Rekonstruktion von 3D-Punkten aus Kamerabildern : eine anwendungsgeleitete Lernumgebung an der Schnittstelle von analytischer Geometrie der gymnasialen Oberstufe und elementaren Ideen der linearen Algebra

The technical application of 3D reconstruction, which can be built upon the fundamental concepts of analytical geometry and linear algebra as taught in German upper secondary forms, opens up an unfamiliar and challenging perspective on the geometric relationship of photographic images to the real 3D scene which they depict and to one another. Although projections from 3D to 2D space are not injective functions, one aims at reconstructing form and location of a spatial object or the positions of a moving camera by measuring coordinates in one or more images. This is a typical problem from the field of Computer Vision. For instance, a robot navigating in space is reliant on information that can be taken from camera images. In recent decades science has come up with some elegant linear reconstruction algorithms that are based on the use of matrices and linear transformations. Linear equations are therefore the central reconstruction tool and fundamental principle which makes the problem accessible even for high school students. The purpose of this thesis is to didactically analyse the substantial context of 3D point reconstruction from two camera images as found in Photogrammetry and Computer Vision, with the objective to convert it into an applied learning environment. It is aimed at designing authentic teaching material intended to be used with students in the transitional phase between high school and university. The latter constitute the main addressees of this thesis: It is directed towards mathematically-minded students of upper secondary forms in Germany who are well grounded in analytical geometry as well as towards science-oriented first-semester university students who are looking for an applied approach to elementary ideas of linear algebra. The resulting learning environment has been conceptualised as a guided programme (‚Leitprogramm‘) that students can work on self-responsibly and independently. It includes numerous illustrating GeoGebra-3D-applets as well as some MATLAB-tools that enable the students to conduct a computer-facilitated reconstruction project of their own from real photographs. Apart from being material for self-instruction, the programme can be used as a structured store for teachers to configure their own lessons or workshops according to their needs. To that end, the exercises for the central experimental working phases for students have been organised as independent worksheets.The technical application always stays in the focus of the programme - on the one hand, as motivational impulse and, on the other hand, as object of constant mathematical reflection. Fundamental ideas, methods and concepts of linear algebra that are generally applicable, useful and indispensable for many scientific fields are developed along its trajectories. These mathematical ideas are generated on the basis of the concrete technical problem, and their sustainability manifests in their practical implementation. This is supposed to be an incentive especially for those students who are planning to take up scientific or technical degree programmes at university. This thesis offers them valuable insight into the character of mathematics as it is presented to them in university courses. Parts of the developed teaching material have been tested in the context of a two-day workshop with high school students of German senior classes (grammar and comprehensive school), which was part of a week-long RWTH summer school for pupils on various topics of mathematics in the summer of 2014