Strukturelle und quantitative Beurteilung der perspektivischen Beobachtbarkeit linearer zeitinvarianter Systeme

The reconstruction of three-dimensional geometries from their perspective projection, such as a camera image, is a recurring problem in the fields of image processing and robotics. If the object movement can be described by a linear time-invariant system, the property "perspective observability" of the system is relevant for the success of the reconstruction. This thesis studies techniques for testing and assessing this property. First, existing structural criteria are reviewed and partly extended. It is further shown that non perspectively observable systems can be transformed to a canonical form similar to the Kalman decomposition of non-observable systems. For quantitative assessment, the distance measure for observability is extended to the perspective case. This measure is shown to be given by the solution of a quadratic program with quadratic equality constraints. Several modifications of the measure, that are relevant for its practical application, are discussed. For its computation two numerical algorithms are proposed, and their applicability is demonstrated in the course of examples. Finally, an alternative technique for the quantitative assessment of perspective observability is shown. Compared to the distance measure, this technique does not allow a statement about the robustness of the property with respect to perturbations of the system parameters; it can, however, be applied more easily.