Mathematical Methods for Image Based Localization

The underlying question in localization is where am I? In this thesis a purely image based approach is proposed to solve this problem. In order to create a complete image based system, there are many subproblems that have to be addressed. The localization problem can also be solved in other ways, for example, with a GPS. Two advantages with using images compared to GPS are that no open sky is needed and that a higher precision is possible to achieve. The thesis consists of an introductory chapter followed by six papers. In the first paper, enhancements of Gröbner basis techniques to solve systems of polynomial equations are presented. The new strategies improve the numeric stability with several orders of magnitudes, compared to previous state of the art. This framework is then applied in the next three papers to solve several geometrical pose problems relevant for localization. The main difference between the papers is the level of knowledge of the inner calibration of the cameras. The calibration knowledge ranges from completely calibrated cameras to uncalibrated cameras with unknown radial distortion. The fifth paper of the thesis also treats the pose problem, but the method differs from the previous papers. In this paper a method is presented that guarantees a globally optimal solution at the price of computational complexity. To achieve this, the pose problem is reformulated and solved via a minimal vertex cover. The final paper is devoted to large-scale localization. Methods from image retrieval are utilized, and extended, to be able to perform city-scale localization. Moreover geometry is directly incorporated in the retrieval stage.