The following content has been provided by the University of Erlangen-Nürnberg.

Welcome to today's lecture. As I said yesterday, I will replace Professor Horniger today again.

We will again do a repetition of DIMMAP.

So this time we will repeat the homogeneous coordinates and projection models

because you will heavily make use of them also in this lecture.

Before we start, I received an email where to find the content of the lecture slides,

if they are put online or not.

That's why I thought I should show you how to get to our website.

Here in the UniWiz you can find the lecture and then you can also find the link to the lecture here.

Basically you can find all courses here.

Here on the slides you can find the slides, so we will put them online here.

You can also find slides from the last year.

In the exercise link you can find the assignments for the exercises.

We will also put here the solution for the Matlab intro for the tutorial, but only for this intro.

As I said yesterday, no solutions will be put online except this one.

We will also put news and so on on this website.

So if there are some unexpected changes or breakthrough news, something like that,

we will put it online, like here this information,

what I told you yesterday about the zip pool on the Thursday exercises.

All right, so now all of you should know how to get the material.

All right, so let's start with today's lecture.

It's about project models and homogeneous coordinates.

Last semester this was for reconstruction for x-rays and CT and so on.

The principle is basically the same, but we will repeat it so that we can use it for camera systems.

So why do we need projection models?

As I said, if you want to reconstruct a 3D scene, a real world scene,

and we only have, for example, usual cameras or a set of cameras,

then we somehow need to know how the 3D point is projected to the image.

So how the image was generated.

This is why we need to characterize the projection rays, different projection geometries,

how we can mathematically describe this projection,

and how we can get it onto the image, the mathematical background.

Here you can see an example for the first x-ray system where you have a generator and a tube,

and the detector is just a photographic paper.

And somehow we need to know how this is projected onto this photographic paper, for example.

Here is an example for the principle for x-rays.

So for x-rays you have an optical center where the x-rays are emitted,

then they go through the volume you want to reconstruct or you want to project,

and are projected onto a detector.

So now if you have a camera, then of course we don't have x-rays, we have light rays,

and they are not going through our volume, but merely are reflected by our volume.

But the principle stays the same.

So as I said now, we want to somehow mathematically describe those geometries.

So here you can see that in former times already they already did some projections

with a piece of string and pins.

And what we assume now is that our 2D image plane is parallel to the x-y plane of the 3D coordinate system.

This means if I have here my camera and I want to record the scene, then my image plane is parallel to the scene.

And this z equals f and is constant, so this is my z direction.

Okay, so the first and the most simplest one of the projection systems is the orthographic projection.

It means that we just ignore the z value and project it this way on the screen.