Geometric reasoning with uncertain polygonal faces

The reconstruction of urban areas suffers from the dilemma of modeling urban structures in a generic or specific way, thus being too unspecific or too restrictive. One approach to overcome this dilemma is to model and to instantiate buildings as arbitrarily shaped polyhedra and to recognize man-made structures in a subsequent stage by geometric reasoning. Thus, the existence of unconstrained boundary representations for buildings is assumed. To stay generic and to avoid the use of templates for pre-defined building primitives, no assumptions for the buildings' outlines and the planar roof areas are made. Typically, roof areas are derived interactively or in an automatic process based on given point clouds or digital surface models. Due to the measurement process and the assumption of planar boundaries, these planar faces are uncertain. Thus, a stochastic geometric reasoning process with statistical testing is appropriate to detected man-made structures followed by an adjustment to enforce the deduced geometric constraints. Unfortunately, city models usually do not feature information about the uncertainty of geometric entities. We present an approach to specify the uncertainty of the planes corresponding to the planar patches, i.e., polygons bounding a building, analytically. This paves the way to conduct the reasoning process with just a few assumptions. We describe and demonstrate the approach with real data.