Geometric Reasoning for Uncertain Observations of Man-Made Structures

Observations of man-made structures in terms of digital images, laser scans or sketches are inherently uncertain due to the acquisition process. Thus reverse engineering has to be applied to obtain topologically consistent and geometrically correct model instances by feature aggregation. The corresponding spatial reasoning process usually implies the detection of adjacencies, the generation and testing of hypotheses, and finally the enforcement of the detected relations. We present a complete and general work-flow for geometric reasoning that takes the uncertainty of the observations and of the derived low-level features into account. Thereby we exploit algebraic projective geometry to ease the formulation of geometric constraints. As this comes at the expense of an over-parametrization, we introduce an adjustment model which stringently incorporates uncertainty and copes with singular covariance matrices. The size of the resulting normal equation system depends only on the number of established constraints which paves the way to efficient solutions. We demonstrate the usefulness and the feasibility of the approach with results for the automatic analysis of a sketch and for a building reconstruction based on an airborne laser scan.