sense

and then we will derive how we can actually also tackle projective geometry

because this is what we have in the most cases. We have some focal center and we have divergent rays that emerge into the scene.

So once we have found a model to describe this, we will then think about how we can actually calibrate the parameters of the model.

So this is the camera calibration part.

And using those parameters we will then be able to associate pixels with lines in word coordinates.

This will allow us to compute paths for example of X-rays or of lines into the scene.

And another thing that we will think about is how reliable actually the estimates are.

But we will only shortly discuss this.

But of course once you estimate something you want to figure out how robust your estimate is and how reproducible it is.

And if you have numerical instabilities of course you want to know about that and we will look into ways of determining this.

So this is a very simple sketch of an X-ray acquisition system.

And this is actually the version that Röntgen was building over a hundred years ago.

He was building such kind of X-ray imaging systems and you have your tube.

And here you have some film material where you can actually gather the amount of X-rays that arrive at the respective position of the film.

And if you put something in between the X-rays get attenuated and you can see the amount of attenuation in the resulting image.

And interestingly in terms of the projective geometry this system is very similar to what we have today as well.

So we have some tube and some anode where the X-rays emerge from.

And we have essentially if we have an ideal system just one single point that is infinitely small.

And from this point all the X-rays emerge and they go into the scene.

So in CT reconstruction we also call this a cone beam geometry.

Or in computer vision you typically talk about the perspective geometry that is given by such a scene.

So already such a long time ago we already had this kind of setup for acquisition.

And we can actually describe this in the end what we are interested in is some world or volume coordinate system.

That is for example given by a voxel grid like 3D volume elements.

And this voxel grid is traversed by the X-rays.

So this is the optical center.

And from the optical centers all these X-rays traverse our volume of interest.

And then they hit the detector plane and at every point on the detector plane we collect the line integral along this ray.

So this is our imaging geometry and this is still the same as in the first systems.

Good. So what we are interested in today is how we can actually describe the path of the rays very efficiently.