Low Rank Matrix Factorization and Relative Pose Problems in Computer Vision

Abstract: This thesis is focused on geometric computer vision problems. The first part of the thesis aims at solving one fundamental problem, namely low-rank matrix factorization. We provide several novel insights into the problem. In brief, we characterize, generate, parametrize and solve the minimal problems associated with low-rank matrix factorization. Beyond that, we give several new algorithms based on the minimal solvers when the measurement matrix is either sparse, noisy or with outliers. The cost function and the algorithm can easily be adapted to several robust norms, for example, the L1-norm and the truncated L1-norm. We demonstrate our approach on several geometric computer vision problems. Another application is in sensor network calibration, which is also explored. The second part of the thesis deals with the relative pose problem. We solve the minimal problem of estimating the relative pose with unknown focal length and radial distortion. Beyond that, we also propose a brute force approach, which does not suffer from common algorithmic degeneracies. Further, the algorithm achieves a globally optimal solution up to a discretization error and it is easily parallelizable. Finally, we look into the problem of object detection with unknown pose.